# Modified Numerical Method for Improving the Calculation of Rill Detachment Rate

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Data Source

#### 2.1. Rill Erosion Process

^{−3}), x is the rill length (m), A denotes the maximum sediment concentration that reaches sediment transport capacity (kg·m

^{−3}), and B represents the decrease rate of sediment load.

#### 2.2. Numerical Method for Calculating Rill Detachment Rate

^{3}·s

^{−1}), w represents the rill width (m), q represents the unit-width discharge rate (m

^{2}·s

^{−1}), and D

_{r}

_{N}is the RDR calculated using the numerical method (kg·m

^{−2}·s

^{−1}).

#### 2.3. Analytical Method for Calculating Rill Detachment Rate

_{r}represents the RDR calculated using the analytical method (kg·m

^{−2}·s

^{−1}). Substituting Equation (1) into Equation (3) yields the analytical estimation of the RDR:

^{−2}·s

^{−1}), and b represents the decrease rate of RDR (m

^{−1}).

^{−2}·s

^{−1}), and g expresses the decrease rate of the RDR with the increase in sediment concentration (m·s

^{−1}). Lei et al. [24] and Chen et al. [28] introduced detailed derivation processes of the analytical method.

#### 2.4. Principle of Modified Numerical Method

_{1}, c

_{2}, …, c

_{n}) at different rill distances and rill lengths (0, x

_{1}, x

_{2}, …, x

_{n}), the numerical values of RDRs (0, D

_{r}

_{N1}, D

_{r}

_{N2}, …, D

_{r}

_{Nn}) can be calculated as follows:

_{i}is the rill length at location i (m), c

_{i}is the sediment concentration at location i (kg·m

^{−3}), and D

_{r}

_{Ni}denotes the RDR calculated by the numerical method at location i (kg·m

^{−2}·s

^{−1}). The sediment concentration rises from c

_{(i−1)}to c

_{i}with the rill segment increasing from x

_{i−1}to x

_{i}. Thus, the ratio between the sediment delivery rate (q(c

_{i}− c

_{(i−1)})) and the distance of runoff flow (x

_{i}− x

_{i−1}) corresponds to the average numerical RDR of this segment, which is shown as the gradient of L

_{1}(Figure 1). The gradient of L

_{1}can be approximately considered the numerical RDR at x

_{i}when the increment from x

_{i−1}to x

_{i}is relatively low. The gradient of L

_{2}is the tangent line of the sediment delivery curve (c(x)) at x

_{i}, which represents the analytical RDR at x

_{i}. The gradient of L

_{1}is theoretically higher than that of L

_{2}. Thus, the numerical RDR is always higher than the analytical value.

_{i}− x

_{i−1}):

_{i−1}+ (x

_{i}− x

_{i−1})/2) of the interval (x

_{i}− x

_{i−1}) is used to assign the assuming point (x = ξ) (Figure 1). The gradient of L

_{3}represents the modified numerical value of RDR at the midpoint, which is equal to the gradient of L

_{1}based on Equation (8). Meanwhile, the modified numerical RDR is also close to the analytical value (which is shown as L

_{4}) at the midpoint. This modification theoretically results in consistent numerical and analytical values.

_{r}

_{N}represents the numerical RDR (kg·m

^{−2}·s

^{−1}), D

_{r}

_{1}and D

_{r}

_{2}are the analytical RDRs calculated by rill length and sediment concentration (kg·m

^{−2}·s

^{−1}) respectively, D

_{r}

_{M}is the modified numerical RDR (kg·m

^{−2}·s

^{−1}), and h, k, and m are the regression coefficients which reflect the relative errors between each pair of methods.

_{(0–4)}and AE

_{(4–8)}denote the average absolute error between the numerical and the modified numerical RDRs in the range of rill length of 0–4 m and 4–8 m, kg·m

^{−2}·s

^{−1}, respectively.

#### 2.5. Basic Dataset

^{−1}) were set for the experiments. The water flow was introduced into the steady flow flume, and then introduced into the rill 1, 2, 4, and 8 m away from the rill outlet. In each test, four 300 mL steel cups were used to collect runoff samples, after the water flow at the outlet of the rill was stable. Each sampling process took 30~60 s. Subsequently, the samples were left for 24 h, and the supernatant was filtered out. The sediment concentrations of the runoff samples were measured using the oven-drying method at 105 °C. The rill erosion process on saturated loess soil slopes was studied in Huang et al. [33] (Table 1). Subsequently, the analytical RDRs on saturated loess soil slopes could be computed using the rill length (Equation (4)) and sediment concentration (Equation (6)). Meanwhile, the RDRs calculated using the numerical method on saturated loess soil slopes were investigated in Huang et al. [6] (Table 2).

