# Efficient Calibration of Groundwater Contaminant Transport Models Using Bayesian Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Objective Function

#### 2.2. Bayesian Optimization

Algorithm 1. Bayesian optimization for model calibration. |

01 Initialize sample set ${\mathcal{Q}}_{1}$ 02 $\mathrm{f}\mathrm{o}\mathrm{r}t=1,\cdots ,T-1$ 03 Update probabilistic surrogate model $p\left(f|{\mathcal{Q}}_{t},\Theta \right)$ with ${\mathcal{Q}}_{t}$ 04 Update acquisition function $\alpha \left(\Theta ;{\mathcal{Q}}_{t}\right)$ with $p\left(f|{\mathcal{Q}}_{t}\right)$ 05 Select the new model parameters ${\Theta}_{t}$ by maximizing the $\alpha \left(\Theta ;{\mathcal{Q}}_{t}\right)$: ${\Theta}_{t}\leftarrow {\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}_{\Theta}\alpha \left(\Theta ;{\mathcal{Q}}_{t}\right)$ 06 Compute simulation results $M\left(X,{\Theta}_{t}\right)$ 07 Compute the objective function $f\left({\Theta}_{t}\right)$ with $M\left(X,{\Theta}_{t}\right)$ 08 ${\mathcal{Q}}_{t+1}={\mathcal{Q}}_{t}\cup \{{\Theta}_{t},f\left({\Theta}_{t}\right)\}$ 09 end for 10 Output model parameters ${\Theta}^{*}$ with the smallest $f\left(\Theta \right)$ |

**Surrogate model.**Given the sampling results $\mathcal{Q}$ to the objective function, Gaussian process (GP) regression is generally used as the probabilistic surrogate model $p\left(f|\mathcal{Q},\Theta \right)$ for the computationally intensive objective function. The GP regression is flexible and scalable to model an arbitrary nonlinear function and requires a relatively small number of data examples. This is well-suited to model to objective function calibration. The GP extends the multivariate Gaussian distribution over vectors (finite-dimension space) to a distribution over functions (infinite-dimension space), for which any finite collection of dimensions follows a multivariate Gaussian distribution. The distribution of the Gaussian process is the joint distribution of infinite random variables. To surrogate the objective function $f\left(\Theta \right)$, the distribution of the Gaussian process for $f$ is defined by a mean function $m\left(\Theta \right)$ and a covariance function $k\left(\Theta ,\Theta \prime \right)$:

**Acquisition function.**Given the probabilistic surrogate model, as shown in Algorithm 1, the acquisition function is updated for optimization of model parameters and for determining the model parameters for the next sampling. The acquisition function considers the tradeoff between exploitation (i.e., sampling regions with small values of the objective function) and exploration (i.e., sampling regions with high uncertainty). The expected improvement (EI) criterion [61] is used as the acquisition function, which is indicated to be effective theoretically and empirically [62,63]. For our model calibration scenario, given known sampling results $\mathcal{Q}$, the EI function $\alpha \left(\Theta \right)$ is defined as the expected amount of improvement the new model parameters can yield:

#### 2.3. Numerical Model

**Contaminant transport model.**The fate and transport of a contaminant (hexavalent chromium) in 3D transient groundwater flow systems are modeled by the Fickian advection-dispersion transport equation with retardation [64]:

**Groundwater flow model.**The hydraulic head in the 3D groundwater flow system is modeled by the groundwater flow equation:

## 3. Study Area

## 4. Results

#### 4.1. Setup

#### 4.2. Hypothetical Case

#### 4.3. Real Case

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Variation of objective function values and the associated known minima during the BO process in hypothetical case.

