# Optimal Allocation Method of Irrigation Water from River and Lake by Considering the Field Water Cycle Process

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area

#### 2.1. Description of the Study Area

^{3}. However, the Yellow River Water Conservancy Commission plans to reduce water diversion with water-saving projects. The Yellow River enters a freezing period in winter and breaks up in March of the following year with a large number of ice floods hazard downstream. Furthermore, special geological structures formed the unique phenomenon of lake enrichment in the HID, and lake water resources have a large potential for development and utilization. Under such circumstances, ice floods can be transferred to lakes so that farmers can irrigate fields with water brought from nearby lakes under irrigation water shortages. This can not only relieve the pressure of ice floods from the Yellow River but can also effectively improve water use efficiency.

#### 2.2. Data Collection

^{−1}, and 10,500 Yuan ha

^{−1}, respectively. For the water price, the agricultural water price is determined by the times according to the regulation plan, and from 11 April to 30 September, agricultural irrigation water is 0.083 Yuan m

^{−3}. Irrigation water resources from ice floods were not accounted for as adequate ice flood resources. However, a channel needs to be constructed to divert ice flood water to the lake, and the construction costs of the channel are considered in the model. In this study, the lake irrigation cost is 0.03 Yuan m

^{−3}after investigation.

## 3. Methodology

#### 3.1. The Hydrological Cycle Simulation Model

#### 3.1.1. Water Balance Model

_{g}), irrigation infiltration (D

_{inf}), and actual evapotranspiration (ET). Surface runoff was ignored because of low rainfall and relatively flat topography in the study region. In past, there were few irrigation schedule optimization models considering the effect of soil water deficit on crop evapotranspiration. The time step for field water balance is one day. The soil water balance equation for crops can be expressed as:

_{i}is the variation of water storage under 1 m depth soil in day i (mm); z is the depth of the soil, here z = 1 m; θ

_{i}is the volumetric water content in day i (m

^{3}m

^{−3}); A is the irrigation area(m

^{2}), A = 1,000,000 m

^{2}; η is the utilization coefficient of the water supply in the Hetao irrigation area; the efficiency coefficient of irrigation is 0.4; the efficiency coefficient of a sublateral canal is 0.9; and Q

_{jt}is the water amount of water resource j in growth stage t (m

^{3}). The groundwater recharge coefficients were 0.15, the irrigation water for I, and then the recharge of irrigation water to groundwater is D

_{inf}= 0.15I [30].

_{0}/0.53 [32]; and h is phreatic depth (m). Phreatic depths during crop growth were calculated by the groundwater balance Equation (3) for each day.

^{3}), and GA is the surface area that contributes to flow (m

^{2}). The paper sets the discharge of groundwater as the positive direction. The groundwater depth in the region floats approximately 1 m in late April; therefore, the initial value was set as h

_{1}= 1 m. According to Darcy’s law, the exchange capacity of groundwater and lake water is described by the following equation:

_{i}= ΔH

_{i}/L; ΔH

_{i}is the difference of groundwater and lake water levels on day i; L is the length of the cross section; and A is the cross-sectional area of the aquifer, A = 4000 m

^{2}.

^{4}m

^{3}); V is the lake water storage capacity (10

^{4}m

^{3}); LA is the lake surface area (10

^{4}m

^{2}); LH is the lake water level (m); E

_{0}is evaporation from the lake (mm), E

_{0}= 0.55E [34]; and Q

_{2i}is the irrigation from the lake (m

^{3}).

#### 3.1.2. Actual Evapotranspiration Model

_{0}), crop types and growth (by crop coefficient K

_{cb}and the soil evaporation coefficient K

_{e}), soil water supply (by soil water stress coefficient K

_{s}), and the evapotranspiration is calculated by using the dual K

_{c}approach [35]:

_{0}is associated with meteorological factors and calculated in accordance with the FAO-56 recommended Penman-Monteith formula; the crop coefficients K

_{cb}and K

_{e}were determined with the meteorological data from the reference FAO-56; θ

_{fc}is the volumetric water content at field capacity; θ

_{wp}is the volumetric water content at wilting point; θ

_{t}is the volumetric water content at the critical point of water stress, and the measured θ

_{fc}, θ

_{wp}, θ

_{t}were used to calculate soil water stress coefficient K

_{s}: θ

_{fc}= 0.20, θ

_{wp}= 0.08, θ

_{t}= 0.14; and θ

_{1}as initial value equal to θ

_{t}.