#### 2.6. Statistical Analysis

## 3. Results

#### 3.1. Comparison of Numerical, Analytical, and Modified Numerical Methods

^{−1}and slope gradients of 5°, 10°, 15°, and 20° were fitted by Equation (4) to verify the accuracy of the principle of the modified numerical method and study the rill detachment processes determined by three groups of datasets (Figure 2). All determining coefficients were higher than 0.80, and the values of p were lower than 0.05, which indicated the good exponential function relationships among the numerical, analytical, and modified numerical RDRs and rill length. All three methods described the rill detachment processes well. Figure 2 shows that the curves determined using the modified numerical method were closer to those determined using the analytical method than those determined using the numerical method. Both analytical and modified numerical RDRs were significantly lower than the numerical RDRs (p < 0.01). This disparity increased with the slope gradient.

#### 3.2. Improvement Effect of the Modified Numerical Method

## 4. Discussion

#### 4.1. Necessity and Application of Modified Numerical Method

#### 4.2. Measurement Strategy for Improving Accuracy of Rill Detachment Rate Calculation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Comparing three rill detachment processes determined using numerical, analytical, and modified numerical detachment rates (RDRs) under a flow rate of 4 L·min

^{−1}of different slope gradients: (

**a**) 5°, (

**b**) 10°, (

**c**) 15°, and (

**d**) 20°.

**Figure 3.**Comparison between numerical and analytical rill detachment rates over saturated loess soil slopes of different slope gradients: (

**a**) 5°, (

**b**) 10°, (

**c**) 15°, and (

**d**) 20°.

**Figure 4.**Comparison between modified numerical and analytical rill detachment rates over saturated loess soil slopes of different slope gradients: (

**a**) 5°, (

**b**) 10°, (

**c**) 15°, and (

**d**) 20°.

**Table 1.**Fitted rill erosion process with Equation (1) on saturated loess soil slopes [33].

Slope (°) | Flow Rate (L·min^{−1}) | Parameters | Coefficient of Determination (R^{2}) | |
---|---|---|---|---|

A (kg·m^{−3}) | B (m^{−1}) | |||

5 | 2 | 465.34 | 0.32 | 0.92 |

4 | 623.89 | 0.44 | 0.99 | |

8 | 696.29 | 0.37 | 0.97 | |

10 | 2 | 800.81 | 0.32 | 0.98 |

4 | 800.02 | 0.33 | 0.99 | |

8 | 767.42 | 0.48 | 0.99 | |

15 | 2 | 907.93 | 1.09 | 0.99 |

4 | 965.44 | 1.17 | 0.99 | |

8 | 929.38 | 0.99 | 0.99 | |

20 | 2 | 1027.49 | 1.19 | 0.99 |

4 | 1071.35 | 1.08 | 0.99 | |

8 | 1054.62 | 1.23 | 0.99 |

**Table 2.**Numerical RDRs along the eroding rills over saturated loess soil slopes [6].

Slope (°) | Flow Rate (L·min^{−1}) | Rill Detachment Rate (kg·m^{−2}·s^{−1}) | |||
---|---|---|---|---|---|

1 m Rill Length | 2 m Rill Length | 4 m Rill Length | 8 m Rill Length | ||

5 | 2 | 0.047 | 0.007 | 0.036 | 0.006 |

4 | 0.144 | 0.117 | 0.035 | 0.015 | |

8 | 0.363 | 0.148 | 0.066 | 0.049 | |

10 | 2 | 0.069 | 0.053 | 0.036 | 0.017 |

4 | 0.138 | 0.135 | 0.051 | 0.036 | |

8 | 0.413 | 0.198 | 0.121 | 0.045 | |

15 | 2 | 0.210 | 0.046 | 0.018 | 0.006 |

4 | 0.458 | 0.104 | 0.033 | 0.008 | |

8 | 0.783 | 0.279 | 0.082 | 0.006 | |

20 | 2 | 0.241 | 0.067 | 0.015 | 0.002 |

4 | 0.487 | 0.125 | 0.035 | 0.016 | |

8 | 0.994 | 0.294 | 0.033 | 0.026 |

**Table 3.**Proportional coefficients of analytical rill detachment rates (RDRs) compared with numerical RDRs.

Slope (°) | Flow Rate (L·min^{−1}) | Proportional Coefficients | Coefficients of Determination | p_{r} | p_{s} | ||
---|---|---|---|---|---|---|---|

