# Calculation Model of Multi-Well Siphoning and Its Feasibility Analysis of Discharging the Groundwater in Soft Soil

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−8}m/s and the well spacing is 2 m, the decrease in the water level is approximately 9.72 m after 100 days of drainage. It is feasible to apply siphon drainage technology to discharge the groundwater in soft soil.

## 1. Introduction

## 2. Method

#### 2.1. Equivalent Conversion

_{e}, and the well spacing, l, is as follows [26,27]:

#### 2.2. Solution of the Numerical Model

_{0}, and the equivalent radius is r

_{e}. The initial water head is at the top of the aquifer. That is, the initial groundwater level is H

_{0}. The radius of the drainage well is r

_{w}. The water level of the drainage well is fixed at H

_{w}. The bottom of the model is an impervious boundary. The soft clay layer is regarded as a phreatic aquifer. The following basic assumptions are made: (1) The phreatic aquifer is homogeneous, isotropic, uniformly thick, and horizontally distributed. (2) The seepage meets Darcy’s law and Dupuit’s assumption. (3) The cross-flow phenomenon, external infiltration, and water evaporation are not considered. (4) The well is fully penetrated, and the water level of the drainage well is fixed. (5) The water flow at the equivalent radius boundary does not allow seepage in the horizontal direction. (6) The groundwater release caused by the decrease of the water head is completed instantaneously. Based on the above assumptions, an equation of spatial, two-dimensional, axisymmetric groundwater seepage movement can be established [19] as follows:

_{m}/μ, K is the permeability coefficient, h

_{m}is the average thickness of the aquifer, μ is the water supplied by gravity, and φ is the well flow potential function of the phreatic aquifer, φ = H

^{2}/2.

^{i}

_{j}

_{+1}, φ

^{i}

_{j}, and φ

^{i}

_{j}

_{−1}. According to Equations (4) and (5), the calculation formula of the discrete points on the boundary can be obtained as follows:

^{i}

_{N}

_{−1}and φ

^{i}

_{N}

_{−2}can be expanded at φ

^{i}

_{N}by using Taylor’s formula:

^{i}

_{N}can be obtained:

## 3. Validation of the Numerical Model

#### 3.1. The Model Test

^{−4}, 7.9 × 10

^{−4}, and 7.8 × 10

^{−4}m/d. It is estimated that the average value of the permeability coefficient of the soil was approximately 7.93 × 10

^{−4}m/d.

#### 3.2. The Field Test

^{−9}m/s. The parameters used in the calculation of the numerical model are shown in Table 3.

## 4. Feasibility Analysis

^{−8}m/s. The radius of the drainage well is 0.03 m. The water level in the well is fixed at 10 m. The model is calculated with different well spacings. The changes in “minimum drop” (solid line) and “average drop” (dotted line) are shown in Figure 12. Figure 12a shows that both “minimum drop” and “average drop” increased gradually with the increase of time. Initially, the rate of change was fast, but it gradually became slower and then remained stable. Figure 12b shows that both “minimum drop” and “average drop” increased gradually as the well spacing decreased. A smaller well spacing simultaneously results in a greater drop in the level of the groundwater. It can be analyzed from the difference between “minimum drop” and “average drop” that the difference decreased gradually as the well spacing decreased. It is evident that when the well spacing was smaller, the “falling funnel” formed in the drainage process was smoother, resulting in a significant decrease in the level of the groundwater at all locations in the soil. For analysis, take the decrease in the level of the groundwater following a period of 100 days of drainage. When the well spacing was 8 m, the “minimum drop” was 0.89 m, and the “average drop” was 1.91 m. When the well spacing was reduced to 4 m, the “minimum drop” was 4.44 m, and the “average drop” was 5.21 m. When the well spacing was further reduced to 2 m, the “minimum drop” increased to 9.72 m, and the “average drop” increased to 9.77 m. It was about twice as big at 4 m and about nine times as big at 8 m, i.e., approximately equal to the limit drop. This shows that reducing the well spacing effectively can reduce the groundwater level in the soil.

