# Reservoir Routing on Double-Peak Design Flood

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Dimensionless Double-Peak Flood

- (i)
- analytical definition of a double-peak design flood based on the fractal instantaneous unit hydrograph (FIUH) of a river network (i.e., [41]);
- (ii)
- numerical solution of the differential equation that describes the dimensionless water balance of the reservoir, aimed to estimate the reservoir RC as a function of the morphological characteristics of the valley, and the DC of the spillway;
- (iii)
- numerical determination of the “critical” temporal distance between two consecutive peaks of the design flood aimed to find the maximum water level.

_{r}of the basin, according to the following expression:

_{p}, corresponding to a peak flow q

_{e,max}, which is greater than the duration of the precipitation. Assuming that the duration of the rainfall event, d, coincides with the basin lag-time t

_{r}, q

_{e,max}is then obtained by means of Equation (4) for t = t

_{p}and can be expressed by the following equation:

_{r}is supported by Fiorentino et al. [44], who showed that, under different choices of the basin hydrologic response function, the critical duration approaches the lag time; the hypothesis that the time variability of the rainfall intensity can be neglected (e.g., [45,46,47,48,49]) within the critical duration is supported by the above quoted observation, considering that an event with a constant rainfall intensity, during the lag-time, is more precautionary with respect to other kinds of design hyetographs which include rainfall variability. Moreover, the context of application of the proposed model does not require the evaluation of the effects of the rainfall peak propagation as in urban contexts where the rainfall variability is more likely conditioning. Finally, this assumption makes the structure of the analytical solution simpler in order to be used by dam operator or dam designer.

_{e,max}) separated in time by a time interval. The resulting hydrograph, depending on the shift between the two hydrographs, may provide a single larger peak (see Figure 1) or two separate different peaks. The analytical relationship between the largest discharge of the entire flood (Q

_{max}) and q

_{e,max}is written as:

_{r}, D is the fractal dimension of the stream network, and finally the function f(D) = 0.173D + 0.48 was obtained by numerically minimizing (using different values of D) the difference between the maximum value (Q

_{max}) of the resulting flood hydrograph obtained using the analytical relationship in Equation (6), and that was numerically estimated using the function obtained by shifting the two hydrographs, q

_{e}(t), by a time shift equal to Δ*t

_{r}. More specifically, the function f(D) assumes a value equal to 0.78 for D = 1.75 and was obtained by using a range of values between those (ranging between 1.5 and 2) indicated in recent literature (e.g., [43,50,51,52]).

_{r}, it is possible to write Equations (3) and (4) in dimensionless form:

_{max}= Q

_{T}) directly related to the return period; this approach makes the methodology applicable for each assigned return period.

#### 2.2. Dimensionless Reservoir Water Balance

_{e}(t) and q

_{u}(t) represent, respectively, the inflow and outflow discharge, V(t) is the volume stored at time t in the reservoir, h(t) is the water level at time t, evaluated with respect to the fixed-crest spillway and V

_{o}is the SC per units of water depth which is assumed constant and equal to the value of the reservoir surface area at the fixed-crest spillway level. We assume that in the initial conditions, the reservoir surface is at the spillway crest level and the outflow discharge is q

_{u}(0) = 0.

^{−3/2}] and β [m

^{−1}]:

_{max}, and β is the ratio between the volume stored in the valley per 1 m of water level and the volume of a triangular hydrograph with the peak flow equal to the maximum flood peak and a duration equal to 2t

_{r}.

_{u}(τ) and h(τ) functions as those shown, for example, in Figure 2.

## 3. Evaluation of the Dimensionless Design Double-Peak Flood and Discussion

_{r}, the resulting flood is characterized by only one peak, while Δ = 5 produces a resulting flood without any overlap between the two single hydrographs.

_{L}. In other words, the curves display the “critical” dimensionless time interval between consecutive peaks.

