# Modeling the Effects of Spatial Variability of Irrigation Parameters on Border Irrigation Performance at a Field Scale

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Case Study Area

#### 2.1.1. Site Description

^{2}in MW and 43.4 ha

^{2}in YH, respectively. MW is located in the coastal area of the lower reaches of the Yellow River, which has a typical warm temperate zone continental monsoon climate, with an average annual rainfall of 537.4 mm. The soils are alluvial sediments with primary textures of silt loam and sandy loam. YH is located in the North China Plain, which has a typical temperate zone monsoon climate, with the average annual rainfall of 444.1 mm. The experimental soil is classified as sandy loam in the root zone.

#### 2.1.2. Data Collection

#### 2.1.3. Statistical Analysis of Field Physical Properties at a Field Scale

#### 2.2. Numerical Modeling Methodology

#### 2.2.1. Coupled Modeling of Surface-Subsurface Water Flow for Border Irrigation

**U**is the vector of the conserved dependent variables,

**F**is the physical flux, and

**S**is the source vector that comprises bottom elevation vector

**S**, bed roughness vector

_{1}**S**and infiltration vector

_{2}**S**, which are expressed as:

_{3}_{s}is the water depth (m), q denotes the unit discharge along coordinate direction x (m

^{3}/s/m), g is the acceleration due to gravity (m/s

^{2}), z is the bottom elevation (m), n denotes the Manning’s roughness coefficient (m

^{1/6}), and u is the vertically averaged flow velocity (m/s), I represents the infiltration rate (m/s).

_{s}is the saturated soil water content (−), θ

_{r}is the residual soil water content (−), α (1/m), n (−) and m (−) denote the soil water retention curve parameters, m = 1 − 1/n, and K

_{S}is the saturated hydraulic conductivity (m/min), and S

_{e}= (θ(h) − θ

_{r})/(θ

_{s}− θ

_{r}) is the effective saturation (−).

#### 2.2.2. Numerical Solution

**S**at an unknown time (w + 1) and a known time w. Then,

_{2}**S**and

_{1}**S**are discretized using the explicit time scheme at a known time w. At the space discretization of the governing equations, the finite difference scheme is used to discretize the space derivative of the physical flux linear approximation generated by the implicit time scheme. The advection upstream splitting method is used to discretize the space derivative of the advection physical flux,

_{3}**S**and

_{2}**S**are then calculated according to the corresponding space-node values. After spatio–temporal discretization processes of the governing equation of the one-dimensional complete hydrodynamic model, the discretization equations of mass conservation and momentum conservation are distinguished shown in Dong et al. [23]. The final format of surface flow equations can be expressed by:

_{3}**B**is the coefficient matrix consisting of the coefficient of

**ΔU**of cell (j − 1), j, (j + 1), respectively; at a known time w;

**B**is the coefficient matrix consisting of the coefficient of h

^{MA}_{s}of cell (j − 1), j, (j + 1), respectively, at a known time w;

**B**is the coefficient matrix consisting of the coefficient of u of cell (j − 1), j, (j + 1), respectively, at a known time w;

^{MO}**X**is the matrix of

**ΔU**of each cell at a known time w;

**X**is the matrix of h

^{MA}_{s}of each cell; at a known time w;

**X**is the matrix of u of each cell at a known time w;

^{MO}**D**is the coefficient matrix of numerical flux vector of interface of cell j, bed roughness vector

**S**, infiltration vector

_{2}**S**and the averaged form of bottom elevation vector

_{3,}**S**of cell j at a known w;

_{1}**D**is the coefficient matrix about h

^{MA}_{s}and u at a known w; and

**D**is the coefficient matrix of numerical fluxes of interface of cell j, bed roughness, infiltration, and bottom elevation at a known w.

^{MO}_{P}is the residual associated with the Picard iteration. If ${R}_{{}_{P}}^{k+1,m}\le {10}^{-5}$ and $\left|{h}_{i}^{k+1,m+1}-{h}_{i}^{k+1,m}\right|/{h}_{i}^{k+1,m}\le {10}^{-2}$, the iteration stops, the soil pressure heads of every spatial point at the (k + 1)th time step are calculated. The calculation then proceeds to the next time step.