#### 3.1.3. Jensen Model

^{−1}); Y

_{m}is the adequate maximum yield of a crop when the water supply is adequate (kg ha

^{−1}); ET is the actual evapotranspiration in the crop growth stage t (mm); ET

_{m}is the maximum evapotranspiration in the crop growth stage t (mm); and λ

_{t}is the water sensitivity index in the crop growth stage t. Among them, according to the relevant experiment in the HID, the maximum yield of maize is 12,850 kg ha

^{−1}.

_{i−}

_{1}to t

_{i}can be calculated by the formula:

_{m}and λ of maize in the new model in Table 1.

#### 3.2. Uncertainty Optimization Model of Irrigation Schedule with Multi-Water Resources

^{±}is the system net benefit in the study region (Yuan, Chinese monetary currency, 1 US $ ≈ 6.625 Yuan under the present exchange rate); CP is the price of a crop (Yuan kg

^{−1}); TC is the cost of a crop (Yuan ha

^{−1}); A is the area of planting of a crop (ha); and S

_{j}is the cost of water coming from water resource j (Yuan m

^{−3}).

**(1) The actual crop evapotranspiration constraints**

_{m}.

**(2) Water supply capacity from Yellow River constraints**

^{3}at crop growth stages yearly.

**(3) Lake ecological capacity constraints**

_{max}(three hundred and fifty thousand m

^{3}). Lake water storage capacity is more than the lake effective capacity B for ecological sustainable development, but more lake water is often divided for greater benefits in the actual making-decision process, allowing the existence of appropriate risk p. The stochastic chance-constrained programming can effectively reflect the reliability of the system to meet the constraints (or risk) [42]. The model can effectively solve uncertainties, described as intervals and fuzzy characteristics, which exist in the water resources allocation process. However, in many cases, policy makers want to understand different decision plans under different violation probabilities of lake water. Therefore, compared with the IFSOP model, the main advantage of the IFSSOP model is that it provides enough attention on the lake ecological constraints under violation probabilities.

**(4) The amount of water should be positive**

## 4. Results and Discussion

#### 4.1. Evaluation of the Inexact Fuzzy Stochastic Simulation-Optimization Programming (IFSSOP) Model

#### 4.1.1. Water Amount Allocation under Different P in Different Hydrological Years

^{3}for the study district, and under a certain circumstance, more YR water will be saved with the increase of violation probabilities. The main irrigation water in the area is the Yellow River, and this is in accord with the actual situation, with lake water by using flood water, which belongs to extra water. Therefore, to achieve the effect of the Yellow River savings without reducing crop yield, some measures should be taken to use more flood water, such as excavation of artificial lakes for water storage.

#### 4.1.2. System Economic Benefit under Different P in Different Hydrological Years

#### 4.1.3. Water Consumption of Maize in Different Months of Different Hydrological Years

#### 4.1.4. Irrigation Water Allocation in Different Months of Different Hydrological Years

^{3}/ha, [4482.7, 4569.0] m

^{3}/ha, and [4416.1, 4512.1] m

^{3}/ha, respectively. The irrigation amount in the wet year is smaller than the dry year, and the most in the normal year. The irrigation amount is mainly affected by temperature, humidity, and other meteorological factors; therefore, it has no direct relationship with typical hydrological years. The interval results provide policy makers with more options, according to preferences and actual conditions in irrigation systems. For instance, taking the normal year as an example, the total water allocation is 4525.9 m

^{3}/ha applying to the deterministic model, however, policy makers can select any water allocation project from a range (from 4482.7 m

^{3}/ha to 4569.0 m

^{3}/ha) adopting an uncertainty model, instead of a deterministic number. In other words, the policy makers would choose the lower bound of water allocation to address climate warming because climate warming leads to excessive evaporation; therefore, water diversion from the lake becomes less. Meanwhile, lower system benefits may be obtained.

