# Contrast Analysis of Flow-Discharge Measurement Methods in a Wide–Shallow River during Ice Periods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Applications

#### 2.1. Stream-Tube Method

_{y}along the cross-section from the starting point to point y at any position can be written as:

_{y}is equal to the overall cross-sectional flow-discharge Q, as shown in Figure 1a,b.

_{y}to the overall flow in the section can be expressed by Equation (2), proposed by Shen and Ackermann [18]:

_{i}/f

_{b}), as shown in Figure 1a,b. This characteristic coefficient can be expressed as:

- (1)
- Obtain the cross-sectional topography data of rivers, e.g., a double-frequency ground-penetrating radar [31] can be used to quickly obtain the ice thickness and flow-depth distribution along the cross-section.
- (2)
- Obtain the vertical-velocity distribution or depth-averaged velocity at a typical position; for example, the depth-averaged velocity can be obtained using the one-, two-, three-, or the six-point method. Point velocities are usually captured with a SonTek 3D Acoustic Doppler Velocimetry (ADV) or a current meter. The ADV utilizes the principle of acoustic Doppler and uses telemetry for less interference with the flow field at the measurement point. Characterized by high measurement accuracy and sampling frequency, automated data acquisition is possible.
- (3)
- Obtain the section-characteristic coefficient according to the survey-point position and calculate the unit discharge at a typical position based on the depth-averaged velocity and the flow depth at that position. Using Equation (4), we can obtain the relative-unit discharge distribution along the cross-section. Equation (6) can be used to calculate the unit discharge at other measuring points of the section, which can be combined with the flow depth to calculate the depth-averaged velocity. Combined with the deformation of Equation (4), the total-flow discharge can be calculated.

#### 2.2. Characterization Method of the Depth-Averaged Velocity under Ice

#### 2.2.1. Comparison of Characterization Methods

#### 2.2.2. Accuracy of Velocity Estimation of a Single Survey Point

_{b}and m

_{i}are constants related to the flow resistance of the riverbed and ice cover, respectively, H is the total-flow depth, and z is the distance from the riverbed. At z = 0 (riverbed) and z = H (ice-cover bottom), the flow velocity is zero (u = 0).

## 3. Results and Discussion

#### 3.1. Accuracy Analysis of Characterization Method of the Depth-Averaged Velocity

^{3}/s, involving various cross-sectional forms such as natural channels, rectangular cross-sections, and compound sections. These data and associated parameters are shown in Appendix A Table A1. In Figure 3, Figure 4 and Figure 5, we define r

_{m}as the ratio of m

_{b}to m

_{i}, where r

_{m}is represented along the abscissa, and the error percentage is represented along the ordinate; the comparison of the estimation accuracy of each flow-estimation method is discussed below.

- a.
- Comparison of one-point–velocity-estimation methods

_{m}. Values for r

_{m}ranged from 0 to 3.5, and the estimation error of the one-point method at 0.5H was within the range of 6.30%, with an average value of 2.60% and a standard deviation of 1.86. As observed in the figure, the average estimation error of the one-point method at 0.6H is 2.71%, but some values have large deviations; the maximum estimation error is 13.9%, standard deviation is 2.86, and the variation range is large.

- b.
- Comparison of two-point–velocity-estimation methods

_{m}ranged from 0 to 3.5, and the estimation error of the two-point method at 0.2H and 0.8H was within the range of 1.29% to 3.95%, with an average value of 1.98%, satisfying the accuracy requirements. Under the same conditions, the average estimation error of the two-point method at 0.4H and 0.8H was 4.85%, and the maximum estimation error was 20.23%.

- c.
- Comparison of three-point–velocity-estimation methods

_{m}ranged from 0 to 3.5; the average error estimated by the three-point method 1 at 0.15H, 0.5H, and 0.85H was 2.16%; the maximum estimation error was 4.3%, and the standard deviation was 0.52. The average estimation error using the three-point method 2 at 0.2H, 0.6H, and 0.8H was 4.57%; the maximum estimation error was 9.7%, and the standard deviation was 1.56.

