# A Mixed Integer Linear Programming Method for Optimizing Layout of Irrigated Pumping Well in Oasis

^{1}

^{2}

^{3}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Formulation

#### 2.1.1. The Form of the Objective Function

#### 2.1.2. The Form of Constraints

- Well drawdown during pumping is no more than the allowed maximum drawdown:$${s}_{i}<{h}_{\mathrm{max}}$$
- The well irrigation radius is no more than the maximum irrigation radius:$${X}_{i}\le {X}_{\mathrm{max}}$$
- The irrigable area of optimal wells must cover the irrigation area:$$\left\{{F}_{A}\right\}\subseteq \left\{{\displaystyle \sum _{i=0}^{{N}_{r}}{F}_{i}}\right\}$$
- Well flow distribution for the reserved well can be expressed as:$${Q}_{i}={q}_{0}\left({F}_{ib}+{F}_{ic}\right)$$
- The well influence radius is determined by the groundwater exploitable modulus method:$${y}_{i}={Q}_{i}\sqrt{DT\mathrm{t}/\epsilon}$$
- The well influence radius cannot exceed the well irrigation radius:$${y}_{i}\le {X}_{i}$$
- The distance between any two reserved wells is greater than the sum of the influence radius of the two wells:$${D}_{ij}\ge {y}_{i}+{y}_{j}$$

#### 2.2. Solution Methods

#### 2.2.1. Model Reformulation

#### 2.2.2. Model Solving Method

#### 2.3. Study Area

^{2}, of which the irrigation area is nearly 255.5 km

^{2}. In comparison with most other river catchments, the Cele Oasis area is a typical inland river catchment, situated between mountainous areas and among desert plains in an arid region. It is characterized by extremely low precipitation (39 mm/year), strong evaporation (2700 mm/year), and highly vulnerable ecosystems [32]. The Cele Oasis is primarily supplied with water from the Cele River, which originates from the Kunlun Mountains, flows through the Cele Oasis area, and finally discharges into the extremely arid Taklimakan Desert [33].

#### 2.4. Data

^{2}) equals 25, which means that the distance of the grid points is 500 meters. The related parameter was then put into Equation (8), and the results illustrated that Equation (5) was suitable for the Cele Oasis. The groundwater exploitable modulus was equal to the average annual allowable groundwater exploitation amount divided by the irrigation area.

## 3. Results and Discussions

#### 3.1. Model Optimization Results with Different Objective

^{3}/h and 95 m

^{3}/h, respectively, which are higher than the existing condition, and they are both under the well pumping capacity limitation, which illustrates that distributions are both feasible for the Cele Oasis. In comparison with the number of existing wells, the optimal well number of ICO is 134, a reduction of 52.89%, and the optimal well number of ECO is 216, a reduction of 10.74%. The results of two optimizations obviously illustrate that depreciation cost causes a huge difference in optimal wells number due to its weight in the objective function.

#### 3.2. Optimal Well Spatial Layouts

#### 3.3. Influences of Different Objective

^{3}) distribution. Figure 7 shows the related parameter distribution of ECO in Cele Oasis: (1) Figure 7a shows the pumping drawdown distribution, (2) Figure 7b shows the rated pumping flow distribution, (3) Figure 7c shows the single well electricity cost distribution, (4) Figure 7d shows the unit power consumption (kW·h/m

^{3}) distribution. As shown in Figure 6a and Figure 7a, the trend is opposite from the water table distribution in Cele Oasis, which illustrates that the optimal results based on the minimum irrigation cost also have a role in curbing the groundwater decline. The average well drawdown of ICO is obviously higher than that of ECO due to the reduction of pumping wells. Figure 6b and Figure 7b have the same trend as the drawdown distribution. As shown in Figure 6c and Figure 7c, the single well electricity cost is primarily influenced by the groundwater depth and pumping flow. Moreover, the distribution is smoother, and the costly area of well electricity cost is smaller than the deepest area of the water table. As shown in Figure 6d and Figure 7d, the unit power consumption (kW·h/m

^{3}) distribution is primarily affected by groundwater depth and drawdown. Due to the higher average drawdown of ICO, the unit power consumption of ICO is also higher than that of the same area of ECO, and the groundwater depth is obviously the primary indicator of this.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Layout of existing wells and groundwater depths in the Cele Oasis as (

