Water is a key factor that controls the evolution of the ecosystem and restricts social and economic development. The rational allocation of water resources is the basis for the basin's sustainable use of water resources. The article uses fractional differential equations to discuss the increasingly prominent contradictions between supply and demand of agricultural water resources, serious pollution and overexploitation, the decline in groundwater level and water quality, low agricultural water efficiency, and serious waste from the perspective of sustainable development. Second, the article classifies the index system of the water resources carrying capacity evaluation model. Finally, model the resource balance of water supply and water demand in the irrigation area. The study found that the water quantity analysis and determination of the irrigation area will have important guiding significance for implementing the irrigation area's continued construction and water-saving transformation. The allocation and utilization of water resources in farmland irrigation areas will tend to be rationalized. This will greatly improve the water-saving efficiency of irrigation districts.

#### Keywords

- fractional differential equation
- available water
- water demand
- agricultural water use
- water resources balance

#### MSC 2010

- 39A10

Nowadays, China's food security problem is outstanding. As the basic guarantee for increasing food production, the rational use and optimal allocation of water resources have become an important issue in agricultural food production [1]. This paper uses the water balance model to study and apply the specific problems of water resources in farmland irrigation areas. This can better guide the implementation of continuous construction and water-saving transformation in irrigation districts.

The irrigation area we selected is located in the Xinjiang Plain, with flat terrain and fertile soil. The total land area of the irrigation area is 90.73 km^{2}, and the planned irrigation area is 9073hm^{2}. The irrigation area was built in the 1970s, but the project fell into disrepair for a long time and suffered serious damage. The supporting facilities of the project are imperfect, and the leakage is serious. The average irrigation water utilization coefficient is only 0.4. The water productivity is only 0.35 kg/m^{3}. The current water availability and crop water requirements of farmland irrigation areas are relatively large. From a long-term perspective, the irrigation area has great water-saving potential [2]. We should first analyze and calculate the water resources balance in the irrigation area. Formulate water-saving measures based on the irrigation area's conditions, and incorporate the water-saving transformation and continued construction of supporting projects in the irrigation area into the new schedule. In this way, farmland irrigation districts can be built into representative water-saving and high-efficiency medium-scale irrigation districts.

The supply and demand of water resources are contradictory and mutually restrictive. Coordinating the water requirements of various sectors of social and economic development in the production and life of the national economy is an important way to solve this contradiction [3]. The utilization rate of water resources in the irrigation area is the key to saving water and energy in the irrigation area. The water balance analysis and calculation of the irrigation area itself is an effective method to evaluate the utilization rate of water resources. The article puts the economy first as the primary condition and adopts supply based on demand [4]. There are no industrial water requirements in this irrigation area, and only agricultural water and domestic water needs are considered.

The water available for farmland irrigation is KG, which is the water available for the Songhua River Water Source Project [5]. The available water quantity of surface water in this irrigation area is defined as

Under normal circumstances, the recharge of groundwater is equal to the sum of the local precipitation and the groundwater recharge from the side seepage in the front of the mountain. The groundwater replenishment in this area is mainly the replenishment GRB of Songhua River irrigation infiltration [6]. The relationship between the Irrigation Infiltration Replenishment GRB of the Jiseonghua River and the water volume of the Jiseonghua River can be calculated by the following formula:
_{channel} is the water utilization coefficient of the canal system at all levels in the irrigation area. _{field} is the field water utilization coefficient at the end of the irrigation area. _{channel} is related to the anti-seepage structure of the canals at all levels in the irrigation area.

According to the water quota, the rural domestic water consumption NS is defined as the product of the population R and the quota D. The formula for rural domestic water consumption is NS = 365RD. The water sources of rural domestic water include surface water and groundwater [9]. The actual water demand can be estimated to the water source engineering office's water supply based on the planned domestic water consumption in the irrigation area. Since the water supply ratios of different water sources are different, we define _{s}_{g}_{s}_{g}

The planned annual total irrigation area of this irrigation district is 9073 hm^{2}. These are all for water-lifting irrigation. The water source of the farmland irrigation area comes from the Songhua River. According to the irrigation system design calculation, the planned net irrigation quota d = 6090 m^{3}/hm^{2} in the irrigation area [10]. The planned irrigation area is F. According to the calculation formula of the net irrigation water amount, the net irrigation water amount

Due to the circulation of water at the bends of the big river, the sediment transport rate is linked to time to simulate the formation and evolution of river beaches and ridges. In a relatively ideal state, the time-fractional differential equation is used to describe the movement process of quicksand during the formation of the river beach [12]. We give a numerical simulation. It is further organized into a form that is convenient for discussion:

Prove that _{k}^{1−a} − ^{1−a}, so there is
_{k}

According to formula (7), formula (8) and local truncation error calculation:

Use mathematical induction.

Let

Here _{1} = ^{a}^{2})

Assuming that when

The calculation of the water diversion from the Songhua River in this irrigation area can be based on the water resources balance analysis model [13]. We make the total water available to meet the total water demand in the irrigation area to achieve high-efficiency water saving. This can achieve the purpose of rational use of water resources. With the increase of the water volume of the Tisonghua River and the corresponding increase in the supply of groundwater, the water balance analysis result is a surplus of water. When the water volume of the Tisonghua River decreases, the groundwater replenishment will also decrease accordingly, and the water balance analysis result is that the water volume is insufficient. A conclusion can be drawn based on the above analysis. There is an optimal balance for the amount of water in the Jacques Songhua River. The water supply and water demand of this irrigation area are related to the water diversion volume of the Songhua River [14]. After the simulation analysis and calculation of specific data, this paper draws the relationship between the water supply, the water demand, and the Jacquard River's water volume (Figure 2).

According to the above theory and specific calculation and analysis, it can be seen that it is more reasonable and feasible to raise the Songhua River water volume to 83.21 million m^{3} in the farmland irrigation area. See Table 1 for detailed results after water resources balance in this irrigation area.

Water resources balance analysis results for the planning level year (2020) of farmland irrigation districts

^{3} |
^{3} |
^{3} |
|||
---|---|---|---|---|---|

Surface water | 8596 | 8595 | 8589 | −6.4 | |

Groundwater | 0.32 | 267.37 | 267.69 | 267.69 | 0 |

Total | 0.32 | 8863.1 | 8863.1 | 8856.73 | −6.4 |

The article analyzes and calculates the balance of water resources in farmland irrigation areas. The analysis and determination of the water volume of the irrigation area will have important guiding significance for the implementation of the continuous construction and water-saving transformation of the irrigation area. The article uses a water resource balance model to analyze the relationship between water supply and water demand in the region and find the best point of water balance. In this way, a reasonable amount of water supply can be determined so that the irrigation area can achieve the long-term goal of water-saving and energy-saving. The irrigation area is located in the arid and semiarid areas of China. It will be the next specific goal to analyze the water consumption balance of the irrigation area while achieving the balance of water supply and demand. Its research and application have put forward new requirements for the modern management of irrigation districts. This is also a new task that we will continue to study and realize in the future.

#### Water resources balance analysis results for the planning level year (2020) of farmland irrigation districts

^{3} |
^{3} |
^{3} |
|||
---|---|---|---|---|---|

Surface water | 8596 | 8595 | 8589 | −6.4 | |

Groundwater | 0.32 | 267.37 | 267.69 | 267.69 | 0 |

Total | 0.32 | 8863.1 | 8863.1 | 8856.73 | −6.4 |

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