#### 3.2. Analysis of Position Selection of the Typical Survey-Point

#### 3.2.1. Relative Unit Discharge Distribution of Common River Cross-Sections

#### 3.2.2. Influence of the Survey-Point Position on the Cross-Section Discharge Estimation Accuracy

## 4. Conclusions

- Contrast analysis of commonly used estimation methods of depth-averaged velocity under ice cover. Based on the selected sixty sets of measured data, the depth-averaged velocity-estimation errors obtained by applying the one-point method at 0.5H, two-point method at 0.2H and 0.8H, three-point method proposed by Shan et al., and six-point method, were calculated as 2.60%, 1.98%, 1.22%, and 0.45%, respectively, and the corresponding standard deviations were 1.86, 0.44, 0.62, and 0.35, respectively. The one-point method at 0.5H is appropriate for estimating the depth-averaged velocity of a single line, depending on the workload. If the workload increases, the two-point method at 0.2H and 0.8H may be chosen. If the measurement conditions can meet the arrangement of three measuring points, depending on the measurement efficiency and accuracy, the new three-point method proposed by Shan et al. is recommended. The reasonable and accurate selection of the estimation methods of depth-averaged velocity under ice cover further reduces the workload in the application of the stream-tube method.
- By analyzing the parameter sensitivity of the flow-discharge measurement accuracy to the cross-section characteristic coefficient α and the typical survey-point position, the latter was found to have less influence on the flow-discharge measurement accuracy of the stream-tube method compared to the typical survey-point position. The cross-section characteristic coefficient was taken to be 0.5 and 0.25 for natural rivers and artificial channels, respectively. By analyzing the relationship between the relative unit discharge distributions of common river cross-sections and the cross-sectional flow-depth distributions, the survey point should be set at the thalweg of the section in the mainstream area. Using the proposed method at this suggested survey-point position, the percentage of measurement points with an estimated error of unit discharge less than 20% within the selected cross-section, including the laboratory flume model tests and natural-river data, reached 90.5% of all the measuring points.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Hydraulic parameters and flow characteristics for error estimation in ice-covered channels.

Data Source | Test | Discharge (m ^{3}/s) | Width Depth Ratio (B/H) | Resistance Parameter | ||
---|---|---|---|---|---|---|

m_{b} | m_{i} | r_{m} | ||||

Tatinclaux and Gogus [37] | Athabasca R., AL | 1850.00 | 85.00 | 5.73 | 2.44 | 2.35 |

Athabasca R., AL | 1230.00 | 106.00 | 5.47 | 2.44 | 2.24 | |

Athabasca R., AL | 850.00 | 121.00 | 5.04 | 1.85 | 2.72 | |

Engmann [38] | 101 | 7.10 × 10^{−3} | 1.22 | 3.10 | 6.46 | 0.48 |

102 | 15.60 × 10^{−3} | 1.22 | 4.12 | 8.08 | 0.51 | |

103 | 12.70 × 10^{−3} | 1.22 | 4.60 | 7.54 | 0.61 | |

104 | 11.40 × 10^{−3} | 1.22 | 4.60 | 7.54 | 0.61 | |

Parthasarathy and Muste [39] | R1 | 50.10 × 10^{−3} | 4.20 | 4.59 | 7.65 | 0.60 |