_{s,}

_{initial}and water flow velocity u

_{initial}in each cell are zero before surface irrigation modeling is initiated. However, the numerical procedure requires all depths to be greater than zero to avoid undefined terms in Equation (1). Therefore, a small and positive value (0.0001 m) is initially assigned to the depth of flow at all cells in the flow domain. At the upstream end, the unit inflow rate is q = q

_{0}. If irrigation stops, the water flow velocity at the field head is zero. The field end is closed. The initial time step for the subsurface flow system is given by users and the size of time step changes in accordance with the one used for the surface flow model. Time-variable surface pressure head h

_{s}is set as the upper boundary condition; the bottom boundary conditions can be assumed as free drainage. The uniform pressure head throughout the soil profile is set as the initial condition in the computational domain.

#### 2.2.3. Monte Carlo Modeling

#### Probability Distribution Functions of Model Parameters

_{d}and the spatial distribution of SED between the actual and the target design elevations. Under the same field tillage and management conditions, all of the S

_{d}values of SED at a field scale are assumed as the same [26]. For a same S

_{d}the spatial distribution of SED may vary, which makes it difficult to describe the impacts of the spatial variability of SED on surface irrigation performance. Therefore, surface micro-topography at a field scale can be represented by a set of elevation distributions of SED for the same S

_{d}. Based on the field geometry (length and width) and the statistical characteristics of observed SED, the surface elevation nodes can be expressed by the following distribution:

_{mt}is SED (cm), u

_{mt}is the mean value of SED (cm), and σ

_{mt}is the standard deviation of SED (cm).

#### Monte Carlo Simulation

_{d}of surface micro-topography at a field scale are the same under the same filed tillage and management conditions and the spatial variability of soil properties, border length, unit discharge, and the spatial distribution of SED are the main factors that affect surface irrigation performance of field scale and the spatial distribution characteristics. The Monte Carlo simulation process of border irrigation at a field scale shown in Figure 4 is discussed as follows:

_{i}+ C + S + SOC = 100%

_{i}is soil silt content, %, C is soil clay content, %, S is soil sand content, %, and SOC is soil organic matter content, %.

_{x}is the grid number along the basin width direction (−), and N

_{y}is the grid number along the basin length direction (−).

_{a}) of each irrigation event are calculated based on the simulation values of soil water content along the border length direction. In the proposed paper, the maximum number of randomly selected samples is set to 272, which is the field-observed number.

_{a}of the iteration (n + 1) and mean values of CU and E

_{a}of the iteration n cannot satisfy the convergence criterion of Equations (14) and (15) (shown in Section 2.3), the iterative process continues. Otherwise, the iteration is terminated.

#### 2.3. Evaluation Indicators of Surface Irrigation Performance

_{a}) and the uniformity coefficient (CU), as defined by Walker (1987), are selected as surface irrigation performance indicators. The application efficiency E

_{a}(%) is defined as:

_{a}and θ

_{b}are the average soil volumetric water content in the root zone before and after irrigation event, cm

^{3}/cm

^{3}, RD is the depth of root zone, cm, Z

_{avg}is the average infiltrated depth of water applied to the field (mm) and determined from the cut-off time and the inflow rate, Z

_{i}is the infiltrated depth of water in the ith field unit along the field length direction. N is the number of field units along the border length direction.

#### 2.4. Model Performance Criteria of Field Scale

_{a}and CU resulting from the simulation data of the different irrigation event through a set of stochastic parameter sample combinations. M sets of irrigation performance indicators can be obtained by modeling the different irrigation event with M sets of stochastic parameter samples for a given condition. The number m (m < M) of stochastic parameter sample generations required to characterize the spatial variability of field physical properties may be determined by analyzing the changes of border irrigation performance with the number of stochastic parameter samples generations. The irrigation performance simulation experiment is conducted through the hybrid coupled model. Statistical analysis is carried out by using Equations (16) and (17) (m < 272), which can reflect the effect of the stochastic parameter distribution on the border irrigation performance. If the MARE

_{k}< 0.5% and RARE

_{k}< 5% of six consecutive data points are used as the discriminant indicators that the mean value and standard deviation reach the steady state, the mean values of border irrigation performance indicators and the corresponding standard deviations can be determined. MARE

_{k}and RARE

_{k}can be calculated as follows:

_{k}is the relative error between the (k + 1)th and the kth mean values of border irrigation performance indicators, RARE

_{k}is the relative error between the (k + 1)th and the kth standard deviations of border irrigation performance indicators, ${S}_{d}^{k}$ represents the kth standard deviations of border irrigation performance indicators, and ${S}_{d}^{k+1}$ represents the (k + 1)th standard deviations of border irrigation performance indicators.

