Publication n 11? de. 1'AaadcUi.on InteAnatlonale. du ScJ.mc.tA Hyd/iotogiquu Sympoiitm de Tokyo (Décembre 1975)

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1 Publication n 11? de. 1'AaadcUi.on InteAnatlonale. du ScJ.mc.tA Hyd/iotogiquu Sympoiitm de Tokyo (Décembre 1975) DESIGN OF FLOODWATER RETARDING STRUCTURES UNDER UNCERTAIN WATERSHED CONDITIONS Martin M. FOGEL Professor, University of Arizona, Tucson, Arizona, USA Istvan BOGARDI Head, Water Resources Centre, Budapest, Hungary Louis H. HERMAN Research Associate, University of Arizona, Tucson, Arizona, USA SUMMARY Floodwater retarding structures are often designed for ungaged watersheds, for conditions of limited hydrologie data or where the characteristics of the basin itself are changing. These difficulties may be overcome by a stochastic analysis of rainfall events which serves as an input into a deterministic rainfall-runoff formula calibrated with experimental basin data. Experimental data from southern Arizona are used to estimate parameters of a stochastic rainfall model and to calibrate the U.S. Soil Conservation Service formula for runoff. Two approaches are used to estimate the probability density functions of runoff volume and peak flow rate: first, an analytic method based on transformations of random variables, and second, a Monte Carlo simulation that can take antecedent moisture and changing basin conditions into account. Also discussed, is the use of benefit-risk analysis, which combines the probability distributions of runoff and economic data for overdesign and underdesign, to select an optimum size of structure. This decision procedure represents an improvement over the use of a fixed return period or of the related concept of probable maximum flood. RESUME DESSEIN DES STRUCTURES QUI RETARDENT L'INONDATION SOUS DES CONDITIONS INCERTAINES DES BASSINS Les structures qui retardent les inondations sont souvent dessinées pour les bassins sans gages, pour les conditions des données limitées de hydrologie ou bien, ou les caractéristiques du bassin lui-même sont en train de changer. Ces difficultés peuvent être surmontées par une analyse stochastique des averses qui sert d'une entree dans une formule déterministe de la pluie et de l'écoulement. Cette formule est calibrée avec des données du bassin expérimental. Des données expérimentales du sud de 1'Arizona sont utilisées pour estimer les paramètres d'un modèle stochastique de la pluie et pour calibrer la formule du U.S. Soil Conservation Service pour l'écoulement. Deux approches sont employees pour estimer la distribution de probabilité du volume de l'écoulement et du taux de l'écoulement maximum: d'abord, la méthode analytique basée sur transformations des variables aléatoires, et puis, une méthode de Monte-Carlo qui peut tenir compte de la pluie précédente et des conditions changeantes du bassin. Aussi discutée est l'utilisation de l'analyse de bienfait et^de risque, qui combine les distributions de probabilité de l'écoulement et les données économiques pour le dessein trop haut ou le dessein sous-estime pour qu'on puisse choisir une dimension optimum de la structure. Cette procedure determinative représente une amelioration a l'utilisation d'une période fixée de retour ou du concept allie de l'inondation maximum probable. 681

2 71.2 INTRODUCTION For many of the world's developing countries, hydrologie data is a luxury. Information of this type is simply not available or it is limited to the extent that the design of water control structures becomes a formidable task. The purpose of this paper is to present a methodology to be used in the design of floodwater retarding structures for ungaged watersheds or where streamflow records are not available but precipitation records are. The procedure consists of a stochastic analysis of rainfall events which serves as an input into a deterministic rainfall-runoff relationship calibrated with experimental basin data. This analysis results in a probability density function for both runoff volumes and peak flow rates. In sizing a floodwater retarding structure, that is, determining the storage volume and spillway capacity, the 100-year flood is often used without regard to the economic optimum. The probability density functions, together with economic data that relate to the penalties for overdesigning and underdesigning, forms the basis for selecting the size of the floodwater retarding structure that maximizes the difference between the expected benefits to be derived from the structure and the costs for constructing and operating such a structure. The emphasis of this paper will be on the hydrologie aspects of the problem. HYDROLOGIC CRITERIA FOR DESIGNING FLOODWATER RETARDING STRUCTURES The Soil Conservation Service (SCS) of the United States Department of Agriculture is the prime operational agency that is concerned with the design and installation of floodwater retarding structures constructed on streams draining moderately-sized watersheds, e.g., up to a few hundred square kilometers. For this purpose, they have developed one of the most widely used set of procedures for analyzing hydrologie data (U.S. Soil Conservation Service, 1972). The design of water control structures is generally divided into two major components, one concerned with the hydrologie design for the storage volume and principal spillway and the other for the emergency spillway. The principal spillway is a concrete or metal conduit that conveys discharges coming into the reservoir in a safe, non-erosive manner. SCS criteria requires the principal spillway capacity and the associated floodwater retarding storage volume to be such that project objectives are met and the frequency of emergency spillway operation is within specified limits. In many instances, project objectives are satisfied with containing the estimated runoff volume from a 100- year storm within the structure without resorting to the use of the emergency spillway. Flows larger than those completely controlled by the principal spillway and retarding storage are safely conveyed past an earth dam by an emergency spillway. The emergency spillway capacity is based on a hydrograph developed from what is known as the basic 6-hour design storm. The return period of this storm varies with the hazard class of the structure ranging from an amount slightly greater than the 100-year return period to the probable maximum precipitation. The reason for using a 6-hour storm for emergency spillways rather than the 10-day storm which is used in the principal spillway design is not clear 682