**Figure 9.**Evolution of simulated concentrations (

**left column**), their differences from observed concentrations (

**middle column**), and scatters of simulated versus observed concentrations (

**right column**) during the BO process in hypothetical case.

**Figure 11.**Simulated concentrations (

**left column**), their differences from observed concentrations (

**middle column**), and scatters of simulated versus observed concentrations (

**right column**) obtained by removing the 13 monitoring wells in the north of the study area.

**Figure 12.**Comparisons of model calibration resulting from the calibration targets with different levels of Gaussian noise (Sigma/$\mathrm{m}\mathrm{g}{\mathrm{L}}^{-1}$: standard deviation of the Gaussian noise added to calibration targets). Simulated concentrations (

**left column**); differences from observed concentrations (

**middle column**); scatters of simulated concentrations versus observed concentrations (

**right column**) in hypothetical case.

**Figure 13.**Comparison of objective function values and the associated known minima during the BO and the PSO iteration processes in hypothetical case.

**Figure 14.**Simulated concentrations (

**left column**), the differences from observed concentrations (

**middle column**), and scatters of simulated versus observed concentrations (

**right column**) resulting from the PSO method after 200 objective function evaluations.

**Figure 16.**Evolution of simulated concentrations (

**left column**), their differences from observed concentrations (

**middle column**), and scatters of simulated versus observed concentrations (

**right column**) during the BO process in real case.

**Figure 17.**Simulated concentrations (

**left column**), their differences from observed concentrations (

**middle column**), and scatters of simulated versus observed concentrations (

**right column**) obtained by removing the 23 monitoring wells in the north of the study area.

**Figure 18.**Comparisons of model calibration resulting from the calibration targets with different levels of Gaussian noise (Sigma/$\mathrm{m}\mathrm{g}{\mathrm{L}}^{-1}$: standard deviation of the Gaussian noise added to calibration targets). Simulated concentrations (

**left column**); differences from observed concentrations (

**middle column**); scatters of simulated concentrations versus observed concentrations (

**right column**) in real case.

**Figure 19.**Comparison of objective function values and the associated known minima during the BO and the PSO iteration processes in real case.

**Figure 20.**Simulated concentrations (

**left column**), the differences from observed concentrations (

**middle column**), and scatters of simulated versus observed concentrations (

**right column**) resulting from PSO after 200 object function evaluations.

Layer | Thickness | Water Permeability |
---|---|---|

Mixed fill | 3~4 m | Strong permeability |

Silty clay | 2~4 m | Weak permeability |

Sandy gravel | 3~4 m | Medium permeability |

Weathered argillaceous siltstone | 9~10 m | Weak permeability |

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

Releasing water (m^{−1}) | 0.0$0\times {10}^{-3}$~$1.00\times {10}^{-3}$ |

Effective porosity | 0.10~0.60 |

Dispersity (m^{2} s^{−1}) | 0.0$0\times {10}^{-3}$~$1.00\times {10}^{-3}$ |

Retardation factor | 1.00~5.00 |

Horizontal permeability (m s^{−1}) | $6.90\times {10}^{-5}$~$8.30\times {10}^{-5}$ |

Vertical permeability (m s^{−1}) | $6.90\times {10}^{-7}$~$8.30\times {10}^{-7}$ |

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

Releasing water (m^{−1}) | 1.0 $0\times {10}^{-4}$ |

Effective porosity | 0.30 |

Dispersity (m^{2} s^{−1}) | 1.00 $\times {10}^{-4}$ |

Retardation factor | 2.00 |

Horizontal permeability (m s^{−1}) | 6.97 $\times {10}^{-10}$ |

Vertical permeability (m s^{−1}) | 4.37 $\times {10}^{-12}$ |

**Table 4.**Statistics of iterations and minima when a better solution of the objective function minimum is found.

Iteration | 1 | 3 | 5 | 8 | 26 | 74 | 150 |
---|---|---|---|---|---|---|---|

Minimum | 2284.01 | 383.84 | 297.55 | 191.93 | 34.18 | 28.09 | 1.31 |

**Table 5.**Comparison of the model parameters identified with partial data and the full data shown in Figure 7. The partial data, compared with the full data, only include observations at eight monitoring wells in the south.