**a**) well layout in the Cele Oasis; (

**b**) groundwater depth contours in the Cele Oasis.

**Figure 4.**Well layouts of implicit cost optimization (ICO) in the Cele Oasis: (

**a**) well optimization layout, (

**b**) well optimization irrigation radius layout, (

**c**) well optimization influence radius layout, and (

**d**) recommended closed well layout.

**Figure 5.**Well layout of explicit cost optimization (ECO) in the Cele Oasis: (

**a**) well optimization layout, (

**b**) well optimization irrigation radius layout, (

**c**) well optimization influence radius layout, and (

**d**) recommended closed well layout.

**Figure 6.**Well related parameter comparisons of ICO in the Cele Oasis: (

**a**) the pumping drawdown distribution, (

**b**) the well pumping flow distribution, (

**c**) the electricity cost (CNY/a) distribution, and (

**d**) the unit power consumption (kW·h/m

^{3}) distribution.

**Figure 7.**Well related parameter comparisons of ECO in the Cele Oasis as (

**a**) the pumping drawdown distribution, (

**b**) the well pumping flow distribution, (

**c**) the electricity cost (CNY/a) distribution, (

**d**) the unit power consumption (kW·h/m

^{3}) distribution.

Crop | Irrigation Area (hm^{2}) | Proportion of Total Area (%) | Net Irrigation Water Demand during Irrigation Period (m^{3}/hm^{2}) |
---|---|---|---|

Jujube | 15177 | 59.4 | 7695 |

Walnut | 5877 | 23 | 5190 |

Pomegranate | 1456 | 5.7 | 6000 |

Cotton | 1048 | 4.1 | 4995 |

Other crop | 1993 | 7.8 | 5250 |

Total | 25,550 | 100 | 6721 |

Parameter | Value | Parameter | Value |
---|---|---|---|

W_{im} (CNY) | 500 | t (h) | 15 |

W_{w} (CNY) | 60,000 | r (m) | 0.3 |

n_{y} | 15 | T (m^{2}/day) | 866 |

W_{id} (CNY) | 4000 | K (m/day) | 45 |

p (CNY/kw-h) | 0.2 | D_{a} (m) | 19.7 |

D (days) | 100 | S | 0.005 |

m | 0.1831 | ε (m^{3}/(km^{2}·a)) | 361,722 |

n | 0.02 | q_{0} (m^{3}/(h·hm^{2})) | 0.8 |

X_{max} (m) | 1500 | F_{j} (hm^{2}) | 25 |

Q_{max} (m^{3}/h) | 230 | F_{A} (hm^{2}) | 25,550 |

Implicit Cost Optimization | Implicit Cost without Optimization | Explicit Cost Optimization | Explicit Cost without Optimization | |
---|---|---|---|---|

Total Cost of Irrigation (CNY) | 2,822,498 | 3,056,122 | 2,130,534 | 2,209,122 |

Electricity Cost (CNY) | 2,366,498 | 2,088,122 | 2,022,534 | 2,088,122 |

Total Cost Difference (%) | 7.64% | - | 3.56% | - |

Average Well Flow (m^{3}/h) | 179 | 84 | 95 | 84 |

Existed well number | 242 | 242 | 242 | 242 |

Optimized well number | 114 | - | 216 | - |

Well Number Difference (%) | 52.89% | - | 10.74% | - |

^{3}/h) is average value of reserved well pumping flow.

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**MDPI and ACS Style**

Ma, T.; Wang, J.; Liu, Y.; Sun, H.; Gui, D.; Xue, J. A Mixed Integer Linear Programming Method for Optimizing Layout of Irrigated Pumping Well in Oasis. *Water* **2019**, *11*, 1185.
https://doi.org/10.3390/w11061185

**AMA Style**

Ma T, Wang J, Liu Y, Sun H, Gui D, Xue J. A Mixed Integer Linear Programming Method for Optimizing Layout of Irrigated Pumping Well in Oasis. *Water*. 2019; 11(6):1185.
https://doi.org/10.3390/w11061185

**Chicago/Turabian Style**

Ma, Teng, Jinwen Wang, Yi Liu, Huaiwei Sun, Dongwei Gui, and Jie Xue. 2019. "A Mixed Integer Linear Programming Method for Optimizing Layout of Irrigated Pumping Well in Oasis" *Water* 11, no. 6: 1185.
https://doi.org/10.3390/w11061185