h | k | R_{r}^{2} | R_{s}^{2} | ||||

5 | 2 | 1.468 | 1.523 | 0.92 | 0.84 | 0.043 | 0.081 |

4 | 1.304 | 1.293 | 0.98 | 0.97 | <0.001 | 0.002 | |

8 | 1.305 | 1.360 | 0.94 | 0.90 | 0.006 | 0.014 | |

10 | 2 | 1.176 | 1.144 | 0.98 | 0.96 | 0.001 | 0.003 |

4 | 1.222 | 1.193 | 0.97 | 0.94 | 0.002 | 0.006 | |

8 | 1.293 | 1.295 | 0.98 | 0.97 | 0.001 | 0.001 | |

15 | 2 | 1.831 | 1.843 | 0.98 | 0.94 | <0.001 | 0.033 |

4 | 1.917 | 1.948 | 0.99 | 0.97 | <0.001 | 0.017 | |

8 | 1.714 | 1.719 | 0.99 | 0.99 | <0.001 | 0.003 | |

20 | 2 | 1.930 | 1.952 | 0.99 | 0.99 | <0.001 | 0.002 |

4 | 1.816 | 1.834 | 0.99 | 0.97 | <0.001 | 0.016 | |

8 | 1.970 | 1.952 | 0.99 | 0.99 | <0.001 | 0.004 |

**Table 4.**Proportional coefficients of analytical rill detachment rates (RDRs) compared with modified numerical RDRs.

Slope (°) | Flow Rate (L·min^{−1}) | Proportional Coefficient | Coefficient of Determination | p |
---|---|---|---|---|

m | R^{2} | |||

5 | 2 | 0.930 | 0.68 | 0.054 |

4 | 1.028 | 0.97 | 0.001 | |

8 | 1.062 | 0.91 | 0.007 | |

10 | 2 | 0.988 | 0.99 | <0.001 |

4 | 1.016 | 0.96 | 0.002 | |

8 | 1.007 | 0.98 | <0.001 | |

15 | 2 | 1.062 | 0.98 | <0.001 |

4 | 1.068 | 0.99 | <0.001 | |

8 | 1.044 | 0.99 | <0.001 | |

20 | 2 | 1.064 | 0.99 | <0.001 |

4 | 1.057 | 0.99 | <0.001 | |

8 | 1.064 | 0.99 | <0.001 |

**Table 5.**Proportional coefficients of numerical/modified numerical rill detachment rates compared with analytical RDRs.

Study | Soil Type | Rill Width (m) | Slope | Flow Rate (L·min ^{−1}) | Proportional Coefficient | Coefficient of Determination | p_{h} | p_{m} | ||
---|---|---|---|---|---|---|---|---|---|---|

h | m | R_{h}^{2} | R_{m}^{2} | |||||||

Chen et al. [28] | Purple soil | 0.1 | 10° | 4 | 1.06 | 1.00 | 0.93 | 0.93 | <0.001 | <0.001 |

8 | 0.93 | 0.86 | 0.95 | 0.96 | <0.001 | <0.001 | ||||

15° | 2 | 1.22 | 1.08 | 0.96 | 0.96 | <0.001 | <0.001 | |||

4 | 1.19 | 1.05 | 0.97 | 0.98 | <0.001 | <0.001 | ||||

8 | 1.16 | 0.98 | 0.97 | 0.99 | <0.001 | <0.001 | ||||

20° | 2 | 1.19 | 1.01 | 0.96 | 0.97 | <0.001 | <0.001 | |||

4 | 1.11 | 0.94 | 0.99 | 0.99 | <0.001 | <0.001 | ||||

8 | 1.10 | 0.95 | 0.96 | 0.97 | <0.001 | <0.001 | ||||

25° | 2 | 1.16 | 1.01 | 0.97 | 0.97 | <0.001 | <0.001 | |||

4 | 1.24 | 1.07 | 0.97 | 0.98 | <0.001 | <0.001 | ||||

8 | 1.09 | 0.93 | 0.95 | 0.97 | <0.001 | <0.001 | ||||

Lei et al. [40] | Loess soil | 0.1 | 10° | 4 | 1.00 | 1.00 | 0.95 | 0.95 | <0.001 | <0.001 |

