# The Hydraulic and Boundary Characteristics of a Dike Breach Based on Cluster Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Object and Analysis Methods

#### 2.1. Research Object

- (1)
- Embankment breach and developing characteristics

- (2)
- Dike collapse characteristic value

#### 2.2. Analytical Research Methods

#### 2.2.1. Research Ideas

#### 2.2.2. Specific Analysis Methods

_{i}and y

_{i}are the values of the two variables, which were standardized with Equation (1).

_{i}, y

_{i}), i = 0, 1, 2, …, m}, the best curve y = S*(x) can be found from a specific curve to ensure that the curve can fit those data most reasonably.

_{i}, y

_{i}), i = 0, 1, 2, …, m}, let y

_{i}= f(x

_{i}) (i = 0, 1, 2, …, m). Let y = S*(x) be the fitting function of the given data, and record the error δ

_{i}= S*(x

_{i}) − y

_{i}(i = 0, 1, 2, …, m), δ = (δ

_{0}, δ

_{1}, …, δ

_{m})

^{T}. Let φ

_{0}(x), φ

_{1}(x), …, φ

_{n}(x) be a family of linear independent functions on the continuous function space C[a,b]. Find a function S*(x) from φ = span{φ

_{0}(x), φ

_{1}(x), …, φ

_{n}(x)} to minimize the sum of squared errors:

_{n}}.

_{i}) [21,22,23]. To find the fitted curve by the least squares method, we find a function y = S*(x) in S(x) shown as (4), which minimizes the sum of squared errors of the samples. This is needed to determine the minimum point of the multifunction (a

_{0}

^{*},a

_{1}

^{*},…,a

_{n}

^{*}). Let the multivariate function I be:

_{1}≠ 0; otherwise, accept the null hypothesis. Here, $\widehat{\sigma}=\sqrt{\frac{S{S}_{e}}{n-2}}$,${S}_{xx}={\displaystyle \sum {({x}_{i}-\overline{x})}^{2}}$, where SS

_{e}is called the sum of squared residuals, $S{S}_{e}={\displaystyle \sum _{i=1}^{m}{({y}_{i}-{\widehat{y}}_{i})}^{2}}$.

^{2}[24]:

^{2}means how much stronger the linear correlation between y and x is characterized by the regression curve.

## 3. Results and Discussion (Analysis of Eigenvalues of Generalized Hydraulic Boundary)

#### 3.1. Cluster Analysis Results for the Eigenvalues of the Breaches

^{3}/s, even if the maximum value of the dike breaking sample reaches 4200 m

^{3}/s.

#### 3.2. Results of Fitting Regression Analysis

- (1)
- Fitting relationship between Q and B

^{2}is 0.664, which is in accordance with the goodness of fit test requirement.

- (2)
- Fitting relationship between H and Q

_{1}is 0.00165, and B

_{2}is −1.069 × 10

^{−8}. Therefore, the fitting equation is

^{2}is 0.783, which is also in accordance with the goodness of fit test requirement.

#### 3.3. Fitting Complement and Dimensionless Parameter Analysis

## 4. Discussion

^{2}). The probability density function f(B/H) of width-to-depth can be obtained by lognormal distribution fitting and is shown as follows with μ = 1.742 and σ = 0.633.

## 5. Conclusions

^{3}/s and a maximum of 4198 m

^{3}/s; the drop is generally not more than 5.66 m. These analytical values make up for the shortcomings of the characteristic parameters of the hydraulic boundary of the breach and can provide basic data for scientific research, such as dam break model tests and plugging technology design.

^{3}/s. Correlation analysis between variables shows that these models meet the goodness of fit test requirements.

^{2}), μ = 1.64, σ = 0.434. The maximum probability density is 0.137; the value of B/H is mainly in the range of 3~8, and the cumulative frequency of the interval is approximately 55%, which has characteristics proving that the mouth width is larger than the mouth water depth. The Froude number in the fracture zone also conforms to the normal distribution: Fr~N(μ, σ

^{2}) μ = 0.476, σ = 0.204, the maximum probability density is 1.956; Fr is mainly in the range of 0.4~0.8, and the corresponding cumulative frequency is approximately 60%. The upstream and downstream water head difference decreases in the middle and late stages of the collapse, the water flow energy in the fracture zone is smaller than the potential energy, and the flow state is mostly slow flow. Based on the above two dimensionless parameters, B/H and Fr are selected to further determine the hydraulic boundary conditions of the generalized breach, and a simulation test of the dike collapse is carried out to study the hydraulic characteristics of the breach and the plugging technique.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of occurrence characteristics of embankment breach. Bi, Hi and vi denotes the width, water level and velocity of the breach at moment I, respectively, while B, H, and v denotes the maximum value of the width, water level and velocity of the breach, respectively.

**Figure 2.**The transverse and longitudinal section of the generalizes breach. The left figure (

**a**) shows that the cross section of the breach is generalized as a trapezoid shape with side slope m = 1.0 and height of the dyke as h. The right figure (

**b**) shows that there is a water head drop Δz between two sides of the breach along the flood direction. The meaning of other symbols in the figure are same as above of the respective characteristic value, and the abscissa is time breach developing.

**Figure 3.**Research logic chart by means of cluster analysis method. During the process, the feature values are considered dimensionless, and the probability density distribution characteristics of each dimensionless parameter are analysed. On this basis, the hydraulic-boundary feature value of the generalized breach are determined.

**Figure 4.**Hydraulic boundary eigenvalues of embankment breach. Considering the similarities and differences in hydraulic boundary conditions, these three kinds of data were classified into 6 types.

**Figure 6.**Schematic diagram of calculation process of fitted and imputed value. The missing value estimation of each eigenvalue can be sequentially performed.

