# Hydraulic Geometry and Theory of Equilibrium Water Depth of Branching River

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{w}to supplement the formula. Leopold and Maddock [12] proposed a hydraulic geometry relationship as a power function between width, depth, velocity and flow discharge. They used data from 119 measurement stations in nine different basins of America in their study and indicated that the hydraulic geometry of the channel can be expressed as a function of flow discharge.

_{50}to represent the three factors of meteorology, topography and geological structure, respectively. Using the dimensional analysis method, the expression of river width and water depth is obtained, which is widely used as $\sqrt{Bd}/H=\zeta $ after simplification.

## 2. Methods

#### 2.1. Hydraulic Geometry

_{i}is the bed-forming discharge, B

_{i}is the river width, h

_{i}(y) is the water depth, u

_{i}(y) is the vertical average velocity, and y represents the lateral direction, that is, the river width direction. The subscript i = 0 represents the main stream, and i = 1, 2 represents a distributary channel, as shown in Figure 1.

_{i}is the hydraulic gradient and C

_{i}is the dimensionless Chezy coefficient, which reflects the river resistance.

_{i}is the average water depth, U

_{i}is the average velocity in section, and k

_{0}is a dimensionless constant.

_{m}is the average particle size, which reflects the influence of the composition of non-uniform sediment on the cross-section morphology. It is usually assumed that the average particle size D

_{m}and morphological coefficient $\xi $ remain unchanged before and after branching.

_{max}is the maximum water depth in the section, as shown in Figure 2.

_{1}is an integral related to the lateral distribution of relative water depth, which can be obtained by numerical integration of terrain curves or calculated using actual terrain data. It is applicable to arbitrary terrain curve, and its value varies with terrain.

_{2}is the integral coefficient of the terrain curve, generally between 0.5 and 1. The average velocity of the main stream and distributary channel is:

#### 2.2. Theory of Equilibrium Water Depth

_{2}= 2/3, then

_{L}within the channel width is:

_{s}is:

_{L}can still be calculated according to the channel position by Equation (22).

_{r}is the distance between the regulation lines, B

_{j}is the total width, h

_{u}is the water depth of the upper rectangular, determined according to the average water depth on the regulation line, and h

_{d}is the maximum water depth of lower parabolic, then the main channel section area is:

_{max}is expressed as:

## 3. Results

#### 3.1. Equilibrium Water Depth of North Passage

#### 3.1.1. Before the Project

_{max}before the project. The positions of the sections are shown in Figure 5. The shape function of the sections was set as a parabola.

_{max}is 6.9 m, which is basically consistent with the actual situation, wherethe water depth at the top of the bar sand section of the Yangtze Estuary channel was maintained at 6–7 m for a long time before the project. This shows that the hydraulic geometry proposed in this paper is suitable for estimating the equilibrium water depth of a branching river.

#### 3.1.2. After the Project

^{3}. Considering the influence of spur dikes, we used the modified hydraulic geometry relationship to calculate the navigable water depth in the channel. The layout of the regulating structure is shown in Figure 6.

#### 3.2. Equilibrium Water Depth of Fujiangsha Waterway

#### 3.2.1. Before the Project

_{L}before the project. The shape function of the sections was set as a parabola. The positions of the sections are shown in Figure 9.

#### 3.2.2. After the Project

_{L}after the project. Compared with the design navigable depth of 12.5 m, the effect of channel regulation can be predicted and analyzed, which will provide follow-up ideas and scientific basis for channel regulation. The positions of the calculated sections, which are located between spur dikes, are shown in Figure 11.

#### 3.3. Equilibrium Water Depth of Shiyezhou Waterway

#### 3.3.1. Before the Project

_{L}before the project. The shape function of the sections was also set as a parabola. The positions of the sections are shown in Figure 12.

#### 3.3.2. After the Project

_{L}after the project. The positions of the sections are shown in Figure 14. The navigable depth calculated according to the hydraulic geometry is shown in Table 6.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Locations of waterways in the 12.5 m deep-water channel below Nanjing of the Yangtze River.