8 | 1.03 | 0.99 | 0.91 | 0.91 | <0.001 | <0.001 | ||||

12 | 1.00 | 0.99 | 0.99 | 0.99 | <0.001 | <0.001 | ||||

15° | 4 | 1.16 | 1.01 | 0.98 | 0.98 | <0.001 | <0.001 | |||

8 | 0.99 | 0.99 | 0.96 | 0.96 | <0.001 | <0.001 | ||||

12 | 0.99 | 0.98 | 0.97 | 0.97 | <0.001 | <0.001 | ||||

20° | 4 | 1.04 | 0.98 | 0.94 | 0.94 | <0.001 | <0.001 | |||

8 | 1.03 | 0.98 | 0.89 | 0.9 | <0.001 | <0.001 | ||||

12 | 1.04 | 0.99 | 0.97 | 0.97 | <0.001 | <0.001 | ||||

25° | 4 | 1.06 | 0.97 | 0.91 | 0.91 | <0.001 | <0.001 | |||

8 | 1.06 | 0.99 | 0.95 | 0.95 | <0.001 | <0.001 | ||||

12 | 1.06 | 0.99 | 0.96 | 0.96 | <0.001 | <0.001 | ||||

Zhang et al. [27] | Loess soil | 0.1 | 17.62% | 8 | 1.15 | 1.05 | 0.93 | 0.92 | <0.001 | <0.001 |

12 | 1.12 | 1.09 | 0.99 | 0.98 | <0.001 | <0.001 | ||||

26.78% | 2 | 1.03 | 0.98 | 0.97 | 0.96 | <0.001 | <0.001 | |||

4 | 1.04 | 0.99 | 0.99 | 0.95 | <0.001 | <0.001 | ||||

8 | 1.10 | 1.01 | 0.98 | 0.99 | <0.001 | <0.001 | ||||

36.38% | 2 | 1.08 | 0.97 | 0.96 | 0.96 | <0.001 | <0.001 | |||

4 | 1.06 | 0.94 | 0.96 | 0.97 | <0.001 | <0.001 | ||||

8 | 1.12 | 0.99 | 0.99 | 0.99 | <0.001 | <0.001 | ||||

46.60% | 2 | 1.19 | 1.02 | 0.98 | 0.98 | <0.001 | <0.001 | |||

4 | 1.20 | 1.03 | 0.99 | 0.99 | <0.001 | <0.001 | ||||

8 | 1.02 | 0.92 | 0.95 | 0.96 | <0.001 | <0.001 | ||||

Huang et al. [45] | Russell silt loam | 0.2 | 6.30% | 15.1 | 2.10 | 1.04 | 0.87 | 0.84 | 0.069 | 0.081 |

22.7 | 2.40 | 1.05 | 0.99 | 0.88 | 0.066 | 0.061 | ||||

30.2 | 1.17 | 0.95 | 0.9 | 0.8 | 0.039 | 0.042 | ||||

37.8 | 1.05 | 0.92 | 0.92 | 0.83 | 0.029 | 0.031 | ||||

Saybrook silt loam | 0.2 | 2.40% | 7.6 | 1.55 | 1.20 | 0.94 | 0.87 | 0.065 | 0.066 | |

15.1 | 1.46 | 1.05 | 0.97 | 0.95 | 0.026 | 0.027 | ||||

22.7 | 1.01 | 0.94 | 0.99 | 0.99 | <0.001 | <0.001 | ||||

Polyakov and Nearing [46] | Carmi loam | 0.61 | 7% | 6 | 1.40 | 1.09 | 0.79 | 0.77 | 0.045 | 0.052 |

9 | 1.29 | 0.89 | 0.91 | 0.90 | 0.045 | 0.053 |

_{h}

^{2}and R

_{m}

^{2}denote the coefficients of determination.

