# A Method for Determining the Discharge of Closed-End Furrow Irrigation Based on the Representative Value of Manning’s Roughness and Field Mean Infiltration Parameters Estimated Using the PTF at Regional Scale

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experiments with Closed-End Furrow Irrigation

^{2}and an elevation range of approximately 323–800 m. The annual average temperature is 12.8 °C, and the average annual precipitation is 605 mm. Combined with the actual irrigation experience of local farmers, closed-end furrow irrigation experiments were performed in the Guanzhong plain from July to August 2017, using planted crop maize. The study used 45 experimental sites with minimum and maximum longitudes of 106°55′57″ and 110°29′32″ N, and minimum and maximum latitudes of 33°49′26″ and 35°30′02″ E. The layout of the experimental sites is illustrated in Figure 1. At each experimental site, three to five furrows were laid out. The trapezoidal section most commonly used in northern China was adopted for these irrigation experiments. The downstream end of each furrow was closed, and the cutoff time was recorded when water reached the downstream end. The required water depth was 80 mm. The discharge was measured by a triangle weir. The measurement points were set at 10 m intervals along the furrow length (the last observation point was taken as the actual remaining length if the residual distance was within 10 m). The water advance time was recorded, and the water depths were measured every 1–5 min (every 1 min in the first 6 min, every 2 min from the sixth to the 10th min, and then every 5 min). One day before and one day after irrigation, five to eight measuring points were set along the furrow length. Each point was divided into five layers (i.e., 10, 20, 40, 60, and 80 cm) for soil water content collection, and two or three soil samples were selected at each measuring point to determine the bulk density of each layer. The soil was collected at four depth levels, namely 10, 20, 40, and 60 cm (it would have been difficult to collect soil samples at 80 cm; thus, the bulk density of the layer at 80 cm was assumed to be equal to that of the layer at 60 cm). The soil texture was classified according to the international reclassification standard (clay was ≤0.002 mm, silt was 0.002–0.02 mm, and sand was 0.02–2 mm). The detailed measurements of closed-end furrow irrigation at each experimental site are listed in Table 1 (these detailed measurements are the mean values for each experimental site). Because the opportunity time is relatively short for completing furrow irrigation, soil infiltration capacity mainly depends on the upper-layer soil properties. Therefore, the bulk density, initial soil water content, and soil particle proportions for each experimental site were recorded as mean values for soil layers from 0 to 40 cm (Table 1). These provided the basic values for PTF that were used to estimate the normalization factor.

#### 2.2. SIPAR_ID and WinSRFR Descriptions

^{−a}), α is the infiltration index, and τ is the opportunity time (min). The basic data concerning field length, bottom slope, cross-sectional parameters, inflow discharge, advance trajectory, and water depth in the upstream (an optional item) obtained through the field experiments were the inputs for SIPAR_ID, and the differences between simulated and measured values of advance trajectory were minimal. Thus, this was used as the objective function for estimating the infiltration parameters and Manning’s roughness. Note that, Furrow irrigation involves a two-dimensional infiltration (in lateral and vertical directions; i.e., simultaneous infiltration of both the width and depth of the furrow); in this study, the infiltration coefficient was obtained by SIPAR_ID as the inversion of the cumulative infiltration per unit length, which was divided by the furrow spacing (to match the infiltration characteristic item in a simulation of furrow irrigation by WinSRFR); that is, the infiltration coefficient k in Equation (1), under the cumulative infiltration per unit area [3,46]. Unlike conventional optimization, SIPAR_ID attempts to avoid most typical violations of the mass conservation principle. For example, volume balance methods use a uniform flow equation such as Manning’s equation to describe the cross-sectional area of flow at the field inlet, and then posit assumptions regarding the shape of the flow profile downstream. Typical volume balance methods generally assume that the cross-sectional area is constant, which is known to introduce substantial errors [35]. As the principle of volume balance is the basis of SIPAR_ID, the data used for estimating infiltration parameters and Manning’s roughness of the furrow irrigation comprise the advance trajectory and water depth, with no need for the data collection of recession trajectory, which is a hard work, and the practicability of the SIPAR_ID is then improved. However, it is difficult to fully account for the interaction between the infiltration parameters and Manning’s roughness during the inversion procedure [47]. Therefore, to verify the reliability of infiltration parameters and Manning’s roughness inversed by SIPAR_ID, the inversed results were input into WinSRFR for furrow irrigation simulation, and then the measured data were compared with the simulated advance trajectory and water depth.

