# A Theoretically Derived Probability Distribution of Scour

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hydraulic Assumptions

## 3. The Theoretically Derived Distribution of Scour (TDDS)

#### 3.1. Simplified Rectangular Flood Hydrograph

#### 3.2. Exponential Flood Hydrograph

## 4. Examples Applications

#### 4.1. Parameters of the BRISENT

#### 4.2. Sensitivity Analysis

_{s}) on the main statistics of scour. With this aim, Figure 6 provides a description of the mean and standard deviation of the scour depth as a function of the mentioned parameter, using different cross-section geometries. It can be noticed that both the mean and the variability of scour tend to increase monotonically with D/d

_{s}, except for the case with $\gamma $ = 0.1. Moreover, the graph also clearly highlights the strong role played by the cross-section geometry on the overall dynamics of the process.

#### 4.3. Application to a Real Context

^{2}and has a mean annual discharge of about 141 m

^{3}/s. Figure 7 shows the case study with the location of the watershed and the river.

_{50}= 1.1 mm, which was used as a reference value for the d

_{s}.

- The parameters of the flood distribution were computed from the time series of annual maxima (Figure 8B). Several methods exist to estimate parameters of a Gumbel distribution; we used the method of moments for the sake of simplicity. Given the mean value was $\mu =141$ m
^{3}/s and the standard deviation was $\sigma =49$ m^{3}/s, Gumbel parameters were $b1=119.11$ m^{3}/s and $\alpha =37.99$ m^{3}/s$.$ - Parameter k was estimated from recorded hydrographs. In the present case the duration of the hydrograph was set equal to 10 h.
- BRISENT parameters were estimated following Pizarro et al. [13]. Considering that the pier diameter was $D=1.5$ m and the grain-size was ${d}_{50}=1.1$ mm, the ratio $D/{d}_{50}$ reached a value of 1363.6. Using Equations (24)–(26), the BRISENT parameters were: $\lambda =4236.1$, ${W}_{max}^{*}=196,908.1$, and $S=9.15$.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a [m] | Coefficient of the power law cross-section area function of H |

b[-] | Exponent of the power law cross-section area function of H |

c [1/s] | Coefficient of the power law mean flow velocity function of H |

d[-] | Exponent of the power law mean flow velocity function of H |

$\xi $ [1/m^{2}] | Coefficient of the power law mean flow velocity function of Q |

$\gamma \text{}$[-] | Exponent of the power law mean flow velocity function of Q |

$\alpha $ [-] | location parameter of the Gumbel distribution |

b_{1} [m^{3}/s] | scale parameter of the Gumbel distribution |

D [m] | Pier diameter |

${d}_{s}$ [m] | Sediment grain-size |

g [m/s] | Acceleration of gravity |

H [m] | Flow depth |

k [s] | Duration of a rectangular hydrograph |

$\lambda $ [-] | BRISENT coefficient |

$\omega $ [sec] | Characteristic time |

$\Omega $ [m^{2}] | Cross-section area |

p_{q}(Q) [-] | Probability density function of floods |

p_{v}(V) [-] | Probability density function of velocities |

p_{z}_{*}(Z*) [-] | Probability density function of dimensionless scour |

p_{z}(z) [-] | Probability density function of scour |

Q [m^{3}/s] | River discharge |

Q_{max} [m^{3}/s] | Maximum river discharge in a flood event |

$\rho \text{\u2019}$ [kg/m^{3}] | Relative density |

${\rho}_{s}$ [kg/m^{3}] | Sediment density |

${\rho}_{w}$ [kg/m^{3}] | Water density |

S [-] | Entropic-scour parameter |

t [s] | Time |

t_{end} [s] | Time in which a hydrograph is able to make work |

t_{R} [s] | Reference time |

u_{c} [m/s] | Critical velocity for the initiation of sediment motion |

u_{cs} [m/s] | Critical velocity for the incipient scour |

u_{R} [m/s] | Reference velocity |

V [m/s] | Cross-section-averaged velocity |

W^{*} [-] | Dimensionless, effective flow work |

W^{*}_{max} [-] | Maximum possible W*, according to BRISENT formulation |

Z^{*} [-] | Normalized scour depth |

Z^{*}_{max} [-] | Maximum possible Z*, according to BRISENT formulation |

z_{R} [m] | Reference scour depth |

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**Figure 1.**Example of different cross-sections described with different values of $\gamma $ (

**A**), the corresponding flow rating curve (

**B**) and the velocity as a function of the discharge (

**C**). Parameter a and parameter b are defined on the basis of the assigned value of $\gamma $, while the remaining parameters are a = 2$\text{}\gamma $, b = 1 and c = 0.2.

