# Uncertainty Analysis of Numerical Simulation of Seawater Intrusion Using Deep Learning-Based Surrogate Model

^{1}

^{2}

^{*}

## Abstract

**:**

^{−}concentration and the area of seawater intrusion were conducted at typical observation wells. The work that is here provides a reliable reference for decision making in the area.

## 1. Introduction

## 2. Factors Influencing Uncertainty in Numerical Models

## 3. Methods of Surrogate Model-Artificial Intelligence-Based Deep Learning

- (1)
- Normalizing the target:

- (2)
- Using DBNN to replace the setting of the model parameters:
- ①
- Parameter Initialization

- ②
- Numbers of Hidden Layers

- ③
- Determining the Number of odes in the Hidden Layers

- (3)
- Unsupervised learning:

- (4)
- Fine-tuning the learning process:

- (5)
- Reverse data normalization:

- (1)
- Principle of forward conduction:

- (2)
- Principle of back-propagation:

## 4. Establishing a Seawater Intrusion Simulation Model

^{2}.

^{3}per year, and the evaporation is about 1200 mm per year. The model consisted of models of flow and water quality [32]). It included partial differential equations and definite solution conditions, where the two were coupled based on the equations of motion. The model was solved by the SEAWAT program [33]. The measured data were substituted into the model for calculation, and the combination of the parameters was adjusted to satisfy the accuracy-related requirements. The inputs for the model included the sea level and the intensity of exploitation of groundwater, and the outputs consisted of the area of seawater intrusion and the Cl

^{−}concentration in the typical observation wells. The concentrations of Cl

^{−}in five observation wells and the area of seawater intrusion (Cl

^{−}> 250 mg/L) in the study area were selected as the outputs to analyze the impact of uncertainty in the rise in sea levels on seawater intrusion. Figure 7 shows the results of the model.

## 5. Establishment of the Surrogate Model

## 6. Results and Discussion

^{−}and areas of seawater intrusion in the five observation wells were calculated. The outputs (Cl−concentration) of each well were then statistically analyzed.

^{2}, which was close to the result that was predicted by the deterministic model (68.5 km

^{2}). The standard deviation of the area of the seawater intrusion was 7.31 km

^{2}, which was lower than the standard deviation of the chloride ion concentration in each well. This indicated that the randomness of the rise in sea level had a smaller impact on the area of the seawater intrusion in the study area than it did on the chloride ion concentration in the wells in the area.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Range of values and characteristics of the distribution of input variables to the surrogate model.

Variable Name | Sea Level Rise (mm) | Pumping Capacity of Well Group 1 (10 ^{4} m^{3}/a) | Pumping Capacity of Well Group 2 (10 ^{4} m^{3}/a) | Pumping Capacity of Well Group 3 (10 ^{4} m^{3}/a) |
---|---|---|---|---|

Distribution characteristics | Random variable (normal distribution) | Deterministic variable | ||

Value range | 80.00–170.00 | 273.00 | 161.00 | 230.00 |

Name | DBNN | RBF | DCNN |
---|---|---|---|

Max relative error (%) | 3.961 | 6.720 | 4.114 |

Mean relative error (%) | 1.658 | 4.009 | 4.013 |

Root mean-squared error | 3.707 | 8.162 | 5.179 |

Coefficient of determination | 0.989 | 0.804 | 0.920 |

Wells | ObW-1 | ObW-2 | ObW-3 | ObW-4 | ObW-5 | |
---|---|---|---|---|---|---|

Value (mg/L) | ||||||

Max. value | 332.92 | 364.44 | 419.71 | 278.02 | 254.00 | |

Min. value | 166.04 | 151.38 | 89.57 | 104.02 | 11.96 | |

Average value | 250.63 | 251.65 | 253.95 | 183.26 | 73.38 | |

Standard deviation | 28.29 | 37.77 | 66.45 | 37.49 | 43.90 |

Well Name | ObW-1 | ObW-2 | ObW-3 | ObW-4 | ObW-5 |
---|---|---|---|---|---|

Risk of seawater intrusion (Cl^{−} > 250 mg/L) | 52.00% | 49.50% | 58.00% | 5.50% | 1.00% |

Well | Confidence Level (%) | Confidence Interval (mg/L) | Confidence Level (%) | Confidence Interval (mg/L) |
---|---|---|---|---|

ObW-1 | 80 | 211.55−284.77 | 50 | 232.42−270.73 |

ObW-2 | 80 | 205.65−302.82 | 50 | 225.33−279.90 |

ObW-3 | 80 | 167.26−346.96 | 50 | 200.65−295.49 |

ObW-4 | 80 | 132.79−233.51 | 50 | 157.49−210.38 |

ObW-5 | 80 | 30.71−121.56 | 50 | 45.98−90.67 |

Median km ^{2} | Standard Deviation km ^{2} | Average km ^{2} | Coefficient of Variation | Confidence Interval (km^{2}) | |
---|---|---|---|---|---|

80% | 50% | ||||

70.27 | 7.31 | 69.39 | 10.40 | 68.01−72.55 | 69.16−71.41 |

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**MDPI and ACS Style**

Miao, T.; Huang, H.; Guo, J.; Li, G.; Zhang, Y.; Chen, N.
Uncertainty Analysis of Numerical Simulation of Seawater Intrusion Using Deep Learning-Based Surrogate Model. *Water* **2022**, *14*, 2933.
https://doi.org/10.3390/w14182933

**AMA Style**

Miao T, Huang H, Guo J, Li G, Zhang Y, Chen N.
Uncertainty Analysis of Numerical Simulation of Seawater Intrusion Using Deep Learning-Based Surrogate Model. *Water*. 2022; 14(18):2933.
https://doi.org/10.3390/w14182933

**Chicago/Turabian Style**

Miao, Tiansheng, He Huang, Jiayuan Guo, Guanghua Li, Yu Zhang, and Naijia Chen.
2022. "Uncertainty Analysis of Numerical Simulation of Seawater Intrusion Using Deep Learning-Based Surrogate Model" *Water* 14, no. 18: 2933.
https://doi.org/10.3390/w14182933