**Figure 7.**Fitting regression curve of Q~B. According to the t test method, the standard error of the fitting line is 3.733, and the t value is 10.098.

**Figure 8.**Fitting regression curve of H-Q. The intercept C of the fitting line is 5.486, and the standard error is 0.974.

**Figure 9.**Comparison of breach water head before and after fitting. The fitted data of water depth is more uniform, and is centred in about 6 m.

**Figure 10.**Comparison of gate flow rate before and after fitting. The fitted data of velocity ranges mainly in 2 m/s and 7 m/s.

**Figure 11.**Fitting error distribution of breach rate and water head. The relative deviation e of the data is decreased largely after fitting.

**Figure 13.**Probability density of width-to-depth ratio (B/H) and its percentile distribution. The probability density distribution of the width–depth ratio basically conforms to the lognormal distribution.

**Figure 15.**Probability density and percentile distribution of Froude number (Fr). The probability density distribution of the Fr approximates the general normal distribution, i.e., Fr~N(μ, σ

_{2}).

Eigenvalues | B /m | H /m | v /(m·s ^{−1}) | ΔZ /m | Q /(m ^{3}·s^{−1}) |
---|---|---|---|---|---|

B/m | 1.000 | 0.535 | −0.232 | 0.275 | 0.811 |

H/m | 1.000 | −0.191 | −0.093 | 0.865 | |

v/(m·s^{−1}) | 1.000 | −0.226 | −0.068 | ||

ΔZ/m | 1.000 | −0.307 | |||

Q/(m^{3}·s^{−1}) | 1.000 |

Number | B/H | Fr | Number | B/H | Fr | Number | B/H | Fr | Number | B/H | Fr |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 16.45 | 0.07 | 20 | 15.00 | 0.82 | 39 | 4.55 | 0.73 | 60 | 4.82 | 0.57 |

2 | 15.13 | 0.10 | 21 | 4.55 | 0.73 | 40 | 55.00 | 1.00 | 61 | 6.20 | 0.22 |

3 | 12.67 | 0.20 | 22 | 3.08 | 0.58 | 41 | 4.65 | 0.31 | 62 | 22.22 | 4.16 |

4 | 6.63 | 0.33 | 23 | 9.23 | 0.43 | 42 | 32.80 | 0.57 | 63 | 11.35 | 0.17 |

5 | 7.29 | 0.55 | 24 | 6.09 | 0.47 | 43 | 13.62 | 0.15 | 64 | 5.00 | 0.49 |

6 | 20.00 | 0.61 | 25 | 4.33 | 0.52 | 44 | 13.33 | 0.41 | 65 | 4.40 | 0.52 |

7 | 3.98 | 0.71 | 26 | 1.45 | 0.27 | 45 | 6.67 | 0.08 | 66 | 3.38 | 0.26 |

8 | 14.10 | 0.13 | 27 | 6.84 | 0.32 | 46 | 5.30 | 0.73 | 67 | 2.45 | 0.76 |

9 | 11.15 | 0.29 | 28 | 3.35 | 0.64 | 47 | 7.34 | 0.24 | 69 | 1.36 | 0.62 |

10 | 10.00 | 0.42 | 29 | 8.36 | 0.50 | 48 | 6.40 | 0.65 | 70 | 4.33 | 0.52 |

11 | 9.44 | 0.41 | 30 | 3.98 | 0.71 | 49 | 4.93 | 0.57 | 72 | 5.69 | 0.11 |

12 | 7.75 | 0.55 | 31 | 13.33 | 0.52 | 50 | 9.44 | 0.41 | 74 | 4.13 | 0.53 |

13 | 10.00 | 0.42 | 32 | 2.35 | 0.15 | 51 | 3.35 | 0.64 | 76 | 7.67 | 0.06 |

14 | 6.65 | 0.17 | 33 | 3.98 | 0.71 | 52 | 12.12 | 0.23 | 77 | 2.17 | 0.87 |

15 | 4.55 | 0.73 | 34 | 13.62 | 0.15 | 53 | 4.39 | 0.31 | 78 | 4.29 | 0.52 |

16 | 8.04 | 0.52 | 35 | 15.16 | 0.10 | 54 | 3.86 | 0.71 | 80 | 3.75 | 0.56 |

17 | 4.55 | 0.73 | 36 | 5.99 | 0.67 | 55 | 12.06 | 0.57 | 81 | 1.84 | 0.23 |

18 | 4.55 | 0.73 | 37 | 33.33 | 2.09 | 56 | 1.25 | 0.57 | 82 | 9.26 | 0.43 |

19 | 8.04 | 0.52 | 38 | 3.35 | 0.64 | 57 | 4.50 | 0.51 | 83 | 5.75 | 0.37 |

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**MDPI and ACS Style**

Liu, M.; Luo, Y.; Qiao, C.; Wang, Z.; Ma, H.; Sun, D.
The Hydraulic and Boundary Characteristics of a Dike Breach Based on Cluster Analysis. *Water* **2023**, *15*, 2908.
https://doi.org/10.3390/w15162908

**AMA Style**

Liu M, Luo Y, Qiao C, Wang Z, Ma H, Sun D.
The Hydraulic and Boundary Characteristics of a Dike Breach Based on Cluster Analysis. *Water*. 2023; 15(16):2908.
https://doi.org/10.3390/w15162908

**Chicago/Turabian Style**

Liu, Mingxiao, Yaru Luo, Chi Qiao, Zezhong Wang, Hongfu Ma, and Dongpo Sun.
2023. "The Hydraulic and Boundary Characteristics of a Dike Breach Based on Cluster Analysis" *Water* 15, no. 16: 2908.
https://doi.org/10.3390/w15162908