^{2}), q is the discharge (m

^{3}min

^{−1}), x is the distance from the field inlet (m), h is the water depth (m), t is time (min), S

_{0}is the bottom slope of the furrow, S

_{f}is the friction slope, and I is the unit length of infiltration volume (m

^{3}m

^{−1}). Here, I is calculated from furrow spacing, which is the transverse width when infiltration is independent of the wetted perimeter, in the simulation of furrow irrigation by WinSRFR. Thus, Equation (1) is multiplied by the furrow spacing.

#### 2.3. Influence of Manning’s Roughness on Advance Trajectory and Performance Indicators of Furrow Irrigation

_{a}), the storage efficiency (E

_{s}) [12], and Christiansen’s uniformity (CU); CU indicates the uniformity distributed along the furrow length. The mathematical expressions for these indices are:

_{s}is the average depth of the infiltrated water stored in the root zone, Z

_{f}is the depth of total water applied (mm), Z

_{r}is the required water depth (all the Z

_{r}were set at 80 mm in this study), Z

_{mi}is the infiltrated depth of the m-th measurement point for the i-th furrow, ${\overline{Z}}_{i}$ is the average depth of infiltration, and N is the number of measurement points. Note that Z

_{s}, Z

_{mi}, and ${\overline{Z}}_{i}$ can be calculated from the soil water content and bulk density measured at multiple points along the length of the furrow one day before and after the irrigation. At least five measurement points (i.e., N) were set for soil water content along the length of each furrow.

#### 2.4. Functional Normalization of the Kostiakov Equation and PTF Development

_{C}) is calculated from the following equation:

_{Ci}is the calculated normalization factor for the i-th furrow, Z

_{ij}is the cumulative infiltration from Equation (1) for the i-th furrow, ${\overline{Z}}_{j}$ is the average cumulative infiltration of j-th opportunity time for all furrows, and M is the sequence number of opportunity time. Then, $\overline{Z}$ can be written as:

_{C}calculations. Therefore, the maximum advance time was considered for the furrow irrigation (i.e., F23 with 66 min (Table 1) as a basis) and all opportunity times were adjusted to 70 min in this study. To develop PTF in MATLAB, multiple linear regression analyses were applied to correlate F

_{C}with available soil physical parameters (i.e., soil particle proportions, bulk density, and initial soil water content), which were based on the results of a correlation analysis run in SPSS.

#### 2.5. Optimization of Inflow Discharge Based on Manning’s Roughness Representative Values and PTF to Estimate the Mean Infiltration Parameters

^{−1}.

#### 2.6. Criteria for Evaluation

_{sf}is the f-th simulated value; X

_{mf}is the f-th measured or simulated value (the simulation results of Sim.4 served as a benchmark). Note that, in the PTF estimations, X

_{sf}represents the estimated PTF results and X

_{mf}represents the results of Equation (7). MAPRE enables a quantitative comparison of the simulated values with the measured values for the advance trajectory of furrow irrigation, indicators of irrigation performance, and normalization factor F

_{C}. MAPRE is a measure of the accuracy of the models with the simulated and measured values, and low values of MAPRE indicate a favorable simulation accuracy [53].