R2 | 50.10 × 10^{−3} | 3.70 | 4.90 | 5.83 | 0.84 | |

R3 | 50.10 × 10^{−3} | 3.10 | 4.70 | 4.56 | 1.03 | |

Smith and Ettema [40] | S2 | 78.70 × 10^{−3} | 4.90 | 7.02 | 8.46 | 0.83 |

M2 | 75.50 × 10^{−3} | 4.70 | 6.63 | 6.38 | 1.04 | |

R2 | 75.40 × 10^{−3} | 4.40 | 5.73 | 4.74 | 1.21 | |

S4 | 76.30 × 10^{−3} | 5.00 | 4.51 | 7.52 | 0.60 | |

M4 | 75.30 × 10^{−3} | 4.80 | 4.79 | 5.70 | 0.84 | |

R4 | 74.50 × 10^{−3} | 4.40 | 4.70 | 4.56 | 1.03 | |

Wei and Huang [41] | Case 1 | 50.10 × 10^{−3} | 2.10 | 9.68 | 8.27 | 1.17 |

Case 2 | 50.10 × 10^{−3} | 2.10 | 9.68 | 8.27 | 1.17 | |

Case 3 | 50.10 × 10^{−3} | 2.10 | 9.68 | 8.27 | 1.17 | |

Case 4 | 69.90 × 10^{−3} | 2.30 | 9.58 | 8.19 | 1.17 | |

Case 5 | 60.00 × 10^{−3} | 2.50 | 9.48 | 8.10 | 1.17 | |

Case 6 | 40.10 × 10^{−3} | 3.00 | 9.26 | 7.91 | 1.17 | |

Case 7 | 30.30 × 10^{−3} | 3.50 | 9.26 | 7.91 | 1.17 | |

Case 8 | 50.70 × 10^{−3} | 2.30 | 9.58 | 8.12 | 1.18 | |

Case 10 | 50.00 × 10^{−3} | 2.60 | 8.00 | 3.15 | 2.54 | |

Case 11 | 60.20 × 10^{−3} | 2.40 | 8.00 | 3.15 | 2.54 | |

Case 12 | 50.20 × 10^{−3} | 2.50 | 8.00 | 3.15 | 2.54 | |

Case 13 | 50.20 × 10^{−3} | 2.50 | 8.00 | 3.15 | 2.54 | |

Case 15 | 50.70 × 10^{−3} | 2.10 | 3.45 | 3.05 | 1.13 | |

Case 16 | 41.20 × 10^{−3} | 2.30 | 3.45 | 3.05 | 1.13 | |

Attar and Li [33] | Salmon R., NB | 12.00 | 3.70 | 3.36 | 4.93 | 0.68 |

S.W. Miramichi R., NB | 51.00 | 3.10 | 3.59 | 7.39 | 0.49 | |

R. John, NS | 2.00 | 4.90 | 8.52 | 8.17 | 1.04 | |

Kaministiquia R., ON | 43.00 | 4.70 | 4.10 | 6.01 | 0.68 | |

Saugeen R., ON | 29.00 | 4.40 | 2.89 | 5.55 | 0.52 | |

Nith R., ON | 1.50 | 5.00 | 4.54 | 6.76 | 0.67 | |

Burnt R., ON | 10.00 | 4.80 | 3.20 | 5.48 | 0.58 | |

Eels Cr.,ON | 1.94 | 4.40 | 3.78 | 5.08 | 0.74 | |

Moira R., ON | 2.22 | 25.97 | 2.95 | 7.70 | 0.38 | |

Salmon R., ON | 4.73 | 23.53 | 2.45 | 5.47 | 0.45 | |

Upper Humber R., NF | 64.00 | 69.44 | 2.79 | 7.58 | 0.37 | |

Terra Nova R., NF | 25.00 | 33.50 | 2.61 | 7.19 | 0.36 | |

Groundhog R., ON | 86.00 | 51.03 | 3.50 | 4.66 | 0.75 | |

Oldman R., AB | 2.33 | 136.00 | 2.94 | 7.14 | 0.41 | |

Red Deer R., AB | 18.00 | 97.96 | 2.79 | 7.64 | 0.37 | |

N.SaskatchewanR.,SK | 116.00 | 204.00 | 3.75 | 10.51 | 0.36 | |

Ou’Appelle R., SA | 1.14 | 37.50 | 5.70 | 6.25 | 0.91 | |

Beaver R., AB | 2.69 | 41.82 | 2.36 | 7.14 | 0.33 | |

Pembina R., AB | 12.00 | 105.71 | 3.23 | 6.25 | 0.52 | |

Halfway R., BC | 7.40 | 72.22 | 2.77 | 5.96 | 0.46 | |

Litle Smoky R., AB | 11.50 | 97.50 | 3.22 | 9.02 | 0.36 | |

Peace R., NWT | 1111.00 | 116.70 | 5.44 | 9.22 | 0.59 | |

Yellowknife R., NWT | 24.00 | 24.00 | 3.55 | 5.92 | 0.60 | |

Fraser R., BC | 32.00 | 73.08 | 3.25 | 6.37 | 0.51 | |

Takhini R. YT | 14.00 | 32.86 | 3.12 | 5.96 | 0.52 | |

Yukon R., YT | 246.00 | 58.00 | 3.69 | 7.06 | 0.52 | |

Lu [34] | 2.1–2.3 floodplain | 0.05–0.09 | 4.50–9.00 | 7.28 | 2.66 | 2.74 |

3.1–3.3 floodplain | 0.05–0.09 | 4.50–9.00 | 9.08 | 2.85 | 3.19 | |