_{max}= Q

_{T}, the peak discharge of a given return period, and Δ assumes the critical value shown in Figure 4, for $\mathsf{\alpha}=\frac{\mathsf{\mu}\text{}\mathrm{l}\text{}\sqrt{2\mathrm{g}}}{{\mathrm{Q}}_{\mathrm{T}}}$ and $\mathsf{\beta}=\frac{{\mathrm{V}}_{0}}{{\mathrm{Q}}_{\mathrm{T}}\text{}{\mathrm{t}}_{\mathrm{r}}}$.

_{L}) obtained for different fixed values of α and a variable β. It is worth noting that h

_{L}decreases while increasing β. This negative trend is more pronounced for smaller values of α. For a fixed value of α, a reservoir characterized by a small storage capacity generates a higher water level than one with a large storage capacity. On the other hand, for a fixed value of β, an increase of the spillway DC reduces the maximum water level.

_{L}(see Figure 5) and the routing capacity η (see Figure 6); in this case, the routing effect may be evaluated using Figure 5 and Figure 6, starting from the hydrologic behavior of the basin (t

_{r}, Q

_{max}), the SC of the reservoir (V

_{o}) and the length (l) of the spillway (see Equation(14)). On the other hand, the performed sensitivity analysis may also be useful for dam designing, evaluating in particular the DC of the spillway α (see Figure 4), starting from the SC of the reservoir (V

_{o}), the hydrologic behavior of the basin (t

_{r}, Q

_{max}) and the shift between the two peaks Δ. This gives a strong contribution to applied research, allowing the realistic application of the proposed research.

## 4. Conclusions

_{T}of a given return period. In this way the present research provides the most dangerous conditions that may occur in a river basin resulting from the combined effects of the hydrological behavior of the basin, the reservoir SC and the spillway discharge capacity.