#### 2.5. Data Analysis

## 3. Results and Discussion

#### 3.1. Total Water Volume of Surface Irrigation

_{s}is the total water volume of surface irrigation (m

^{3}), V

_{i}is the irrigation water volume of the ith basin (m

^{3}), A is the total area of the experimental field (m

^{2}), L is the basin length (m), W is the basin width (m), and n is the number of basins in the experimental field (−).

_{V}) is calculated as,

_{s}is the simulation value of the total water volume of surface irrigation (m

^{3}), V

_{o}is the observed value of the total water volume of the surface irrigation (m

^{3}).

^{3}cm

^{−3}, respectively. Table 5 presents the relative error of the total water volumes of the border irrigation between the model-predicted values and the field-observed values. The RE

_{V}of the PNM is lower than that of the DNM. This result indicates that the PNM for scaling up the border irrigation performance at a field scale exhibits the more satisfactory simulation accuracy.

#### 3.2. Surface Flow Advance

_{p}is the relative error of the average p value from the model-predicted data and the field-observed data, p represents the fitting parameters a and b, and P

_{s}and P

_{o}are the average values of the fitting parameters from the field-observed data and the model-predicted data, respectively (−).

#### 3.3. The Irrigation Time of Surface Flow Advance

_{t}) is calculated as,

_{s}is the average FT value under the model-predicted conditions (min) and t

_{o}is the average FT value under the field-observed conditions (min).

#### 3.4. Soil Water Content

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgements

## Conflicts of Interest

## References

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**Figure 3.**The frequency distribution of border length and width at the Mawan and Yehe irrigation districts. MW indicates the Mawan irrigation district; YH indicates the Yehe irrigation district.

**Figure 4.**Monte Carlo simulation processes of border irrigation at a field scale. E

_{a}indicates application efficiency; CU indicates the uniformity coefficient; SPAW indicates Soil Plant Atmosphere Water model.

**Figure 6.**The spatial distribution of the average soil water content of the field-observed after the ending of surface irrigation. MW indicates the Mawan irrigation district; YH indicates the Yehe irrigation district.

**Figure 7.**The cumulative probability of the average soil water content for the field-observed and the model-predicted data with the PNM. MW indicates the Mawan irrigation district; YH indicates the Yehe irrigation district.

**Table 1.**Statistical characteristics and normal test results of unit discharge of border field at a field scale.

Experiment Site | Minimum (L/(s·m)) | Maximum (L/(s·m)) | Mean (L/(s·m)) | S.D. | K-S Test |
---|---|---|---|---|---|

MW | 2.84 | 3.25 | 3.12 | 0.47 | N |

YH | 3.03 | 3.53 | 3.30 | 0.56 | N |

**Table 2.**The parameter values of the pherical semivariogram model for surface relative elevation at a field scale.

Tests | The Pherical Semivariogram Parameters | ||
---|---|---|---|

C_{0} (cm^{2}) | C_{0} + C (cm^{2}) | R (m) | |

MW | 3.46 | 16.48 | 23.13 |

YH | 3.31 | 15.76 | 37.53 |

_{0}indicates the nugget; (C

_{0}+ C) indicates the still; R indicates the range.