## 3. Results and Discussion

#### 3.1. Reliability Analysis of Field Mean Infiltration Parameters and Manning’s Roughness

#### 3.2. Evaluation of the Influence of Manning’s Roughness on Advance Trajectory and Irrigation Performance, and the Determination of its Representative Value in a Maize Field

_{a}, CU, and E

_{s}between the measured and simulated values for all furrow irrigation experiments under Sim.1 were 8.3%, 3.9%, and 4.8%, respectively. Only small differences were evident between the simulated and measured values of irrigation performance under Sim.2. Compared with Sim.1, the MAPRE values of E

_{a}, CU, and E

_{s}under Sim.2 were decreased by 3.4%, 1.7%, and 3.0%, respectively. For Sim.3, the MAPRE values, between the measured and simulated values, were 9.4%, 4.6%, and 2.5%. The typical Manning’s roughness value for the maize fields in this study was 0.075. In general, the advance trajectory and irrigation performance indicators were not sensitive to variations in Manning’s roughness, which can be used as a representative value (i.e., 0.075) in a maize field to simulate the advance trajectory and irrigation performance of a closed-end furrow.

#### 3.3. Establishment of the Normalization Function and PTF and Verification

_{C}for each experimental site were calculated with Equation (7); the results are listed in Table 2. Then, F

_{C}was put into Equation (12) to obtain the estimated infiltration depths, which can be compared with the values calculated from the infiltration parameters in Table 2. The results are presented in Figure 5.

^{2}) from the regression was 0.95, and the MAPRE was 7.3% for all furrow irrigation experiments. The relatively high R

^{2}and relatively low MAPRE indicated the high accuracy achieved using Equation (12) for estimating the infiltration depth, and the reasonable functional normalization of the Kostiakov equation.

_{C}and soil particle proportions, soil bulk density (γ

_{d}), and initial soil water content (θ

_{0}). The results indicated that the variability of F

_{C}was related mainly to sand content (Sa), clay content (Cl), soil bulk density, and initial soil water content; the correlation coefficients were 0.79, −0.73, −0.61, and −0.59 in the study area. However, F

_{C}was not significantly related to silt content, which had a correlation coefficient of −0.19. To develop PTF, multiple linear regression analyses were conducted in MATLAB on the basis of correlation analysis. In the study area, the PTF can be represented as:

_{d}is the soil bulk density (g cm

^{−}

^{3}), and θ

_{0}is the initial soil water content (%). The γ

_{d}, θ

_{0}, Cl, and Sa terms in Equation (13) were set to the mean values of the 0–40 cm soil layers measured at each experimental site. To ensure reliability, the F

_{E}estimates from Equation (13) were compared with the values of F

_{C}calculated using Equation (7). As illustrated in Figure 6, the estimated F

_{E}values were comparatively consistent with the calculated values, and the regression lines were notably close to the 1:1 line (Figure 6). The R

^{2}and the MAPRE values were 0.56 and 7.7% in the study area, respectively. The results demonstrated that using Equation (13) to estimate F

_{C}generated reliable results in the study area.

#### 3.4. Reliability Verification of the Proposed Method for Determining the Inflow Discharge at a Regional Scale

_{a}, CU, and E

_{s}) were simulated using WinSRFR under Sim.4 and Sim.5, and then the comprehensive indicator Y was estimated using Equation (10). In the WinSRFR simulations, the furrow bottom slopes were 0.001, 0.0025, and 0.004, the furrow lengths were 80, 105, and 130 m, and the discharges were 2.0, 3.0, 4.0, 5.0, and 6.0 L m

^{−1}; a total of 2025 simulations were run for all furrow irrigation sites under Sim.4 and Sim.5.

^{2}was 0.76 for all experimental sites. The MAPRE of optimal discharge between Sim.4 and Sim.5 for all furrow irrigation experiments was 9.2%; the simulation results of Sim.4 served as a benchmark. From the optimal results of discharge depicted in Figure 7, the irrigation performance indicators were estimated using WinSRFR, and then the comprehensive indicator, Y, was obtained according to Equation (10). The results are displayed in Table 3.

_{C}, and that it could be used for determining the optimized discharge of closed-end furrow irrigation of maize fields in the study areas. A properly optimized design can ensure high irrigation performance.