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RC | routing coefficient |

SC | storage capacity |

DC | discharge capacity |

SFA | synthetic flood attenuation |

## References

- Bagatur, T.; Onen, F. Computation of design coefficients in ogee-crested spillway structure using GEP and regression models. KSCE J. Civ. Eng.
**2016**, 20, 951–959. [Google Scholar] [CrossRef] - Savage, B.; Johnson, M. Flow over ogee spillway: Physical and numerical model case study. J. Hydraul. Eng.
**2001**, 127, 640–649. [Google Scholar] [CrossRef] - U.S. Army Corp of Engineers (USACE). Hydraulic Design of Spillways; EM 1110-2-1603; Department of the Army: Washington, DC, USA, 1990. [Google Scholar]
- Gonzalez, C.A.; Chanson, H. Hydraulic Design of Stepped Spillways and Downstream Energy Dissipators for Embankment Dams. Dam Eng.
**2007**, 17, 223–244. [Google Scholar] - Ohtsu, I.; Yasuda, Y.; Takahashi, M. Flow Characteristics of Skimming Flows in Stepped Channels. J. Hydraul. Eng.
**2004**, 130, 860–869. [Google Scholar] [CrossRef][Green Version] - Johnson, M.; Savage, B. Physical and Numerical Comparison of Flow over Ogee Spillway in the Presence of Tailwater. J. Hydraul. Eng.
**2006**, 132, 1353–1357. [Google Scholar] [CrossRef] - Graf, W.L. Downstream hydrologic and geomorphic effects of large dams on American rivers. Geomorphology
**2006**, 79, 336–360. [Google Scholar] [CrossRef] - Cheng, C.T.; Chau, K.W. Flood control management system for reservoirs. Environ. Model. Softw.
**2004**, 19, 1141–1150. [Google Scholar] [CrossRef] - Chang, L.C. Guiding rational reservoir flood operation using penalty-type genetic algorithm. J. Hydrol.
**2008**, 354, 65–74. [Google Scholar] [CrossRef] - Magilligan, F.J.; Nislow, K.H. Changes in hydrologic regime by dams. Geomorphology
**2005**, 71, 61–78. [Google Scholar] [CrossRef] - Collier, M.; Webb, R.H.; Schmidt, J.C. Dams and Rivers: Primer on the Downstream Effects of Dams; U.S. Geological Survey Circular; U.S. Geological Survey: Tucson, AZ, USA, 1996; Volume 1126, p. 94.
- Mediero, L.; Jiménez-Álvarez, A.; Garrote, L. Design flood hydrographs from the relationship between flood peak and volume. International Workshop. Hydrol. Earth Syst. Sci.
**2010**, 14, 2495–2505. [Google Scholar] [CrossRef][Green Version] - Klein, B.; Schumann, A.; Pahlow, M. Stochastic Generation of Hydrographs for the Flood Design of Dams. In Proceedings of the 3rd International Symposium on Integrated Water Resources Management, Ruhr-University Bochum, Bochum, Germany, 26–28 September 2006.
- Klein, B.; Pahlow, M.; Hundecha, Y.; Schumann, A. Probability Analysis of Hydrological Loads for the Design of Flood Control Systems Using Copulas. J. Hydrol. Eng.
**2010**, 15, 360–369. [Google Scholar] [CrossRef] - Flores-Montoya, I.; Sordo-Ward, Á.; Mediero, L.; Garrote, L. Fully Stochastic Distributed Methodology for Multivariate Flood Frequency Analysis. Water
**2016**, 8, 225. [Google Scholar] [CrossRef] - Blazkova, S.; Beven, K. Flood frequency estimation by continuous simulation of subcatchment rainfalls and discharges with the aim of improving dam safety assessment in a large basin in the Czech Republic. J. Hydrol.
**2004**, 292, 153–172. [Google Scholar] [CrossRef] - Micovic, Z.; Hartford, D.N.; Schaefer, M.G.; Barker, B.L. A non-traditional approach to the analysis of flood hazard for dams. Stoch. Environ. Res. Risk Assess.
**2016**, 30, 559–581. [Google Scholar] [CrossRef] - Liu, D.; Xie, B.; Li, H. Design Flood Volume of the Three Gorges Dam Project. J. Hydrol. Eng.
**2011**, 16, 71–80. [Google Scholar] [CrossRef] - Marone, V. Calcolo di massima dell’effetto di laminazione di un serbatoio sulle piene. L’energia Elettrica
**1964**, 41, 693–698. (In Italian) [Google Scholar] - Marone, V. Calcolo di massima di un serbatoio di laminazione. L’Energia Elettrica
**1971**, 48, 561–567. (In Italian) [Google Scholar] - Damiani, P.; Di Santo, A. Valutazione del volume di laminazione di un’onda di piena per il dimensionamento degli scarichi di superficie. Idrotecnica
**1986**, 4, 231–238. (In Italian) [Google Scholar] - Scarrott, R.M.J.; Reed, D.W.; Bayliss, A.C. Indexing the attenuation effect attributable to reservoirs and lakes. In Statistical Procedures for Flood Frequency Estimation; Robson, A., Reed, D., Eds.; Flood Estimation Handbook; Centre for Ecology & Hydrology: Wallingford, UK, 1999; Volume 5, pp. 19–26. [Google Scholar]
- Piga, E.; Saba, A.; Salis, F.; Sechi, G.M. Distribuzione probabilistica delle portate massime annue laminate da un invaso con sfioratore superficiale. In Proceedings of the XXVII Convegno di Idraulica e Costruzioni Idrauliche, Genova, Italy, 12–15 September 2000; Volume 3, pp. 85–92. (In Italian)
- Miotto, F.; Laio, F.; Claps, P. Metodologie speditive per la valutazione dell’effetto di laminazione dei grandi invasi. In Proceedings of the XXX Convegno di Idraulica e Costruzioni Idrauliche, Roma, Italy, 10–15 September 2006. (In Italian)
- Pianese, D.; Rossi, F. Curve di possibilità di laminazione delle piene. Giornale Genio Civile
**1986**, 4–6, 131–148. (In Italian) [Google Scholar] - Fiorentino, M.; Margiotta, M.R. La valutazione dei volumi di piena e il calcolo semplificato dell’effetto di laminazione di grandi invasi. In Proceedings of the XIV Corso di Aggiornamento Sulle Tecniche per la Difesa Dall’inquinamento, Cosenza, Italy, 28–30 June 1998. (In Italian)
- Fiorentino, M. La valutazione dei volumi di piena nelle reti di drenaggio urbano. Idrotecnica