Tests | Indicators | Soil Properties | ||||
---|---|---|---|---|---|---|

SOC | Clay (<0.002 mm) | Silt (0.002–0.05 mm) | Sand (0.05–2 mm) | BD | ||

% | % | % | % | g cm^{−3} | ||

MW | Minimum | 0.36 | 6.99 | 56.77 | 3.42 | 1.15 |

Maximum | 2.59 | 35.74 | 79.04 | 36.23 | 1.68 | |

Mean | 1.13 | 19.91 | 65.97 | 14.12 | 1.38 | |

SD | 0.42 | 5.18 | 4.22 | 4.66 | 0.36 | |

K-S test | N | N | N | N | N | |

YH | Minimum | 0.75 | 9.17 | 18.47 | 15.38 | 1.14 |

Maximum | 1.78 | 33.87 | 61.38 | 64.89 | 1.62 | |

Mean | 1.11 | 22.34 | 38.14 | 39.52 | 1.37 | |

S.D. | 0.37 | 4.47 | 7.89 | 8.67 | 0.30 | |

K-S test | N | N | N | N | N |

**Table 4.**Statistical characteristics and normal test results of surface roughness coefficient of border field at a field scale.

Experiment Site | Minimum (−) | Maximum (−) | Mean (−) | S.D. | K-S Test |
---|---|---|---|---|---|

MW | 0.09 | 0.17 | 0.13 | 0.03 | N |

YH | 0.08 | 0.14 | 0.10 | 0.02 | N |

**Table 5.**The relative errors of the total water volumes from the model-predicted and the field-observed data.

Tests | V_{o}/m^{3} | PNM | DNM | ||
---|---|---|---|---|---|

V_{s}/m^{3} | RE_{V}/% | V_{s}/m^{3} | RE_{V}/% | ||

MW | 127,100 | 115,200 | 9.36 | 109,900 | 13.53 |

YH | 72,300 | 65,900 | 8.85 | 62,300 | 13.83 |

_{o}indicates the observed value of the total water volume of the surface irrigation; V

_{s}indicates the simulation value of the total water volume of surface irrigation; RE

_{v}indicates the relative error of the total water volumes of surface irrigation between the simulation values and the observed values.

**Table 6.**The relative error of the fitting parameter values from the model-predicted and the field-observed data.

Tests | Parameters | P_{o} | PNM | DNM | ||
---|---|---|---|---|---|---|

P_{s} | RE_{p} (%) | P_{s} | RE_{p} (%) | |||

MW | a | 0.1054 | 0.1017 | 3.51 | 0.1145 | 8.63 |

b | 1.3324 | 1.2928 | 2.97 | 1.2484 | 6.30 | |

YH | a | 0.1826 | 0.1742 | 4.60 | 0.1982 | 8.54 |

b | 1.2137 | 1.1718 | 3.45 | 1.1287 | 7.01 |

_{p}indicates the relative error of the average fitting parameter value from the model-predicted data and the field-observed data; P

_{s}and P

_{o}are the average values of the fitting parameters from the field-observed data and the model-predicted data.

**Table 7.**The relative error of the average irrigation time from the field-observed and the model-predicted data.

Tests | t_{o} (min) | PNM | DNM | ||
---|---|---|---|---|---|

t_{s} (min) | RE_{t} (%) | t_{s} (min) | RE_{t} (%) | ||

MW | 62.70 | 57.12 | 8.90 | 54.21 | 13.54 |

YH | 116.50 | 108.36 | 6.99 | 102.86 | 11.71 |

_{s}indicates the average irrigation time under the model-predicted conditions; t

_{o}indicates the average irrigation time under the field-observed conditions; RE

_{t}indicates relative error of the average irrigation time of the field- observed and model-predicted data.

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## Share and Cite

**MDPI and ACS Style**

Dong, Q.; Zhang, S.; Bai, M.; Xu, D.; Feng, H.
Modeling the Effects of Spatial Variability of Irrigation Parameters on Border Irrigation Performance at a Field Scale. *Water* **2018**, *10*, 1770.
https://doi.org/10.3390/w10121770

**AMA Style**

Dong Q, Zhang S, Bai M, Xu D, Feng H.
Modeling the Effects of Spatial Variability of Irrigation Parameters on Border Irrigation Performance at a Field Scale. *Water*. 2018; 10(12):1770.
https://doi.org/10.3390/w10121770

**Chicago/Turabian Style**

Dong, Qin’ge, Shaohui Zhang, Meijian Bai, Di Xu, and Hao Feng.
2018. "Modeling the Effects of Spatial Variability of Irrigation Parameters on Border Irrigation Performance at a Field Scale" *Water* 10, no. 12: 1770.
https://doi.org/10.3390/w10121770