Model Parameter | Partial Data | Full Data | Target Parameters |
---|---|---|---|

Releasing water (m^{−1}) | 9.00 $\times {10}^{-5}$ | 1.00 $\times {10}^{-4}$ | 1.0 $0\times {10}^{-4}$ |

Effective porosity | 0.31 | 0.28 | 0.30 |

Dispersity (m^{2} s^{−1}) | 1.10 $\times {10}^{-4}$ | 1.10 $\times {10}^{-4}$ | 1.00 $\times {10}^{-4}$ |

Retardation factor | 1.90 | 1.92 | 2.00 |

Horizontal permeability (m s^{−1}) | 3.58 $\times {10}^{-10}$ | 7.35 $\times {10}^{-10}$ | 6.97 $\times {10}^{-10}$ |

Vertical permeability (m s^{−1}) | 1.82 $\times {10}^{-12}$ | 6.07 $\times {10}^{-12}$ | 4.37 $\times {10}^{-12}$ |

Objective function value | 9.87 | 1.31 | - |

**Table 6.**Calibrated model parameters resulting from calibration targets with different levels of noise (sigma/$\mathrm{m}\mathrm{g}{\mathrm{L}}^{-1}$ : standard deviation of the Gaussian noise added to calibration targets).

Model Parameter | Sigma = 0.05 | Sigma = 0.10 | Sigma = 0.25 | Sigma = 0.50 | Target Parameters |
---|---|---|---|---|---|

Releasing water (m^{−1}) | 1.10 $\times {10}^{-4}$ | 1.0 $0\times {10}^{-4}$ | 1.10 $\times {10}^{-4}$ | 1.1 $0\times {10}^{-4}$ | 1.0 $0\times {10}^{-4}$ |

Effective porosity | 0.30 | 0.31 | 0.30 | 0.31 | 0.30 |

Dispersity (m^{2} s^{−1}) | 1.00 $\times {10}^{-4}$ | 1.20 $\times {10}^{-4}$ | 1.0 $0\times {10}^{-4}$ | 1.00 $\times {10}^{-4}$ | 1.00 $\times {10}^{-4}$ |

Retardation factor | 1.96 | 1.98 | 1.95 | 2.08 | 2.00 |

Horizontal permeability (m s^{−1}) | 5.71 $\times {10}^{-10}$ | 5.88 $\times {10}^{-10}$ | 7.84 $\times {10}^{-10}$ | 6.02 $\times {10}^{-10}$ | 6.97 $\times {10}^{-10}$ |

Vertical permeability (m s^{−1}) | 4.80 $\times {10}^{-12}$ | 6.06 $\times {10}^{-12}$ | 3.98 $\times {10}^{-12}$ | 4.87 $\times {10}^{-12}$ | 4.37 $\times {10}^{-12}$ |

Objective function value | 1.79 | 2.64 | 1.63 | 1.92 | - |

Iteration | 1 | 3 | 11 | 15 | 19 | 153 |
---|---|---|---|---|---|---|

Known minimum | 2975.10 | 1614.30 | 1470.65 | 210.01 | 46.03 | 1.58 |

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## Share and Cite

**MDPI and ACS Style**

Deng, H.; Zhou, S.; He, Y.; Lan, Z.; Zou, Y.; Mao, X.
Efficient Calibration of Groundwater Contaminant Transport Models Using Bayesian Optimization. *Toxics* **2023**, *11*, 438.
https://doi.org/10.3390/toxics11050438

**AMA Style**

Deng H, Zhou S, He Y, Lan Z, Zou Y, Mao X.
Efficient Calibration of Groundwater Contaminant Transport Models Using Bayesian Optimization. *Toxics*. 2023; 11(5):438.
https://doi.org/10.3390/toxics11050438

**Chicago/Turabian Style**

Deng, Hao, Shengfang Zhou, Yong He, Zeduo Lan, Yanhong Zou, and Xiancheng Mao.
2023. "Efficient Calibration of Groundwater Contaminant Transport Models Using Bayesian Optimization" *Toxics* 11, no. 5: 438.
https://doi.org/10.3390/toxics11050438