# 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

- Kennedy, R.G. The prevention of silting in irrigation canals. Minutes Proc. Inst. Civ. Eng.
**1895**, 119, 281–290. [Google Scholar] [CrossRef][Green Version] - Ackers, P. Experiments on small streams in alluvium. J. Hydraul. Div. Proc. Am. Soc. Civ. Eng.
**1964**, 90, 1–37. [Google Scholar] [CrossRef] - Brush, L.M. Drainage Basins, Channels and Flow Characteristics of Selected Streams in Central Pennsylvania; US Government Printing Office: Washington, DC, USA, 1961; Volume 282-F. [CrossRef]
- Fahnestock, R.K. Morphology and Hydrology of a Glacial Stream: White River, Mount Rainier, Washington; US Government Printing Office: Washington, DC, USA, 1963; Volume 422-A. [CrossRef]
- Leopold, L.B.; Miller, J.P. Ephemeral Streams: Hydraulic Factors, and Their Relation to the Drainage Net; US Government Printing Office: Washington, DC, USA, 1956; Volume 282-A. [CrossRef]
- Lewis, L.A. The adjustment of some hydraulic variables at discharges less than one cfs. Prof. Geogr.
**1966**, 18, 230–234. [Google Scholar] [CrossRef] - Miller, J.P. High Mountain Streams: Effect of Geology on Channel Characteristics and Bed Material; New Mexico Bureau of Mines and Mineral Resources Memoir: Socorro, NM, USA, 1958; Volume 4, Available online: https://docslib.org/doc/5832274/high-mountain-streams-effects-of-geology-on-channel-characteristics-and-bed-material (accessed on 23 October 2022).
- Stall, J.B.; Fok, Y.S. Hydraulic Geometry of Illinois Streams; Research Report; University of Illinois Water Resources Center: Champaign, IL, USA, 1968; Volume 15, Available online: https://catalog.library.tamu.edu/Record/in00000588148 (accessed on 23 October 2022).
- Wolman, M.G. The Natural Channel of Brandywine Creek, Pennsylvania; US Government Printing Office: Washington, DC, USA, 1955; Volume 271. [CrossRef]
- Lacey, G. Stable channels in alluvium. Minutes Proc. Inst. Civ. Eng.
**1930**, 229, 259–292. [Google Scholar] [CrossRef] - Blench, T. Regime Behaviour of Canals and Rivers; Butterworths Scientific Publications: Waltham, MA, USA, 1957; Available online: https://www.semanticscholar.org/paper/Regime-behaviour-of-canals-and-rivers-Blench/1a9d784e82aed05fbfa042b5e1bd64c50c5ca5d7 (accessed on 23 October 2022).
- Leopold, L.B.; Maddock, T.J. The Hydraulic Geometry of Stream Channels and Some Physiographic Implication; US Government Printing Office: Washington, DC, USA, 1953; Volume 252. [CrossRef][Green Version]
- Gleason, C.J. Hydraulic geometry of natural rivers: A review and future directions. Prog. Phys. Geogr. Earth Environ.
**2015**, 39, 337–360. [Google Scholar] [CrossRef] - Julien, P.Y. Downstream hydraulic geometry of alluvial rivers. Proc. Int. Assoc. Hydrol. Sci.
**2014**, 367, 3–11. [Google Scholar] [CrossRef][Green Version] - Velikanov, M.A. Alluvial Process. In Alluvial Process: Findamental Principles; Velikanov, M.A., Ed.; State Publishing House for Physical and Mathematical Literature: Moscow, Russia, 1958; pp. 241–245. [Google Scholar]
- Taylor, E.H. Flow characteristics at rectangular open-channel junctions. Trans. Am. Soc. Civ. Eng.
**1944**, 109, 893–902. Available online: https://www.semanticscholar.org/paper/Flow-Characteristics-at-Rectangular-Open-Channel-Taylor/4c6add5c5c25136cbb5ba6549573757978f0af96 (accessed on 23 October 2022). [CrossRef] - Marra, W.A.; Parsons, D.R.; Kleinhans, M.G.; Keevil, G.M.; Thomas, R.E. Near-bed and surface flow division patterns in experimental river bifurcations. Water Resour. Res.
**2014**, 50, 1506–1530. [Google Scholar] [CrossRef] - Wang, B.; Xu, Y. Estimating bed material fluxes upstream and downstream of a controlled large bifurcation—the MississippiAtchafalaya River diversion. Hydrol. Process.
**2020**, 34, 2864–2877. [Google Scholar] [CrossRef] - Ding, S.J.; Qiu, F.L. Calculation of branch sediment diversion. J. Sediment Res.
**1981**, 1, 58–64. [Google Scholar] [CrossRef] - Xie, J.H. Riverbed Evolution and Regulation; China Water Resources and Hydropower Publishing House: Beijing, China, 1990. [Google Scholar]
- Wang, Z.B.; De Vries, M.; Fokkink, R.J.; Langerak, A. Stability of river bifurcations in ID morphodynamic models. J. Hydraul. Res.