#### 3.5. General Discussion

_{C}, in this study area was 0.156; however, in 72 double-ring infiltration experiments in the typical fields of the first and third terraces (Yangling, Shaanxi province, China), the values of CV of F

_{C}were 0.334 and 0.271 [40]. The results demonstrated that the variability may be overestimated by point infiltration measurement, which obtained results similar to those of Bautista and Wallender [28]. Therefore, it can be inferred that using the field mean infiltration parameters may be helpful for obtaining a more robust design of a furrow irrigation system and managing such an irrigation system.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Map of study area and experimental layout of closed-end furrow irrigation. Notes: ■ represents the study area. ● represents an experimental site.

**Figure 2.**Comparisons of measured and simulated advance time and water depth in the upstream using WinSRFR.

**Figure 3.**Cumulative frequency of Manning’s roughness for all closed-end furrow irrigation experiments.

**Figure 4.**Comparisons of measured and simulated advance time and irrigation performance indicators in WinSRFR under the three scenarios. Notes: Sim.1: Manning’s roughness cumulative frequency = 5%; Sim.2: Manning’s roughness cumulative frequency = 50%; Sim.3: Manning’s roughness cumulative frequency = 95%.

**Figure 5.**Comparisons of the estimated infiltration depths determined using Equation (12), and calculated values based on the infiltration parameters in Table 2.

**Figure 6.**Calculated normalization factor from Equation (7) and estimated values from Equation (13).

**Figure 7.**Optimal discharges were compared for Sim.4 and Sim.5. Notes: Sim.4: Inverted field mean infiltration parameters and Manning’s roughness of each furrow; Sim.5: Field mean infiltration parameters by PTF of each furrow and Manning’s roughness cumulative frequency as 50%.

No. | q (L s^{−1}) | t (min) | L (m) | BW (m) | FD (m) | FS (m) | S_{0} (‰) | γ_{d} (g cm^{−3}) | θ_{0} (%) | Soil Particle Proportions (%) | Soil Texture | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cl | Si | Sa | |||||||||||

F1 | 2.95 | 42.0 | 90 | 0.28 | 0.16 | 0.68 | 2.5 | 1.33 | 10.7 | 28.6 | 29.5 | 41.9 | Loamy clay |

F2 | 2.42 | 34.0 | 82 | 0.15 | 0.15 | 0.60 | 1.9 | 1.40 | 11.0 | 29.7 | 29.3 | 41.0 | Loamy clay |

F3 | 2.95 | 31.0 | 88 | 0.25 | 0.14 | 0.70 | 1.7 | 1.38 | 12.8 | 28.0 | 32.7 | 39.3 | Loamy clay |

F4 | 2.69 | 59.0 | 115 | 0.27 | 0.13 | 0.65 | 2.7 | 1.26 | 10.7 | 27.8 | 29.0 | 43.2 | Loamy clay |

F5 | 2.51 | 40.0 | 86 | 0.26 | 0.15 | 0.65 | 1.5 | 1.34 | 16.1 | 32.8 | 34.7 | 32.5 | Loamy clay |

F6 | 2.46 | 29.0 | 88 | 0.25 | 0.16 | 0.63 | 2.9 | 1.32 | 13.3 | 32.7 | 32.5 | 34.8 | Loamy clay |

F7 | 3.31 | 38.0 | 97 | 0.26 | 0.12 | 0.67 | 1.9 | 1.25 | 10.9 | 20.7 | 33.2 | 46.1 | Clay loam |

F8 | 2.09 | 51.0 | 89 | 0.20 | 0.15 | 0.63 | 2.3 | 1.25 | 12.1 | 25.0 | 33.4 | 41.6 | Clay loam |

F9 | 2.33 | 34.0 | 86 | 0.24 | 0.12 | 0.60 | 1.8 | 1.35 | 15.5 | 30.1 | 33.2 | 36.7 | Loamy clay |

F10 | 2.29 | 58.0 | 93 | 0.29 | 0.12 | 0.68 | 2.4 | 1.40 | 13.5 | 27.0 | 34.5 | 38.5 | Loamy clay |

F11 | 2.78 | 53.0 | 89 | 0.21 | 0.13 | 0.70 | 2.3 | 1.26 | 14.9 | 30.3 | 31.2 | 38.5 | Loamy clay |

