# 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 |

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**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

**AMA Style**

Gioia A.
Reservoir Routing on Double-Peak Design Flood. *Water*. 2016; 8(12):553.
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