2.1–2.3 main channel | 0.05–0.09 | 3.00–5.00 | 7.62 | 4.52 | 1.69 | |

3.1–3.3 main channel | 0.05–0.09 | 3.00–5.00 | 8.25 | 4.14 | 1.99 |

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**Figure 1.**Flow profile is divided into bed- and ice-dominated layers: (

**a**) entire cross-section; (

**b**) partial cross-section.

**Figure 6.**Comparison of velocity-estimation errors obtained from various depth-averaged velocity characterization methods (the one-point method at 0.5H, two-point method at 0.2H and 0.8H, three-point method 1, six-point method, and the new method proposed by Shan et al. are indicated along the abscissa).

**Figure 7.**Parameter sensitivity analysis: (

**a**) the effect of the cross-section characteristic coefficient α on the estimation error of unit discharge at each measurement point of the cross-section with a constant survey-point position; (

**b**) the effect of the survey-point position on the estimation error of unit discharge at each measurement point of the cross-section with a cross-section characteristic coefficient α of 0.5.

**Figure 8.**Section topography showing ice cover and riverbed distribution, and cross-sectional relative-unit discharge in a rectangular section.

**Figure 9.**Section topography showing ice cover and riverbed distribution, and cross-sectional relative-unit discharge in a trapezoidal compound section.

**Figure 10.**Experimental flume section with main channel and floodplain: (

**a**) section topography showing ice cover, riverbed distribution, and cross-sectional relative-unit discharge; (

**b**) photo of experimental flume.

**Figure 11.**Section topography showing ice cover and riverbed distribution and cross-sectional relative-unit discharge in a channel section with two thalwegs.

**Figure 12.**Section topography showing ice cover and riverbed distribution, and cross-sectional relative-unit discharge: (

**a**) Baotou section on 26 January 2014; (

**b**) Dabusutai section on 21 February 2014; (

**c**) Baotou section on 14 December 2014; (

**d**) Baotou section on 23 January 2015.

**Figure 14.**Section topography showing ice-cover and riverbed distribution, and the comparison of estimated and measured unit discharge: (

**a**) Baotou section on 10 March 2015; (

**b**) Baotou section on 18 February 2015; (

**c**) Sanhuhekou section on 21 December 2013.

Method | Selection Point Position | Coefficient | |

One-point method | one-point method 1 | 0.5H | 0.88 |

one-point method 2 | 0.6H | 0.92 | |

Two-point method | two-point method 1 | 0.2H, 0.8H | |

two-point method 2 | 0.4H, 0.8H | 0.32, 0.68 | |

Three-point method | three-point method 1 | 0.15H, 0.5H, 0.85H | |

three-point method 2 | 0.2H, 0.6H, 0.8H | ||

method proposed by Shan et al. | 0.2H, 0.5H, 0.8H | 0.67, −0.34, 0.67 | |

Six-point method | six-point method | 0.03H, 0.2H, 0.4H, 0.6H, 0.8H, 0.95H [32] |