F12 | 2.12 | 35.0 | 87 | 0.28 | 0.20 | 0.70 | 4.3 | 1.49 | 16.0 | 35.6 | 33.4 | 31.0 | Loamy clay |

F13 | 4.43 | 17.0 | 136 | 0.27 | 0.13 | 0.78 | 3.0 | 1.35 | 13.7 | 29.8 | 31.6 | 38.6 | Loamy clay |

F14 | 3.53 | 53.0 | 128 | 0.20 | 0.19 | 0.62 | 3.3 | 1.23 | 9.1 | 20.5 | 33.8 | 45.7 | Clay loam |

F15 | 3.10 | 40.0 | 104 | 0.25 | 0.12 | 0.65 | 2.3 | 1.36 | 13.9 | 30.2 | 34.8 | 35.0 | Loamy clay |

F16 | 2.98 | 53.0 | 113 | 0.25 | 0.17 | 0.75 | 2.3 | 1.28 | 16.2 | 28.2 | 32.8 | 39.0 | Loamy clay |

F17 | 3.40 | 39.0 | 127 | 0.27 | 0.10 | 0.62 | 3.3 | 1.34 | 17.0 | 28.0 | 31.8 | 40.2 | Loamy clay |

F18 | 3.98 | 29.0 | 134 | 0.21 | 0.13 | 0.65 | 3.4 | 1.41 | 14.8 | 30.9 | 35.8 | 33.3 | Loamy clay |

F19 | 2.09 | 40.0 | 84 | 0.15 | 0.15 | 0.65 | 2.1 | 1.46 | 16.7 | 29.6 | 32.2 | 38.2 | Loamy clay |

F20 | 2.16 | 34.0 | 92 | 0.23 | 0.13 | 0.63 | 2.0 | 1.38 | 14.0 | 28.0 | 32.0 | 40.0 | Loamy clay |

F21 | 2.70 | 43.0 | 106 | 0.23 | 0.12 | 0.60 | 2.1 | 1.26 | 15.4 | 25.4 | 33.0 | 41.6 | Loamy clay |

F22 | 2.95 | 49.0 | 114 | 0.26 | 0.13 | 0.62 | 0.9 | 1.40 | 11.4 | 27.4 | 30.6 | 42.0 | Loamy clay |

F23 | 2.17 | 66.0 | 107 | 0.26 | 0.12 | 0.70 | 2.6 | 1.39 | 11.4 | 22.3 | 33.8 | 43.9 | Clay loam |

F24 | 2.43 | 60.0 | 106 | 0.19 | 0.15 | 0.70 | 1.2 | 1.38 | 11.3 | 27.4 | 29.8 | 42.8 | Loamy clay |

F25 | 2.85 | 43.0 | 93 | 0.25 | 0.17 | 0.65 | 3.6 | 1.27 | 13.3 | 28.2 | 30.9 | 40.9 | Loamy clay |

F26 | 2.20 | 47.0 | 82 | 0.28 | 0.15 | 0.68 | 4.1 | 1.30 | 12.3 | 30.5 | 31.3 | 38.2 | Loamy clay |

F27 | 2.78 | 42.0 | 86 | 0.23 | 0.15 | 0.68 | 1.8 | 1.33 | 14.3 | 30.9 | 33.1 | 36.0 | Loamy clay |

F28 | 2.97 | 25.0 | 87 | 0.19 | 0.12 | 0.65 | 2.6 | 1.33 | 14.2 | 30.0 | 31.7 | 38.3 | Loamy clay |

F29 | 2.81 | 48.0 | 96 | 0.23 | 0.12 | 0.65 | 3.1 | 1.30 | 12.5 | 22.0 | 34.0 | 44.0 | Clay loam |

F30 | 2.40 | 53.0 | 99 | 0.25 | 0.13 | 0.65 | 2.0 | 1.42 | 14.9 | 30.1 | 31.3 | 38.6 | Loamy clay |