#### 4.2. Comparison between the IFSSOP Model and the IFSSOP-NG Model

^{3}/ha, 4526 m

^{3}/ha, and 4464 m

^{3}/ha for the wet year, normal year, and dry year respectively, with 4649 m

^{3}/ha, 4581 m

^{3}/ha, and 4579 m

^{3}/ha for the IFSSOP-NG model. Compared with the two models, the IFSSOP-NG model has a preference to use as much lake water as possible for the greater system benefit; however, the IFSSOP-NG model overrated the available lake water amount that would lead to inaccurate irrigation scheduling. The system benefit for maize obtained from the IFSSOP model is higher than the IFSSOP-NG model solution.

#### 4.3. Analysis of Agriculture Water Saving Potential with IFSSOP Model

^{3}in agriculture and industry, respectively [45]. The redistribution of saved water from agriculture to industry or service industry will be a huge augmenter in most developing countries [46]. On paper, agricultural water saving potential is defined as the Yellow River water saving amount adopting lake water to irrigate for the total benefit of agriculture. According to local government, the Yellow River water amount for irrigation in agriculture can be transferred to industry at a certain price (water right transfer price). The water right transfer price is 0.3 Yuan/m

^{3}in the study region [47]. Different water saving potential scenarios are considered in order to study the relationship between agriculture water saving potential and system economic benefit more directly. In the scenarios, the percentage of water amount accounting for the total water supply amount was set up (100%, 90%, 80%, 70%, 60%, 50%, 40%, and the water deficit irrigation ratio without LW).

^{4}m

^{3}of YR water with maximum benefit at the upper bound, but no appropriate result at the lower bound. The results suggest that there is not enough YR irrigation water converting to industrial water under certain water shortages. The results also show that by maintaining the original crop water deficit irrigation ratio, the region can save 30–34% of the Yellow River, approximately [44.98, 45.67] × 10

^{4}m

^{3}.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Schematic of the simulation-optimization model for irrigation system: P is precipitation; ET, E

_{0}and ET

_{g}are actual evapotranspiration, lake evaporation and groundwater evaporation; YI and LI are irrigation from the Yellow River and lake, respectively; G is ice flood; D

_{inf}is irrigation infiltration; R is the exchange capacity of groundwater and lake.

**Figure 3.**The optimal irrigation water allocation of two water sources under various p in different hydrological years.

**Figure 5.**Monthly crop water consumption and the proportion of the actual crop consumption to the maximum crop consumption: wet year (

**a**); normal year (

**b**) and dry year (

**c**).

**Figure 6.**Monthly irrigation water allocation schemes in different hydrological years: (

**a**) is wet year; (

**b**) is normal year; (

**c**) is dry year. Lower bound and upper bound of water allocation from left to right.

**Figure 8.**Optimization solutions under different scenarios (100%, 90%, 80%, 70%, 60%, 50%, 40% of total water amount and original water deficit irrigation ratio).

Hydrological Year | Month | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|

λ | 0.01 | 0.05 | 0.14 | 1.31 | 0.03 | 0.01 | |

Wet year | ET_{m} (mm) | [21.54, 23.80] | [62.84, 69.45] | [179.03, 197.87] | [204.78, 226.34] | [158.79, 175.51] | [41.81, 46.21] |

Normal year | ET_{m} (mm) | [19.79, 21.08] | [60.53, 64.46] | [169.51, 80.58] | [236.02, 251.35] | [161.86, 172.39] | [48.07, 51.20] |

Dry year | ET_{m} (mm) | [19.60, 21.67] | [63.92, 70.65] | [167.69, 185.34] | [211.30, 233.55] | [158.49, 175.18] | [40.21, 44.44] |

_{m}means the maximum crop evapotranspiration in a month (interval number).

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**MDPI and ACS Style**

Li, X.; Huo, Z.; Xu, B.
Optimal Allocation Method of Irrigation Water from River and Lake by Considering the Field Water Cycle Process. *Water* **2017**, *9*, 911.
https://doi.org/10.3390/w9120911

**AMA Style**

Li X, Huo Z, Xu B.
Optimal Allocation Method of Irrigation Water from River and Lake by Considering the Field Water Cycle Process. *Water*. 2017; 9(12):911.
https://doi.org/10.3390/w9120911

**Chicago/Turabian Style**

Li, Xuemin, Zailin Huo, and Bing Xu.
2017. "Optimal Allocation Method of Irrigation Water from River and Lake by Considering the Field Water Cycle Process" *Water* 9, no. 12: 911.
https://doi.org/10.3390/w9120911