# An Integrated Bayesian and Machine Learning Approach Application to Identification of Groundwater Contamination Source Parameters

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## Abstract

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## 1. Introduction

_{(D)}) [37]. For GCSP identification, the unknown parameters are both continuous (such as contamination source intensity) and discrete (such as contamination source location). However, many studies assume the contamination source location as a continuous variable [23,24,38,39]. To identify GCSPs more accurately and effectively, the DREAM

_{(D)}-MCMC approach, which can consider both discrete and continuous variables, is used for GCSP identification in this study.

## 2. Theoretical Framework

#### 2.1. Simulation Model

^{−1}); $h$ represents the hydraulic head (L); $c$ is the concentration of a contaminant dissolved in groundwater (ML

^{−3}); $t$ is time (T); $q$ is the volumetric flow rate per unit area of the aquifer representing fluid sources (positive) (LT

^{−1}); c

_{s}symbolizes the concentration of the source or sink (ML

^{−3}); $n$ denotes the porosity of the porous medium; $b$ symbolizes the aquifer thickness (L); ${D}_{ij}$ is the hydrodynamic dispersion tensor (L

^{2}T

^{−1}); and ${u}_{i}$ represents the actual flow velocity (LT

^{−1}). ${D}_{ij}$ and ${u}_{i}$ can be written as:

_{L}and ${\alpha}_{T}$ represent the longitudinal and transversal dispersivities (L), respectively; ${u}_{x}$ and ${u}_{y}$ are the components of the actual flow velocity (LT

^{−1}); and $\left|u\right|$ denotes the modulus of $u$, such that $\left|u\right|=\sqrt{{u}_{x}^{2}+{u}_{y}^{2}}$.

#### 2.2. Optimal Observation Well Location Design

#### 2.3. Parameter Identification

#### 2.3.1. Bayesian Inversion

#### 2.3.2. MCMC

_{(D)}), which can consider both discrete and continuous variables, is used to identify GCSPs. The DREAM

_{(D)}approach is not described in detail here; interested readers are referred to Vrugt and Ter Braak [37].

#### 2.4. Multi-Layer Perceptron

## 3. Numerical Applications

#### 3.1. Case studies

#### 3.1.1. Case 1

#### 3.1.2. Case 2

#### 3.2. Application of the Surrogate Model

#### 3.3. Optimal Observation Well Location Design for Case Studies

#### 3.4. Computational Time Analysis

## 4. Results and Discussion

#### 4.1. Analysis of the Surrogate Model

#### 4.2. Analysis of the Optimal Observation Well Locations

#### 4.3. Analysis of the Parameter Identification Results

_{(D)}algorithm; q is the number of Markov chains; B represents the variance of the average value of the q Markov chains; W denotes the average value of the intrachain variance of the q Markov chains. Generally, if the value of R is less than 1.2, it is considered that the Markov chain has attained a stable convergence state; that is, the sampling process of the algorithm has converged.

## 5. Conclusions

_{(D)}-MCMC approach.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Outputs and relative error of simulation model and surrogate model for Case 1 (the x-axis and y-axis represent the length and width of the flow field, respectively).

**Figure 5.**Output and relative error of simulation model and surrogate model for Case 2 (the x-axis and y-axis represent the length and width of the flow field, respectively).

**Figure 6.**The OWLs of the optimal design and 3 random designs for Case 1 (

**upper**) and Case 2 (

**lower**).

**Figure 9.**Comparison results of the posterior probability distributions for the optimal design and 3 other random designs for Case 1 (

**Upper**) and Case 2 (

**Lower**).

Parameters | Values | Unit |
---|---|---|

Hydraulic conductivity, K | 18.00 | LT^{−1} |

Porosity, n | 0.30 | - |

Longitudinal dispersivity, α_{L} | 12.00 | L |

Transverse dispersivity, α_{T} | 3.60 | L |

Parameters | True Values | Prior Ranges | Unit |
---|---|---|---|

S | 3600 | [2000, 5000] | MT^{−1} |

D | 480 | [450, 550] | T |

X | 11 | [10, 18] | L |

Y | 5 | [4, 9] | L |

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**MDPI and ACS Style**

An, Y.; Zhang, Y.; Yan, X. An Integrated Bayesian and Machine Learning Approach Application to Identification of Groundwater Contamination Source Parameters. *Water* **2022**, *14*, 2447.
https://doi.org/10.3390/w14152447

**AMA Style**

An Y, Zhang Y, Yan X. An Integrated Bayesian and Machine Learning Approach Application to Identification of Groundwater Contamination Source Parameters. *Water*. 2022; 14(15):2447.
https://doi.org/10.3390/w14152447

**Chicago/Turabian Style**

An, Yongkai, Yanxiang Zhang, and Xueman Yan. 2022. "An Integrated Bayesian and Machine Learning Approach Application to Identification of Groundwater Contamination Source Parameters" *Water* 14, no. 15: 2447.
https://doi.org/10.3390/w14152447