F31 | 3.53 | 17.0 | 97 | 0.20 | 0.15 | 0.62 | 1.9 | 1.37 | 16.7 | 32.8 | 32.2 | 35.0 | Loamy clay |

F32 | 2.53 | 44.0 | 93 | 0.23 | 0.15 | 0.65 | 2.1 | 1.27 | 16.0 | 31.5 | 32.1 | 36.4 | Loamy clay |

F33 | 4.43 | 21.0 | 118 | 0.22 | 0.12 | 0.60 | 1.8 | 1.40 | 17.7 | 30.1 | 36.3 | 33.6 | Loamy clay |

F34 | 3.21 | 28.0 | 109 | 0.20 | 0.14 | 0.60 | 2.9 | 1.38 | 14.1 | 28.0 | 34.6 | 37.4 | Loamy clay |

F35 | 3.51 | 35.0 | 107 | 0.16 | 0.12 | 0.60 | 2.4 | 1.43 | 14.9 | 34.6 | 31.4 | 34.0 | Loamy clay |

F36 | 2.31 | 35.0 | 93 | 0.18 | 0.15 | 0.63 | 1.8 | 1.37 | 14.1 | 35.2 | 30.2 | 34.6 | Loamy clay |

F37 | 3.24 | 42.0 | 116 | 0.18 | 0.13 | 0.64 | 2.5 | 1.34 | 12.0 | 27.3 | 33.4 | 39.3 | Loamy clay |

F38 | 2.33 | 58.0 | 104 | 0.20 | 0.13 | 0.63 | 0.8 | 1.30 | 15.6 | 31.6 | 32.9 | 35.5 | Loamy clay |

F39 | 2.61 | 39.5 | 86 | 0.20 | 0.15 | 0.62 | 1.7 | 1.31 | 16.3 | 32.0 | 33.0 | 35.0 | Loamy clay |

F40 | 2.50 | 39.0 | 94 | 0.15 | 0.16 | 0.61 | 1.3 | 1.33 | 15.3 | 31.3 | 31.3 | 37.4 | Loamy clay |

F41 | 2.51 | 35.5 | 83 | 0.25 | 0.15 | 0.68 | 1.9 | 1.42 | 11.6 | 23.2 | 36.0 | 40.8 | Clay loam |

F42 | 2.20 | 51.0 | 96 | 0.19 | 0.12 | 0.60 | 2.7 | 1.35 | 11.9 | 30.4 | 32.1 | 37.5 | Loamy clay |

F43 | 2.24 | 35.0 | 92 | 0.20 | 0.15 | 0.60 | 0.8 | 1.46 | 16.4 | 32.7 | 35.2 | 32.1 | Loamy clay |

F44 | 2.90 | 27.0 | 89 | 0.20 | 0.15 | 0.65 | 4.5 | 1.43 | 16.4 | 32.2 | 32.8 | 35.0 | Loamy clay |

F45 | 2.54 | 42.0 | 84 | 0.23 | 0.15 | 0.64 | 2.6 | 1.39 | 11.4 | 27.8 | 31.3 | 40.9 | Loamy clay |

Mean value | 98.8 | 0.23 | 0.14 | 0.65 | 2.4 | 1.35 | 13.9 |

_{0}: furrow bottom slope; γ

_{d}: bulk density; θ

_{0}: initial soil water content; Cl: clay content; Si: silt content; Sa: sand content; the γ

_{d}, θ

_{0}, and soil particle proportions were the mean values of soil layers from 0 to 40 cm.