3 in SCS literature. Contact with SCS personnel, however, indicate that the reason for this is that experience and studies have indicated that the longer duration storms provide the larger volume of runoff in most cases while the short duration storms provide the highest peak inflow. Since the principal spillway design is mainly one of retaining a given amount of volume and releasing it slowly over a long period of time, volume is the most critical factor. In emergency spillway design, safely passing the peak flows which come into the reservoir the peak rate is more important. Design rainfall is selected, for the most part, in regards to safety of the reservoir and to life and property downstream rather than solely on economic considerations. PROPOSED METHODOLOGY The suggested procedure for determining the optimum size of a floodwater retarding structure for ungaged watersheds consists of first obtaining parameter estimates from precipitation data for a probabilistic rainfall model. Then, rainfall is converted to runoff by means of a rainfall-runoff equation, and finally, a form of benefit-risk analysis is made. Rainfall Model In earlier papers (Fogel and Duckstein, 1969; Duckstein et.al., 1972), an event-based stochastic model of warm-season rainfall was presented. The model was composed of two basis distributions, a Poisson distribution of the number of storm events per season and a distribution for the amount of rainfall per event, generally a gamma variate. If a random number of thunderstorms are assumed to occur in a fixed time interval, a Poisson variate can be used to describe this number of events per given period. Furthermore, the time interval between events that follow a Poisson arrival process should be exponentially distributed (Feller, 1967). With this latter distribution and the oi.e for the amount of rainfall per event, it becomes possible through Monte Carlo simulation to generate a sequence of storm events, in other words a synthetic historical record of rainfall for as long a record as is desirable. A frequency analysis can then be made on the synthetic data to determine the return periods for various sizes of storms. The event-based precipitation model has the flexibility of incorporating the effects of elevation (Duckstein et.al., 1973) and can also consider storm duration as a random variable. From the two basic distributions, two additional distributions are readily obtained. One is an annual or seasonal maximal distribution similar to what is obtained by a frequency analysis of maximum annual flows. The second distribution is for total seasonal runoff, which is the sum of a random number of independent, identically distributed random rainfalls (Duckstein et.al., 1972). Transformation of Rainfall into Runoff Two procedures can be used to convert distributions of rainfall into runoff. 683

4 71.4 One uses the previously mentioned Monte Carlo simulation of rainfall events. Each rainfall event is then transformed into a storm runoff amount using an appropriate rainfall-runoff equation which, in this case, is the SCS formula (R - A) 2 V = _ for R>A (1) R - A + S where V is storm runoff in mm, R is storm rainfall in mm and A and S are watershed parameters. The initial abstractions A are generally taken to be 20 percent of the potential maximum retention S (Kent, 1973). For sake of convenience, values of S in mm are converted to curve numbers or a watershed index W by the equation 250, S In a similar fashion, a distribution function for peak flow rates can be obtained. The SCS formula for estimating peak flows is of the form K x V QP = _ (3) TP where QP is the peak rate of discharge per unit watershed area, K is a constant that reflects conversion of units and TP is time to reach peak flow. The time to reach peak discharge in the SCS procedure is a function of storm duration and an empirical relationship that estimated the time of concentration of a particular watershed. If storm duration for a given class of storms is assumed to be a constant, then QP will have a distribution similar to that for V differing only by a scale factor. On the other hand, storm durations are not a constant and have a distribution of their own. The proposed procedure can readily incorporate this variability as shown in earlier efforts by the authors (Fogel et.al. 1974). In the above instance, storm durations can also be simulated along with rainfall amounts to produce a synthetic time series of rainfall amounts and durations. These pairs can then be transformed into storm volumes and peak discharge rates according to the SCS formulas. A conventional frequency analysis can then be made from the simulated time series of runoff volumes and peak discharge rates. In the second method, the distribution function of R is transformed directly into storm runoff volumes (Duckstein et.al., 1972) and to peak discharge rates (Fogel et.al., 1974). These distribution functions are then combined with a distribution that describes the arrival of a given number of runoff events in a specified time interval to give extreme value distributions. Briefly, the procedure is as follows: Let F R (r) be the distribution function (the CDF or cumulative distribution function) for rainfall amounts given that a storm has occurred. The change of random variable R to V is effected by solving (1) for the dummy variable r = R in terms of v = V. Thus (r - A + S)v = (r - A) 2 or r 2 - (2A + v)r + (A 2 + va - vs) = 0 Solving this quadratic equation, the only possible solution is KC > r = A + v/2 + 1/2 (v 2 + 4vS) 1/2 (4) 684