Section | A_{0} (m^{2}) | B_{i} (m) | Q_{i}/Q_{0} | A_{i} (m^{2}) | h_{max} (m) |
---|---|---|---|---|---|

b-b’ | 69,539 | 8303 | 0.793 | 57,002 | 6.9 |

Section | A_{0} (m^{2}) | B_{r} (m) | Q_{i}/Q_{0} | A_{i} (m^{2}) | h_{L} (m) | $\Delta \mathit{H}$ (m) |
---|---|---|---|---|---|---|

① | 73,012 | 3030 | 0.439 | 36,053 | 11.46 | −1.04 |

② | 3044 | 0.455 | 37,176 | 11.43 | −1.07 | |

③ | 3068 | 0.473 | 38,433 | 11.40 | −1.10 | |

④ | 2942 | 0.376 | 31,586 | 9.66 | −2.84 | |

⑤ | 3023 | 0.404 | 33,585 | 9.52 | −2.98 | |

⑥ | 3211 | 0.442 | 36,215 | 9.78 | −2.72 | |

⑦ | 3724 | 0.501 | 40,344 | 10.36 | −2.14 | |

⑧ | 4118 | 0.564 | 44,693 | 10.39 | −2.11 |

Section | A_{0} (m^{2}) | Q_{i}/Q_{0} | A_{i} (m^{2}) | B_{i} (m) | h_{L} (m) | $\Delta \mathit{H}$ (m) |
---|---|---|---|---|---|---|

② | 40,389 | 0.802 | 33,429 | 3409 | 12.10 | −0.40 |

③ | 3385 | 12.20 | −0.30 | |||

④ | 3396 | 12.14 | −0.36 | |||

⑤ | 3405 | 12.11 | −0.39 | |||

⑥ | 0.376 | 17,463 | 1859 | 7.59 | −4.91 | |

⑦ | 1836 | 7.77 | −4.73 | |||

⑧ | 1844 | 7.71 | −4.79 |

Section | A_{0} (m^{2}) | Q_{i}/Q_{0} | A_{i} (m^{2}) | B (m) | h_{L} (m) | $\Delta \mathit{H}$ (m) |
---|---|---|---|---|---|---|

② | 43,089 | 0.749 | 33,634 | 2625 | 13.21 | +0.71 |

③ | 0.746 | 33,519 | 2531 | 13.22 | +0.72 | |

④ | 0.748 | 33,596 | 2590 | 13.05 | +0.55 | |

⑤ | 0.754 | 33,826 | 2795 | 13.06 | +0.56 | |

⑥ | 0.282 | 12,161 | 1344 | 10.71 | −1.79 | |

⑦ | 0.301 | 13,003 | 1159 | 10.63 | −1.87 | |

⑧ | 0.312 | 13,454 | 1271 | 10.66 | −1.84 |

Section | A_{0} (m^{2}) | Q_{i}/Q_{0} | A_{i} (m^{2}) | B (m) | h_{L} (m) | $\Delta \mathit{H}$ (m) |
---|---|---|---|---|---|---|

② | 38,900 | 0.609 | 25,429 | 1772 | 10.79 | −1.71 |

③ | 1802 | 11.12 | −1.38 | |||

④ | 1848 | 11.56 | −0.94 | |||

⑤ | 1869 | 11.90 | −0.60 |

Section | A_{0} (m^{2}) | Q_{i}/Q_{0} | A_{i} (m^{2}) | B (m) | h_{L} (m) | $\Delta \mathit{H}$ (m) |
---|---|---|---|---|---|---|

② | 38,900 | 0.491 | 21,143 | 1252 | 12.95 | +0.45 |

③ | 0.509 | 21,805 | 1226 | 13.30 | +0.80 | |

④ | 0.493 | 21,217 | 1216 | 13.36 | +0.86 | |

⑤ | 0.512 | 21,916 | 1234 | 14.15 | +1.65 |

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**MDPI and ACS Style**

Gao, Y.; Lv, Y.; Li, Y.; Pan, Y.; Yang, E. Hydraulic Geometry and Theory of Equilibrium Water Depth of Branching River. *Water* **2023**, *15*, 430.
https://doi.org/10.3390/w15030430

**AMA Style**

Gao Y, Lv Y, Li Y, Pan Y, Yang E. Hydraulic Geometry and Theory of Equilibrium Water Depth of Branching River. *Water*. 2023; 15(3):430.
https://doi.org/10.3390/w15030430

**Chicago/Turabian Style**

Gao, Yun, Yufeng Lv, Ying Li, Yun Pan, and Enshang Yang. 2023. "Hydraulic Geometry and Theory of Equilibrium Water Depth of Branching River" *Water* 15, no. 3: 430.
https://doi.org/10.3390/w15030430