No. | Field Mean Infiltration Parameters | n | F_{c} | MAPRE (%) | ||
---|---|---|---|---|---|---|

A | k (mm h^{−α}) | Advance Time | Water Depth | |||

F1 | 0.756 | 147.37 | 0.142 | 1.10 | 5.1 | 7.2 |

F2 | 0.642 | 129.67 | 0.096 | 1.03 | 10.3 | 5.9 |

F3 | 0.434 | 102.39 | 0.075 | 0.93 | 5.4 | 6.2 |

F4 | 0.443 | 125.26 | 0.103 | 1.13 | 4.0 | 13.9 |

F5 | 0.521 | 103.75 | 0.123 | 0.89 | 4.1 | 8.2 |

F6 | 0.435 | 98.06 | 0.038 | 0.89 | 5.1 | 17.7 |

F7 | 0.417 | 137.34 | 0.042 | 1.27 | 4.5 | 10.3 |

F8 | 0.464 | 135.56 | 0.079 | 1.21 | 3.2 | 7.0 |

F9 | 0.735 | 138.07 | 0.074 | 1.04 | 4.4 | 7.5 |

F10 | 0.438 | 128.51 | 0.078 | 1.17 | 2.9 | 14.5 |

F11 | 0.605 | 137.04 | 0.158 | 1.11 | 4.6 | 8.1 |

F12 | 0.223 | 66.83 | 0.057 | 0.72 | 6.0 | 15.3 |

F13 | 0.673 | 117.65 | 0.110 | 0.92 | 5.6 | 15.1 |

F14 | 0.561 | 160.79 | 0.048 | 1.34 | 6.1 | 9.4 |

F15 | 0.550 | 95.71 | 0.186 | 0.81 | 6.5 | 10.7 |

F16 | 0.415 | 101.50 | 0.087 | 0.94 | 5.8 | 16.0 |

F17 | 0.375 | 96.81 | 0.052 | 0.92 | 6.4 | 16.1 |

F18 | 0.613 | 89.71 | 0.071 | 0.73 | 7.1 | 10.3 |

F19 | 0.209 | 82.29 | 0.082 | 0.89 | 6.0 | 7.9 |

F20 | 0.452 | 107.12 | 0.064 | 0.96 | 4.6 | 3.0 |

F21 | 0.374 | 117.06 | 0.074 | 1.12 | 4.6 | 10.0 |

F22 | 0.275 | 106.68 | 0.041 | 1.10 | 6.3 | 23.6 |

F23 | 0.254 | 108.47 | 0.053 | 1.14 | 4.0 | 13.2 |

F24 | 0.468 | 118.16 | 0.062 | 1.05 | 2.2 | 9.2 |

F25 | 0.258 | 115.09 | 0.075 | 1.20 | 3.7 | 8.7 |

F26 | 0.211 | 105.29 | 0.065 | 1.14 | 4.6 | 14.8 |

F27 | 0.308 | 108.10 | 0.096 | 1.08 | 3.7 | 17.9 |

F28 | 0.562 | 104.44 | 0.077 | 0.87 | 7.2 | 8.3 |

F29 | 0.584 | 156.45 | 0.071 | 1.29 | 3.3 | 11.3 |

F30 | 0.225 | 106.57 | 0.047 | 1.14 | 6.0 | 13.2 |

F31 | 0.701 | 103.19 | 0.050 | 0.79 | 7.6 | 7.8 |

F32 | 0.806 | 135.57 | 0.125 | 0.98 | 3.7 | 5.3 |

F33 | 0.592 | 90.09 | 0.057 | 0.74 | 5.5 | 15.1 |

F34 | 0.677 | 127.31 | 0.058 | 0.99 | 5.7 | 10.4 |

F35 | 0.645 | 91.85 | 0.183 | 0.73 | 4.2 | 8.1 |

F36 | 0.412 | 81.36 | 0.094 | 0.75 | 4.4 | 3.5 |

F37 | 0.632 | 127.32 | 0.092 | 1.02 | 5.4 | 10.2 |

F38 | 0.700 | 131.58 | 0.078 | 1.01 | 3.8 | 9.0 |

F39 | 0.673 | 124.89 | 0.179 | 0.98 | 3.5 | 8.0 |

F40 | 0.513 | 129.49 | 0.046 | 1.12 | 3.6 | 8.3 |

F41 | 0.330 | 103.77 | 0.075 | 1.02 | 3.0 | 9.4 |

F42 | 0.299 | 111.13 | 0.073 | 1.12 | 2.8 | 9.2 |

F43 | 0.207 | 74.47 | 0.052 | 0.81 | 3.6 | 10.1 |

F44 | 0.861 | 137.81 | 0.054 | 0.97 | 6.1 | 18.0 |

F45 | 0.311 | 98.07 | 0.163 | 0.98 | 2.0 | 13.1 |

Mean value | 0.485 | 113.68 | 0.085 | 1.00 | 4.8 | 10.8 |