**1985**, 3, 141–152. (In Italian) [Google Scholar] - Montaldo, M.; Mancini, M.; Rosso, R. Flood hydrograph attenuation induced by a reservoir system: Analysis with a distributed rainfall-runoff model. Hydrol. Process.
**2004**, 18, 545–563. [Google Scholar] [CrossRef] - Balistrocchi, M.; Bacchi, B.; Grossi, G. Stima delle prestazioni di una vasca di laminazione: Confronto tra simulazioni continue e metodi analitico-probabilistici. In Proceedings of the XXXII Convegno Nazionale di Idraulica e Costruzioni Idrauliche, Palermo, Italy, 14–17 September 2010. (In Italian)
- Liu, P.; Guo, S.L.; Xiong, L.H.; Li, W.; Zhang, H.G. Deriving reservoir refill operating rules by using the proposed DPNS model. Water Resour. Manag.
**2006**, 20, 337–357. [Google Scholar] [CrossRef] - Hsu, N.S.; Wei, C.C. A multipurpose reservoir real-time operation model for flood control during typhoon invasion. J. Hydrol.
**2007**, 336, 282–293. [Google Scholar] [CrossRef] - Wei, C.C.; Hsu, N.S. Optimal tree-based release rules for real-time flood control operations on a multipurpose multi reservoir system. J. Hydrol.
**2009**, 365, 213–224. [Google Scholar] [CrossRef] - Mediero, L.; Garrote, L.; Martin-Carrasco, F. A probabilistic model to support reservoir operation decisions during flash floods. J. Hydrol. Sci.
**2007**, 52, 523–537. [Google Scholar] [CrossRef] - Karbowski, A.; Malinowski, K.; Niewiadomska–Szynkiewicz, E. A hybrid analytic/rule-based approach to reservoir system management during flood. Decis. Support Syst.
**2005**, 38, 599–610. [Google Scholar] [CrossRef] - Karaboga, D.; Bagis, A.; Haktanir, T. Controlling spillway gates of dams by using fuzzy logic controller with optimum rule number. Appl. Soft Comput.
**2008**, 8, 232–238. [Google Scholar] [CrossRef] - Wardlaw, R.; Sharif, M. Evaluation of genetic algorithms for optimal reservoir system operation. J. Water Resour. Plan. Manag.
**1999**, 125, 25–33. [Google Scholar] [CrossRef] - Akter, T.; Simonovic, S.P. Modeling uncertainties in short-term reservoir operation using fuzzy sets and a genetic algorithm. Hydrol. Sci. J.
**2004**, 49, 1081–1097. [Google Scholar] [CrossRef] - Zahraie, B.; Kerachian, R.; Malekmohammadi, B. Reservoir operation optimization using adaptive varying chromosome length genetic algorithm. Water Int.
**2008**, 33, 380–391. [Google Scholar] [CrossRef] - Negede, A.K.; Horlacher, H.B. Approaches for Dam Overtopping Probability Evaluation. In Proceedings of the 33rd IAHR Congress: Water Engineering for a Sustainable Environment, Vancouver, BC, Canada, 9–14 August 2009.
- Castorani, A.; Piccinni, A.F. The influence of a runoff hydrograph in reservoir routing. In Proceedings of the Fifth Congress Asian and Pacific Regional Division of the International Association for Hydraulic Research, Seoul, Korea, 18–20 August 1986.
- Fiorentino, M.; Oliveto, G.; Rossi, A. Alcuni aspetti del controllo energetico ed idrologico sulla geometria delle reti e delle sezioni fluviali. Parte Prima: Controllo idrologico. In Proceedings of the XXVIII Convegno di Idraulica e Costruzioni Idrauliche, Potenza, Italy, 16–19 September 2002. (In Italian)
- Veneziano, D.; Iacobellis, V. Self-Similarity and Multifractality of Topographic Surfaces at Basin and Sub-Basin Scales. J. Geophys. Res.
**1999**, 104, 12797–12813. [Google Scholar] [CrossRef] - Claps, P.; Oliveto, G. Re-examining the determination of the fractal dimension of river networks. Water Resour. Res.
**1996**, 32, 3123–3135. [Google Scholar] [CrossRef] - Fiorentino, M.; Rossi, F.; Villani, P. Effect of the basin geomorphoclimatic characteristics on the mean annual flood reduction curve. In Proceedings of the 18th Annual Conference on Modeling and Simulation, Pittsburgh, PA, USA, 23–24 April 1987; pp. 1777–1784.
- Iacobellis, V.; Fiorentino, M. Derived distribution of floods based on the concept of partial area coverage with a climatic appeal. Water Resour. Res.
**2000**, 36, 469–482. [Google Scholar] [CrossRef] - Fiorentino, M.; Gioia, A.; Iacobellis, V.; Manfreda, S. Regional analysis of runoff thresholds behavior in Southern Italy based on theoretically derived distributions. Adv. Geosci.
**2011**, 26, 139–144. [Google Scholar] [CrossRef][Green Version] - Gioia, A.; Iacobellis, V.; Manfreda, S.; Fiorentino, M. Influence of infiltration and soil storage capacity on the skewness of the annual maximum flood peaks in a theoretically derived distribution. Hydrol. Earth Syst. Sci.
**2012**, 16, 937–951. [Google Scholar] [CrossRef] - Iacobellis, V.; Castorani, A.; Di Santo, A.R.; Gioia, A. Rationale for flood prediction in karst endorheic areas. J. Arid Environ.
**2015**, 112, 98–108. [Google Scholar] [CrossRef] - Iacobellis, V.; Claps, P.; Fiorentino, M. Climatic control on the variability of flood distribution. Hydrol. Earth Syst. Sci.
**2002**, 6, 229–237. [Google Scholar] [CrossRef] - Fac-Beneda, J. Fractal structure of the Kashubian hydrographic system. J. Hydrol.
**2013**, 488, 48–54. [Google Scholar] [CrossRef] - De Bartolo, S.G.; Gaudio, R.; Gabriele, S. Multifractal analysis of river networks: Sandbox approach. Water Resour. Res.
**2004**, 40, 183–188. [Google Scholar] [CrossRef] - Donadio, C.; Magdaleno, F.; Mazzarella, A.; Kondolf, G.M. Fractal Dimension of the Hydrographic Pattern of Three Large Rivers in the Mediterranean Morphoclimatic System: Geomorphologic Interpretation of Russian (USA), Ebro (Spain) and Volturno (Italy) Fluvial Geometry. Pure Appl. Geophys.
**2015**, 172, 1975–1984. [Google Scholar] [CrossRef]