^{−9}–10

^{−7}m/s. The well spacing is assumed to be 2 m, and the other parameters are unchanged. The model is calculated with different permeability coefficients within the value range. The changes in “minimum drop” (solid line) and “average drop” (dotted line) are shown in Figure 13. It can be seen from Figure 13a that both “minimum drop” and “average drop” gradually increase with the increase in time. Initially, the rate of change was fast, but it gradually became slower and then remained stable. This shows that, after a certain period of siphon drainage, the groundwater level can be reduced to the control height for soils with relatively high permeability. Although it ultimately cannot reduce the groundwater level of low permeability soils to the limit height, it still decreases this level to a certain extent. Figure 13b shows that, in the initial stage of drainage, both “minimum drop” and “average drop” increased uniformly with the increase in the permeability coefficient. In the late stage of the drainage, the two decreases in the groundwater level increased rapidly as the permeability coefficient increased, and then they stabilized at the limit drop value. This indicates that the greater the permeability coefficient becomes, the greater the decrease in the water level will be. However, due to the limitation of the siphon, the groundwater level will remain unchanged after it drops to the limited value. Figure 9 shows that the “minimum drop” and the “average drop” gradually approached each other in size as the permeability coefficient increased. This indicates that the greater the permeability coefficient, the more the groundwater level will decrease at all locations in the soil. Consider the decrease in the groundwater level for analysis with 100 days of drainage time. When the permeability coefficient was 6 × 10

^{−9}m/s, the “minimum drop” was 8.58 m, and the “average drop” was 8.84 m. Obviously, the groundwater level had decreased. When the permeability coefficient was 2 × 10

^{−8}m/s, both the “minimum drop” and the “average drop” were 10 m, which had reached the limit value. This shows that the siphon technology can provide a good drainage effect in soft soils with different permeability coefficients.

## 5. Discussion

^{−3}d. The equivalent radius of the numerical model is assumed to be 1.05 m. The water level drawdown at 0.35 m from the center of the drainage hole is selected for comparative analysis (Figure 14). It can be seen from the figure that when the drainage time is 0 to 30 days, the numerical model and the Jacob model basically provide identical results. When the drainage time exceeds 30 days, the deviation between the numerical model and the Jacob model begins to appear. The difference gradually increases with the increase in time. When the drainage time is 100 days, the decrease in the water level of the numerical model is 3.98 m, and it is 3.09 m in the Jacob model. Compared with the Jacob model, the numerical model shows an increase of 28.8%. This is because the Jacob model considers single-well drainage, and the numerical model in this paper considers multi-well drainage. At the initial stage of drainage, the impact of multi-well drainage is small, and there is no significant difference between single-well drainage and multi-well drainage. However, with the progress of drainage, the impact of multi-well drainage increases, and the increase of the drop of the water level of multi-well drainage is greater than that of single-well drainage. When the drainage duration is 100 days, the numerical model increases by 28.8% compared with the Jacob model.

## 6. Conclusions

- (1)
- When the soil is homogeneous and isotropic, the seepage field between the two drainage wells is distributed symmetrically. The wells can be divided into identical single wells so that the water level change in multi-well siphoning can be calculated. Comparing the numerical model with the model test and the field test, the relative error was within 10%.
- (2)
- It is feasible to apply siphon drainage technology to discharge the groundwater in soft soil. The decrease in the water level increases as the well spacing decreases or the permeability coefficient increases. When the soft soil permeability coefficient is 1 × 10
^{−8}m/s and the well spacing is 2 m, the decrease in the water level can reach 9.72 m after 100 days of drainage. - (3)
- Both Jacob’s model and the numerical model in this paper assume that the water level in the drainage well is fixed. However, the Jacob model considers single-well drainage, and the numerical model in this paper considers multi-well drainage. The two models basically are identical at the initial stage of drainage, after which the deviation begins to appear. The difference increases gradually with time.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Division method under different drainage well layouts: (