5 in which r is greater than A. The CDF, F (v) is then simply Fy(v) = F R r(v) where r(v) is given by (4). Making this substitution results in Fy(v) = F R [A + v / 2 + V 2 (v 2 + AvS) 1/2 ] (5) Presented in earlier works (Duckstein et.al., 1972), the distribution function of V is combined with the distribution for the number of runoff events per season N to produce a distribution for the annual maximum flow volumes, $ v (v). With the assumption that N is a Poisson variate $ v (v) = exp( -m[l - F v (v)]) (6) where m is the mean number of runoff-producing events. The value for m can be obtained directly from rainfall data as once A is selected, a runoff-producing event is simply one in which storm rainfall R is greater than A. The extreme value distribution for peak discharges is readily obtained by transforming the CDF for V into one for QP using equation (3), if storm duration is assumed to be a constant. If not, then a decision has to be made regarding the independence of storm amount and storm duration. If there is little or no correlation between these two random variables, then the procedure described by Fogel et.al. (1974) can be used. Where moderate to strong correlation exists, then a technique such as the one described by Smith et. al. (1974) must be used Optimal Design In most instances, the SCS designs floodwater retarding structures on the basis of a 100-year storm or one that produces probable maximum precipitation. While SCS procedures require economic investigation to be made, no analysis includes determining the optimal capacity of the facility, that is, the size of the structure that will maximize the difference between expected benefits and costs. The proposed procedure which produces distribution functions for runoff volumes and peak discharges allows such an analysis to be made. This is not, however, to infer that economic factors are the only ones considered by the SCS as possible loss of life involves other considerations. With distributions of storm runoff obtained from a simulated set of runoff events or from the transformation of actual rainfall data, benefits resulting from flood prevention and associated construction costs can be examined for structures of varying sizes. Associated with each size of structure is an expected net benefit which is derived by weighting the net benefits for each possible flow with the probability of that flow's occurrence. The purpose of this analysis would be to determine the design that maximizes the expected net benefits. Since a damage-frequency analysis is usually made for these structures, the only additional effort would be to estimate the cost for several economically-feasible structures of different capacities. In effect, the suggested analysis does not design for a single event, such as the 100-year storm, but includes all possible runoff-producing events and selects the size of the structure based on maximizing the expected net benefits from all such events. DATA ANALYSIS AND RESULTS Hydrologie information obtained by analyzing 19 years of record on the Atterbury Experimental Watershed located near Tucson, Arizona USA provided the basic data for modeling the rainfall and runoff processes. The watershed is 685