**Figure 1.**Composition of the dimensionless flood hydrograph, starting from the two single flood hydrographs with a fixed time shift.

**Figure 2.**Inflow and outflow functions of the reservoir balance equation in dimensionless form, for DC α = 4, SC β = 4 and time shift between the two peaks Δ = 1.5. In blue is reported the inflow dimensionless total discharge, in red the outflow discharge from the dam, in green the water level in the reservoir.

**Figure 3.**Flood hydrographs considering the variable temporal distance between the flood peaks, using the spillway DC α = 4 and the SC β = 4: (

**a**) time shift between the two peaks (Δ) equal to 1; (

**b**) time shift between the two peaks (Δ) equal to 5.

**Figure 4.**Trend of the “critical time interval”, Δ, which maximizes the water level, as a function of the spillway DC α and the SC β.

**Figure 5.**Trend of the maximum routing elevation (h

_{L}) as a function of the spillway DC α and the SC β.

**Figure 6.**Trend of the ratio between the downstream and the upstream flood peak discharge (η) as a function of α and β.

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Gioia, A. Reservoir Routing on Double-Peak Design Flood. *Water* **2016**, *8*, 553.
https://doi.org/10.3390/w8120553

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https://doi.org/10.3390/w8120553

**Chicago/Turabian Style**

Gioia, Andrea. 2016. "Reservoir Routing on Double-Peak Design Flood" *Water* 8, no. 12: 553.
https://doi.org/10.3390/w8120553