**a**) equilateral triangle, (

**b**) square.

**Figure 7.**Comparison of the numerical model (curves) and the model test (dots): (

**a**) variation of the decrease in the water level with distance, (

**b**) variation of the decrease in the water level with time.

**Figure 12.**Variation of “minimum drop” (solid line) and “average drop” (dotted line) under different well spacings: (

**a**) variation of the decrease of the water level with time, (

**b**) variation of the decrease of the water level with well spacing.

**Figure 13.**Variation of “minimum drop” (solid line) and “average drop” (dotted line) under different permeability coefficients: (

**a**) variation of water level drop with time, (

**b**) variation of water level drop with permeability coefficient.

**Figure 14.**Comparison of the decrease in the water level between the numerical model and the Jacob model.

r_{e} | r_{w} | H_{0} | H_{w} | K | μ | a | dr | dt |
---|---|---|---|---|---|---|---|---|

0.16 m | 0.033 m | 0.7 m | 0.15 m | 7.93 × 10^{−4} m/d | 0.2 | 2.8 × 10^{−3} m^{2}/d | 0.01 m | 1 × 10^{−3} d |

Layers | Dredger Soil | Mucky Clay | Mucky Clay |
---|---|---|---|

Thickness | 10.2 m | 6.1 m | 9.4 m |

Water content | 47.2% | 35.8% | 42.4% |

Specific gravity | 2.74 | 2.72 | 2.74 |

Void ratio | 1.318 | 1.052 | 1.204 |

Liquid limit | 40.2% | 34.5% | 40.2% |

Plastic limit | 22.2% | 20.6% | 22.2% |

Horizontal permeability coefficient | 5.32 × 10^{−9} m/s | 6.24 × 10^{−9} m/s | 3.97 × 10^{−9} m/s |

Vertical permeability coefficient | 6.95 × 10^{−9} m/s | 8.12 × 10^{−9} m/s | 5.10 × 10^{−9} m/s |

Compressive modulus | 2.89 MPa | 2.57 MPa | 2.42 MPa |

r_{e} | r_{w} | H_{0} | H_{w} | K | μ | a | dr | dt |
---|---|---|---|---|---|---|---|---|

0.51 m | 0.04 m | 18 m | 8 m | 5.31 × 10^{−3} m/d | 0.2 | 4.78 × 10^{−2} m^{2}/d | 0.01 m | 1 × 10^{−4} d |

r_{w} | H_{0} | H_{w} | K | μ | a |
---|---|---|---|---|---|

0.05 m | 15 m | 5 m | 8.64 × 10^{−4} m/d | 0.2 | 6.48 × 10^{−3} m^{2}/d |

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**MDPI and ACS Style**

Shen, Q.; Wu, C.; Wang, J.; Yuan, S.; Shang, Y.; Sun, H.
Calculation Model of Multi-Well Siphoning and Its Feasibility Analysis of Discharging the Groundwater in Soft Soil. *Water* **2023**, *15*, 1319.
https://doi.org/10.3390/w15071319

**AMA Style**

Shen Q, Wu C, Wang J, Yuan S, Shang Y, Sun H.
Calculation Model of Multi-Well Siphoning and Its Feasibility Analysis of Discharging the Groundwater in Soft Soil. *Water*. 2023; 15(7):1319.
https://doi.org/10.3390/w15071319

**Chicago/Turabian Style**

Shen, Qingsong, Chaofeng Wu, Jun Wang, Shuai Yuan, Yuequan Shang, and Hongyue Sun.
2023. "Calculation Model of Multi-Well Siphoning and Its Feasibility Analysis of Discharging the Groundwater in Soft Soil" *Water* 15, no. 7: 1319.
https://doi.org/10.3390/w15071319