**1995**, 33, 739–750. [Google Scholar] [CrossRef] - Bolla Pittaluga, M.; Repetto, R.; Tubino, M. Channel bifurcation in braided rivers: Equilibrium configurations and stability. Water Resour. Res.
**2003**, 39, 1046. [Google Scholar] [CrossRef] - Han, Q.W.; He, M.M.; Chen, X.W. Model of suspended sediment separation in branch channel. J. Sediment Res.
**1992**, 1, 44–54. [Google Scholar] [CrossRef] - Qin, W.K.; Fu, R.S.; Han, Q.W. A model of suspended load diversion in branched channel. J. Sediment Res.
**1996**, 3, 21–29. [Google Scholar] [CrossRef] - Lane, S.N.; Richards, K.S. High resolution, two-dimensional spatial modelling offlow processes in a multi-thread channel. Hydrological Processes.
**1998**, 12, 1279–1298. [Google Scholar] [CrossRef] - Neary, V.S.; Odgaard, A.J. Three-Dimensional Flow Structure at Open-Channel Diversions. J. Hydraul. Eng.
**1993**, 119, 1223–1230. [Google Scholar] [CrossRef] - Barkdoll, B.D.; Hagen, B.L.; Odgaard, A.J. Experimental Comparison of Dividing Open-Channel with Duct Flow in T-Junction. J. Hydraul. Eng.
**1998**, 124, 92–95. [Google Scholar] [CrossRef] - Dargahi, B. Three-dimensional flow modelling and sediment transport in the River Klarälven. Earth Surf. Process. Landforms
**2004**, 29, 821–852. [Google Scholar] [CrossRef] - Yan, Y.X.; Gao, J.; Song, Z.Y.; Zhu, Y.L. Calculation method for stream passing around the Jiuduan sandbank in the river mouth of Yangtze River. J. Hydraul. Eng.
**2001**, 4, 79–84. [Google Scholar] [CrossRef] - Meselhe, E.; Sadid, K.; Khadka, A. Sediment Distribution, Retention and Morphodynamic Analysis of a River-Dominated Deltaic System. Water
**2021**, 13, 1341. [Google Scholar] [CrossRef] - Bertoldi, W.; Tubino, M. River bifurcations: Experimental observations on equilibrium configurations. Water Resour. Res.
**2007**, 43, w10437. [Google Scholar] [CrossRef] - Tong, C.F.; Yan, Y.X.; Meng, Y.Q.; Yue, L.L. Methods for evaluating flow diversion ratio of bifurcated rivers. Adv. Sci. Technol. Water Resour.
**2011**, 31, 7–9. [Google Scholar] [CrossRef] - Yang, X.; Sun, Z.; Deng, J.; Li, D.; Li, Y. Relationship between the equilibrium morphology of river islands and flow-sediment dynamics based on the theory of minimum energy dissipation. Int. J. Sediment Res.
**2022**, 37, 514–521. [Google Scholar] [CrossRef] - Xu, F.; Coco, G.; Townend, I.; Guo, L.; He, Q.; Zhao, K.; Zhou, Z. Rationalizing the Differences Among Hydraulic Relationships Using a Process-Based Model. Water Resour. Res.
**2021**, 57, e2020WR029430. [Google Scholar] [CrossRef] - Zhao, K.; Gong, Z.; Zhang, K.; Wang, K.; Jin, C.; Zhou, Z.; Xu, F.; Coco, G. Laboratory Experiments of Bank Collapse: The Role of Bank Height and Near-Bank Water Depth. J. Geophys. Res. Earth Surf.
**2020**, 125, e2019JF005281. [Google Scholar] [CrossRef] - Barber, C.A.; Gleason, C.J. Verifying the prevalence, properties, and congruent hydraulics of at-many-stations hydraulic geometry (AMHG) for rivers in the continental United States. J. Hydrol.
**2018**, 556, 625–633. [Google Scholar] [CrossRef] - Tran, T.-T.; van de Kreeke, J.; Stive, M.J.; Walstra, D.-J.R. Cross-sectional stability of tidal inlets: A comparison between numerical and empirical approaches. Coast. Eng.
**2011**, 60, 21–29. [Google Scholar] [CrossRef] - D’Alpaos, A.; Lanzoni, S.; Marani, M.; Rinaldo, A. On the tidal prism–channel area relations. J. Geophys. Res. Atmos.
**2010**, 115, F01003. [Google Scholar] [CrossRef][Green Version] - Arkesteijn, L.; Blom, A.; Czapiga, M.J.; Chavarrías, V.; Labeur, R.J. The Quasi-Equilibrium Longitudinal Profile in Backwater Reaches of the Engineered Alluvial River: A Space-Marching Method. J. Geophys. Res. Earth Surf.
**2019**, 124, 2542–2560. [Google Scholar] [CrossRef][Green Version] - Sun, Z.; Xia, S.; Zhu, X.; Xu, D.; Ni, X.; Huang, S. Formula of time-dependent sediment transport capacity in estuaries. J. Tsinghua Univ.
**2010**, 3, 383–386. [Google Scholar] [CrossRef]

**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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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