**Table 6.**Average absolute errors between the numerical and the modified numerical rill detachment rates.

Study | Soil Type | Rill Width (m) | Slope | Flow Rate (L·min^{−1}) | Average Absolute Error (kg·m^{−2}·s^{−1}) | |
---|---|---|---|---|---|---|

0–4 m | 4–8 m | |||||

Huang et al. [6,33] | Loess soil | 0.1 | 5 | 2 | 0.061 | 0.331 |

4 | 0.201 | 0.043 | ||||

8 | 0.411 | 0.073 | ||||

10 | 2 | 0.105 | 0.044 | |||

4 | 0.197 | 0.080 | ||||

8 | 0.499 | 0.063 | ||||

15 | 2 | 0.211 | 0.002 | |||

4 | 0.464 | 0.005 | ||||

8 | 0.835 | 0.018 | ||||

20 | 2 | 0.247 | 0.004 | |||

4 | 0.498 | 0.008 | ||||

8 | 1.053 | 0.008 | ||||

Chen et al. [28] | Purple soil | 0.1 | 10° | 4 | 0.055 | 0.017 |

8 | 0.125 | 0.061 | ||||

15° | 2 | 0.063 | 0.021 | |||

4 | 0.147 | 0.044 | ||||

8 | 0.382 | 0.066 | ||||

20° | 2 | 0.072 | 0.011 | |||

4 | 0.168 | 0.023 | ||||

8 | 0.368 | 0.074 | ||||

25° | 2 | 0.066 | 0.014 | |||

4 | 0.196 | 0.038 | ||||

8 | 0.349 | 0.048 | ||||

Lei et al. [40] | loess soil | 0.1 | 10° | 4 | - | - |

8 | 0.241 | 0.070 | ||||

12 | 0.353 | 0.320 | ||||

15° | 4 | 0.228 | 0.073 | |||

8 | - | - | ||||

12 | 0.416 | 0.507 | ||||

20° | 4 | 0.196 | 0.122 | |||

8 | 0.343 | 0.099 | ||||

12 | 0.555 | 0.067 | ||||

25° | 4 | 0.198 | 0.120 | |||

8 | 0.375 | 0.113 | ||||

12 | 0.583 | 0.095 | ||||

Zhang et al. [27] | Loess soil | 0.1 | 17.62% | 8 | 0.165 | 0.064 |

12 | 0.229 | 0.172 | ||||

26.78% | 2 | 0.086 | 0.048 | |||

4 | 0.165 | 0.095 | ||||

8 | 0.396 | 0.162 | ||||

36.38% | 2 | 0.115 | 0.037 | |||

4 | - | - | ||||

8 | 0.491 | 0.116 | ||||

46.60% | 2 | 0.142 | 0.021 | |||

4 | 0.276 | 0.041 | ||||

8 | 0.477 | 0.112 | ||||

Huang et al. [45] | Russell silt loam | 0.2 | 6.30% | 15.1 | 0.030 | 0.011 |

22.7 | 0.045 | 0.027 | ||||

30.2 | 0.058 | 0.030 | ||||

37.8 | 0.067 | 0.043 | ||||

Saybrook silt loam | 0.2 | 2.40% | 7.6 | 0.016 | 0.014 | |

15.1 | 0.049 | 0.029 | ||||

22.7 | 0.051 | 0.039 | ||||

Polyakov and Nearing [46] | Carmi loam | 0.61 | 7% | 6 | - | - |

9 | 0.005 | 0.002 |

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## Share and Cite

**MDPI and ACS Style**

Huang, Y.; Zhao, M.; Wan, D.; Lei, T.; Li, F.; Wang, W.
Modified Numerical Method for Improving the Calculation of Rill Detachment Rate. *Water* **2023**, *15*, 1875.
https://doi.org/10.3390/w15101875

**AMA Style**

Huang Y, Zhao M, Wan D, Lei T, Li F, Wang W.
Modified Numerical Method for Improving the Calculation of Rill Detachment Rate. *Water*. 2023; 15(10):1875.
https://doi.org/10.3390/w15101875

**Chicago/Turabian Style**

Huang, Yuhan, Mingquan Zhao, Dan Wan, Tingwu Lei, Fahu Li, and Wei Wang.
2023. "Modified Numerical Method for Improving the Calculation of Rill Detachment Rate" *Water* 15, no. 10: 1875.
https://doi.org/10.3390/w15101875