Estimation Method | Estimation Error |
---|---|

One-point method 1 | $\frac{0.88*{0.5}^{1/{m}_{b}}{0.5}^{1/{m}_{i}}}{\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

One-point method 2 | $\frac{0.92*{0.4}^{1/{m}_{b}}{0.6}^{1/{m}_{i}}}{\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

Two-point method 1 | $\frac{{0.8}^{1/{m}_{b}}{0.2}^{1/{m}_{i}}+{0.2}^{1/{m}_{b}}{0.8}^{1/{m}_{i}}}{2*\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

Two-point method 2 | $\frac{0.68*{0.8}^{1/{m}_{b}}{0.2}^{1/{m}_{i}}+0.32*{0.4}^{1/{m}_{i}}{0.6}^{1/{m}_{b}}}{\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

Three-point method 1 | $\frac{{0.85}^{1/{m}_{b}}{0.15}^{1/{m}_{i}}+{0.15}^{1/{m}_{b}}{0.85}^{1/{m}_{i}}+{0.5}^{1/{m}_{b}}{0.5}^{1/{m}_{i}}}{3*\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

Three-point method 2 | $\frac{{0.8}^{1/{m}_{b}}{0.2}^{1/{m}_{i}}+{0.2}^{1/{m}_{b}}{0.8}^{1/{m}_{i}}+{0.4}^{1/{m}_{b}}{0.6}^{1/{m}_{i}}}{3*\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

Method proposed by Shan et al. | $\frac{-0.34*{0.5}^{1/{m}_{b}}{0.5}^{1/{m}_{i}}+0.67*{0.2}^{1/{m}_{b}}{0.8}^{1/{m}_{i}}+0.67*{0.8}^{1/{m}_{b}}{0.2}^{1/{m}_{i}}}{\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

Six-point method | $\frac{{0.97}^{1/{m}_{b}}{0.03}^{1/{m}_{i}}+2*{0.8}^{1/{m}_{b}}{0.2}^{1/{m}_{i}}+2*{0.6}^{1/{m}_{b}}{0.4}^{1/{m}_{i}}}{10*\beta (1+1/{m}_{b},1+1/{m}_{i})}$ |

$+\frac{2*{0.4}^{1/{m}_{b}}{0.6}^{1/{m}_{i}}+2*{0.2}^{1/{m}_{b}}{0.8}^{1/{m}_{i}}+{0.05}^{1/{m}_{b}}{0.95}^{1/{m}_{i}}}{10*\beta (1+1/{m}_{b},1+1/{m}_{i})}-1$ |

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## Share and Cite

**MDPI and ACS Style**

Lu, J.; Guo, X.; Pan, J.; Fu, H.; Wu, Y.; Mao, Z.
Contrast Analysis of Flow-Discharge Measurement Methods in a Wide–Shallow River during Ice Periods. *Water* **2022**, *14*, 3996.
https://doi.org/10.3390/w14243996

**AMA Style**

Lu J, Guo X, Pan J, Fu H, Wu Y, Mao Z.
Contrast Analysis of Flow-Discharge Measurement Methods in a Wide–Shallow River during Ice Periods. *Water*. 2022; 14(24):3996.
https://doi.org/10.3390/w14243996

**Chicago/Turabian Style**

Lu, Jinzhi, Xinlei Guo, Jiajia Pan, Hui Fu, Yihong Wu, and Zeyu Mao.
2022. "Contrast Analysis of Flow-Discharge Measurement Methods in a Wide–Shallow River during Ice Periods" *Water* 14, no. 24: 3996.
https://doi.org/10.3390/w14243996