6 71.6 characterized as being long and narrow with lands lopes ranging between 2 and 3 percent and channel slopes less than one percent. Soils have sandy or gravelly surfaces on the rounded gently sloping ridges while on the watercourses that separate the ridges the surfaces are loamy. Much of the area is underlaid with a lime accumulation 15 to 60 cm below the surface. Vegatative cover is thin, consisting mostly primarily of creosote bush (Larrea tridentata), palo verde (Cercidium microphyllum), mesquite (Prosopis juliflora) and ocotillo (Fouquiera splendens). Average annual precipitation is 280 mm approximately equally divided into two distinct periods. This section first discusses the analysis of rainfall data for the purpose of developing a synthetic time series of rainfall events. Then experimental data is analyzed to relate runoff to rainfall and finally the result of several simulations are presented and compared to actual runoff data. Analysis of Rainfall Data In this study only convective storm rainfall was considered as these localized, highly-intense, relatively short-duration storms produce the record runoff events on the size of watersheds under consideration for southern Arizona. Thunderstorms generally occur during the months of July, August and September, but may also occur during any other time of the year. Only the three summer months were used in this analysis to obtain parameter estimates for frequency distributions. Figure 1 illustrates a gamma distribution fitted to historical data of mean areal rainfall per storm event for a 20 km 2 watershed. The probability density function (PDF) of this distribution is given by.a a-1 -br f R«= b (a-!)! W where a is a shape factor and be is a scale factor. Using the method of monents, values for a and be that gave the best fit were and 3.156, respectively. The distribution for the time between the beginning of one storm event and the start of another is shown in Figure 2. A geometric distribution, the descrete version of an exponential distribution, is fitted to the interarrivai time for storms, T. The PDF for interarrivai times, Mt), is f T (t) = (1-p) t^p t>l (8) in which p is a parameter. This distribution could not be rejected at the 10 percent level of significance using the Kolmogorov-Smirnov test which indicates that a Poisson variate describing the arrival of a number of storm events per given time interval is a reasonable assumption. Relating Runoff to Rainfall The SCS method for estimating storm runoff volume from rainfall is essentially a one-parameter model, the parameter being S the potential maximum retention which is often considered a constant but appears to vary on a storm by storm basis. Analyzing data from a 20.1 km 2 sub-catchment of the Atterbury Experimental Watershed indicated that a considerable portion of the variance in runoff remained unexplained when a constant value for S was used. While a trend was evident that antecedent rainfall had an effect on S, this could not be established statistically. One of the problems was that there were too few events available for analysis. 686

7 Stratifying S according to the maximum 15-minute storm rainfall intensity (iis) did, however, materially improve the rainfall-runoff relationship. Considering extreme runoff events where runoff volumes exceeded a threshold value of 1mm (or 20.1 x 10 3 m 3 ), a regression analysis showed that a single value of S for all these events explained 60 percent of the variance as indicated by the value of R 2 (See Table 1). This is similar to what was reported previously in which a linear rainfall-runoff model was postulated (Fogel and Duckstein, 1970). In this earlier effort it was shown that consideration of the maximum 15-minute intensity significantly increased correlation. Similarly for this study, lumping the storms into two groups, depending on whether or not the maximum 15-minute intensities exceeded 75 mm/hr, significantly reduced the unexplained variance. Since the total number of events were relatively few (28 in 19 years), only two intensity groups were considered. The explained variance (R 2 ) exceeded 70 percent in each group. Table 1 summarizes the results of this analysis. Table 1. Results of Regression Analysis Relating Runoff to Rainfall by the SCS Formula SCS Watershed Parameters Storm Class 7, ë ~ R 2 W a, mm All events ii 5 >75 mm/hr i 15< 75 mm/hr Data Simulation Using the distributions shown in Figures 1 and 2, Monte Carlo techniques were applied to simulate 1000 years of summer rainfall events. Means and variances of the simulated data were found to compare favorably to the statistics obtained from analyzing 19 years of actual data. In transforming rainfall into runoff, four sets of simulations were run. For the first run, (Simulation A) the watershed parameter S was considered to be a constant, which in this case was 51mm as indicated by the regression analysis (Table 1). In the second run (Simulation B), S was considered to be a function of antecedent rainfall and an SCS procedure was used to obtain three classes of the antecedent moisture condition (AMC). If the previous 5-day rainfall total exceeded 35 mm, this watershed was considered to have the highest runoff potential (S = 19mm), while the runoff potential was assumed to be at its lowest level when the antecedent rainfall was less than 15mm (S = 126 mm). The watershed was considered to be in average condition when the 5-day antecedent rainfall was between 15 and 35mm (S = 51 mm). Simulation C considered S to be function of iis. the maximum 15-minute intensity as suggested by the regression analysis. To incorporate this aspect into the simulation, rainfall data was analyzed to determine the frequency of occurrence of the two classes of storms stratified by ii 5. The data indicated that storms with iis>75 mm/hr occurred 35 percent of the time. Each simulated storm was then randomly assigned to an ii 5 class based on the division. The S associated with each class was based on the results of the regression analysis shown in Table 1. For the fourth run, Simulation D, S was allowed to vary according to both the expected ii 5 and AMC. Thus, for each runoff event S took on one of six values. The results of the regression analysis to determine the effects of ii 5 were 687