**Figure 2.**Conceptual diagram illustrating the different steps of TDDS derivation: (1) The natural flood hydrograph; (2) the equivalent exponential hydrograph; (3) the use of the mean flow velocity function to derive the scour energy; (4) estimation of the effective flow work; (5) derivation of the total scour at the event scale; and (6) derivation of the probability distribution of scour.

**Figure 3.**Ratio between equivalent duration k and $\omega $ as a function of the discharge. It can be noticed that the ratio k/$\omega $ tends to stabilize when Q tends to increase.

**Figure 4.**Derived probability density function of the mean flow velocities associated with a given river basin (

**A**) and the corresponding PDFs of scour depth (

**B**) obtained assuming hydrographs with variable equivalent duration k, ranging from 1 to 6 h. Other parameters are: D = 1.07 m; d

_{50}= 0.001 m; ${\rho}_{s}$ = 2.65 t/m

^{3}; u

_{c}= 0.319 m/s; $\lambda $ = 4.7252 × 10

^{3}; W

^{*}

_{max}= 1.8613 × 10

^{5}; S = 8.9026; α = 67.8627 m

^{3}/s; b

_{1}= 143.643; $\gamma $ = 0.33; $\xi $ = 0.36 1/m

^{2}.

**Figure 5.**Derived probability density functions of the mean flow velocities associated with a given river basin (

**A**,

**C**) and the corresponding PDFs of scour depths obtained by modifying the shape parameter of the cross-section γ, with a fixed value of $\xi $ = 0.30 1/m

^{2}and k = 60 min (

**A**,

**B**), or modifying the scale parameter $\xi $ with fixed values of γ = 0.30 and k = 60 min (

**C**,

**D**). Other parameters are: D = 1.07 m; d

_{50}= 0.001 m; ${\rho}_{s}$ = 2.65 t/m

^{3}; u

_{c}= 0.319 m/s; $\lambda $ = 4.7252 × 10

^{3}; W

^{*}

_{max}= 1.8613 × 10

^{5}; S = 8.90; α = 67.86 m

^{3}/s; b

_{1}= 143.64.

**Figure 6.**Mean (

**A**) and standard deviation (

**B**) of scour obtained with the proposed framework for different cross-sections, obtained by modifying the shape parameter of the cross-section γ. Other parameters are: D = 1.07 m; d

_{50}= 0.001 m; ${\rho}_{s}$ = 2.65 t/m

^{3}; u

_{c}= 0.319 m/s; $\lambda $ = 4.7252 × 10

^{3}; W

^{*}

_{max}= 1.8613 × 10

^{5}; S = 8.90; α = 67.86 m

^{3}/s; b

_{1}= 143.64; $\xi $ = 0.30 1/m

^{2}and k = 60 min.

**Figure 7.**Description of the case study adopted: (

**A**) Aerial photo and (

**B**) photo of the bridge (lat: 40.36°; lon: 16.78°).

**Figure 8.**TDDS derived for a real study case: (

**A**) Hydraulic function of velocity and cross-section area as a function of the hydraulic stage H; (

**B**) cumulated probability distribution of flood annual maxima; (

**C**) cumulated probability distribution of scour of the Basento River at SS106.

Description | Data |
---|---|

Cross-section | Basento SS 106 |

River Basin Area (km^{2}) | 1520 |

E[Q] (m^{3}/s) | 141 |

σ[Q] (m^{3}/s) | 49 |

Pier Diameter D (m) | 1.5 |

Sediment Size d_{50} (mm) | 1.1 |

Number of Piers | 4 |

Bridge Length (m) | 15 |

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**MDPI and ACS Style**

Manfreda, S.; Link, O.; Pizarro, A. A Theoretically Derived Probability Distribution of Scour. *Water* **2018**, *10*, 1520.
https://doi.org/10.3390/w10111520

**AMA Style**

Manfreda S, Link O, Pizarro A. A Theoretically Derived Probability Distribution of Scour. *Water*. 2018; 10(11):1520.
https://doi.org/10.3390/w10111520

**Chicago/Turabian Style**

Manfreda, Salvatore, Oscar Link, and Alonso Pizarro. 2018. "A Theoretically Derived Probability Distribution of Scour" *Water* 10, no. 11: 1520.
https://doi.org/10.3390/w10111520