#
The Theoretical Probability Distribution of Peak Outflows of Small Detention Dams^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Theoretically Derived Probability Distribution of Peak Outflows of a Dam

^{3}/s), incoming in the detention basin with the peak outflow, ${Q}_{p,out}$ (m

^{3}/s), is defined.

^{3}) is the volume of water accumulated in the dam at the crest level, ${h}_{s}$ (m) (height of the spillway crest); and ${Q}_{c}$ (m

^{3}/s) is the control value of discharge that is computed using the discharge equation of the submerged opening assuming $h={h}_{s}$.

_{3}of the peak outflow from an in-line detention dam when the water level is at the crest of the spillway. Given the monotonic nature of the above expression, any probability distribution of the undisturbed flood peaks can be adopted. The expression of the TDD

_{3}will be as follows [14]:

_{3}of the outflow when ${Q}_{p,out}>{Q}_{c}$ assumes the following form:

## 3. Application

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notations

α (-) | Scale parameter of the GEV distribution |

β (-) | Location parameter of the GEV distribution |

ξ (-) | Shape parameter of the GEV distribution |

μ_{f} (-) | Coefficient of discharge of the submerged low-level opening |

μ_{s} (-) | Coefficient of discharge of the crest spillway |

b (m) | Width of the low-level opening rectangular section |

d (m) | Height of the low-level opening |

h (m) | Variable water level within the detention basin |

h_{s} (m) | Height of the spillway crest |

h_{f} (m) | Height of the barycenter of the low-level opening |

k_{eq} (s) | Equivalent delay constant of the conceptual reservoir associated with the outflow |

L (m) | Effected crest length |

n (-) | Exponent of the stage-storage capacity curve |

n^{'} (-) | Exponent of the rainfall intensity duration function |

p(Q) (-) | Probability density function of outflows |

Q_{c} (m^{3}/s) | Design outflow from the low-level opening |

Q_{p,out} (m^{3}/s) | Peak outflow from the detention basin |

Q_{max} (m^{3}/s) | Peak flow incoming in the detention basin |

t_{p} (s) | Equivalent flood duration |

t_{r} (s) | Basin lag time |

W_{max} (m^{3}) | Water storage capacity at the crest level |

w_{1} (m^{3−n}) | Parameter of the stage-storage capacity curve |

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**Figure 1.**Example of a detention dam composed of a low-level opening (b × d), a crest spillway of length L and height hs, and with a capacity equal to Wmax. The figure provides the longitudinal (

**A**) and the frontal sections (

**B**).

**Figure 2.**Comparison between three different derived pdfs of the peak outflows obtained using three different flood peak distributions and the empirical pdfs derived via numerical hydraulic simulation (red dots for inflows, blue dots for outflows). The three graphs were obtained by modifying the shape parameter, ξ, of the GEV distribution, which is equal to 0 for Gumbel distribution (

**A**), 0.5 for Fréchet distribution (

**B**), and −0.5 for Weibull distribution (

**C**). Remaining parameters are as follows: the scale parameter of the GEV distribution α = 30 m

^{3}/s; the location parameter of the GEV distribution β = 120 m

^{3}/s; w

_{1}= 5000; h

_{s}= 4 m; b = 1 m; d = 1 m; n = 1.9; h

_{f}= d/2; μ

_{f}= 0.85; μ

_{s}= 0.385; L = 3 m; t

_{p}= 1 h.

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**MDPI and ACS Style**

Manfreda, S.; Miglino, D.; Albertini, C. The Theoretical Probability Distribution of Peak Outflows of Small Detention Dams. *Environ. Sci. Proc.* **2022**, *21*, 90.
https://doi.org/10.3390/environsciproc2022021090

**AMA Style**

Manfreda S, Miglino D, Albertini C. The Theoretical Probability Distribution of Peak Outflows of Small Detention Dams. *Environmental Sciences Proceedings*. 2022; 21(1):90.
https://doi.org/10.3390/environsciproc2022021090

**Chicago/Turabian Style**

Manfreda, Salvatore, Domenico Miglino, and Cinzia Albertini. 2022. "The Theoretical Probability Distribution of Peak Outflows of Small Detention Dams" *Environmental Sciences Proceedings* 21, no. 1: 90.
https://doi.org/10.3390/environsciproc2022021090