8 71.8 assumed to be for average antecedent moisture conditions. Values for the wet and dry conditions were then taken from SCS procedures (U.S. Soil Conservation Service, 1972). All values for S used in this simulation are shown in Table 2. Table 2. Values of the Watershed Parameter S Used in Simulating Runoff Antecedent Moisture Condition Dry Average Wet Maximum 15-minute Intensity iis>75 mm/hr. i 15 <75 mm/hr Each of the four simulations were run for 1000 years from which an annual maximum series was obtained. Table 3 compares the statistics of the simulated series, the mean and standard deviations, with those obtained from the 19 years of runoff data. Table 3. Statistics of Simulated and Observed Annual Maximum Runoff Series Data Set Mean, mm Std. dev., mm Simulation A Simulation B Simulation C Simulation D Observed Standard procedures were then used to fit the simulated and actual annual maximum runoff series to extreme-value (Gumbel) distributions for comparison purposes. Figure 3 illustrates the results of this frequency analysis. DISCUSSION AND CONCLUSIONS The authors feel that the event-based approach to developing stochastic models of rainfall and runoff is well suited to arid and semi-arid conditions and also holds promise for being adapted to the more humid climates. Rainfall models of this type have been demonstrated for a humid as well as an arid climate (Duckstein et.al. 1972). Where sufficient data is available, the use of Monte Carlo simulation for developing probability density functions of runoff has greater flexibility than the analytic method based on transformations of random variables. Simulation models can incorporate the effects of antecedent moisture and another random variable of rainfall, its duration or intensity. They have the added advantage that they can consider the effects of a changing watershed such as due to urbanization. In this study, three of the four simulations resulted in higher storm runoff volumes for a given return period than that obtained from using observed runoff data (see Figure 3). A question arises then as to which is the more realistic distribution. According to Wiesner (1970), 19 years of hydrologie data in semiarid areas is not sufficient to establish stable frequency distributions; thirty years are needed. Yet when Simulation D, the run with the lowest unexplained variance, is compared to the frequency analysis of observed runoff data, there is close agreement. Until demonstrated to the contrary, a conclusion at 688

9 71.9 this time is that the simulation of runoff events as shown herein is an acceptable procedure for forecasting runoff events where the availability of pertinent data is severely limited. Moreover, the long-term effects of a changing watershed can be evaluated by this procedure. In addition, since the proposed methodology produces probability distribution functions of runoff, a basis for selecting the most economical design of water control structure becomes a reality even in the face of uncertain watershed conditions. ACKNOWLEDGEMENT The research was performed under a cooperative research project between the National Water Authority of Hungary and the University of Arizona with support from the National Science Foundation. Earlier efforts were partially supported by funds provided by the U.S. Department of Interior, Office of Water Research and Technology, as authorized under the Water Resources Research Act of REFERENCES Duckstein, L., M.M. Fogel and C.C. Kisiel A stochastic model of runoffproducing rainfall for summer type storms. Water Resources Res. 8(3) : Duckstein, L., M.M. Fogel and J.L. Thames Elevation effects on rainfall: a stochastic model.j. of Hydrology 18(1) : Feller, W An introduction to probability theory and its applications. Vol. 1. John Wiley. New York. 509 pp. Fogel, M.M. and L. Duckstein Point rainfall frequencies in corrective storms. Water Resources Res. J(6) : Fogel, M.M. and L. Duckstein Prediction of convective storm runoff in semiarid regions. Proc. Symp. on Representative and Experimental Watersheds. IASH Publ. No. 96 : Fogel, M.M., L. Duckstein and C.C. Kisiel Modeling the hydrologie effects resulting from land modication. Transactions of the ASAE 17(6) : Kent, K.M A method for estimating volume and rate of runoff in small watersheds. U.S. Dept. of Agric. SCS-TP pp. Smith, J.H., M.M. Fogel and L. Duckstein Uncertainty in sediment yield from a semi-arid watershed. Proc. Hydrology and Water Resources in Arizona and the Southwest. Vol. 4. pp U.S. Soil Conservation Service Hydrology. Sec. 4 SCS National Engineering Handbook. U.S. Dept. of Agric. Wiesner, C.J Hydrometeorology. Chapman and Hall. London. 232 pp. 689

10 71.io GAMMA DISTRIBUTION' (a = 0.767i b= 3.156) E MEAN STORM RAINFALL, MM Fig. 1. Frequency distribution of mean convective storm rainfall OBSERVED DATA ATTERBURY EXPERIMENTAL WATERSHED GEOMETRIC DISTRIBUTION' (j».= 0.333) n p ) INTERVAL TIME, DAYS Fig. 2. Distribution of time between occurrence of convective storms RETURN PERIOD, YEARS Fig. 3. Annual maximum series fit to simulated and observed data using extreme-value distribution 690