_{c}: normalization factor; MAPRE: mean absolute percent relative error; α and k: infiltration parameters in Kostiakov equation. The infiltration coefficient of furrow irrigation inverted by SIPAR_ID is divided by the wetted perimeter to obtain the infiltration coefficient per unit area; that is, the infiltration coefficient k in Equation (1).

**Table 3.**Simulated comprehensive indicator of irrigation performance determined using WinSRFR and according to the optimal results of discharge yielded under Sim.4 and Sim.5.

S_{0} (‰) | L (m) | Simulated Values of Y under Sim.4 | Simulated Values of Y under Sim.5 | ||||
---|---|---|---|---|---|---|---|

Min. | Max. | Mean | Min. | Max. | Mean | ||

1.0 | 80 | 0.87 | 0.98 | 0.93 (0.03) | 0.83 | 0.98 | 0.90 (0.05) |

105 | 0.83 | 0.97 | 0.89 (0.03) | 0.80 | 0.98 | 0.88 (0.05) | |

130 | 0.80 | 0.93 | 0.86 (0.03) | 0.80 | 0.96 | 0.86 (0.06) | |

2.5 | 80 | 0.87 | 0.98 | 0.93 (0.03) | 0.82 | 0.97 | 0.89 (0.05) |

105 | 0.85 | 0.96 | 0.90 (0.03) | 0.81 | 0.96 | 0.88 (0.05) | |

130 | 0.80 | 0.95 | 0.88 (0.03) | 0.81 | 0.97 | 0.87 (0.05) | |

4.0 | 80 | 0.82 | 0.96 | 0.93 (0.03) | 0.84 | 0.97 | 0.90 (0.05) |

105 | 0.85 | 0.95 | 0.90 (0.03) | 0.82 | 0.96 | 0.88 (0.05) | |

130 | 0.82 | 0.93 | 0.88 (0.03) | 0.80 | 0.94 | 0.87 (0.05) |

_{0}: furrow bottom slope; L: furrow length; Y: comprehensive performance indicator of closed-end furrow irrigation; Sim.4: Inverted field mean infiltration parameters and Manning’s roughness of each furrow; Sim.5: Field mean infiltration parameters by PTF of each furrow and Manning’s roughness cumulative frequency as 50%. Values in parentheses are the variance coefficients of comprehensive performance indicator Y.

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nie, W.-B.; Li, Y.-B.; Zhang, F.; Dong, S.-X.; Wang, H.; Ma, X.-Y.
A Method for Determining the Discharge of Closed-End Furrow Irrigation Based on the Representative Value of Manning’s Roughness and Field Mean Infiltration Parameters Estimated Using the PTF at Regional Scale. *Water* **2018**, *10*, 1825.
https://doi.org/10.3390/w10121825

**AMA Style**

Nie W-B, Li Y-B, Zhang F, Dong S-X, Wang H, Ma X-Y.
A Method for Determining the Discharge of Closed-End Furrow Irrigation Based on the Representative Value of Manning’s Roughness and Field Mean Infiltration Parameters Estimated Using the PTF at Regional Scale. *Water*. 2018; 10(12):1825.
https://doi.org/10.3390/w10121825

**Chicago/Turabian Style**

Nie, Wei-Bo, Yi-Bo Li, Fan Zhang, Shu-Xin Dong, Heng Wang, and Xiao-Yi Ma.
2018. "A Method for Determining the Discharge of Closed-End Furrow Irrigation Based on the Representative Value of Manning’s Roughness and Field Mean Infiltration Parameters Estimated Using the PTF at Regional Scale" *Water* 10, no. 12: 1825.
https://doi.org/10.3390/w10121825