# Towards Improving the Efficiency of Bayesian Model Averaging Analysis for Flow in Porous Media via the Probabilistic Collocation Method

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Governing Equations of Groudnwater Flow System

#### 2.2. Bayesian Model Averaging Method

#### 2.3. Unconditional and Conditional Karhunen–Loeve Expansion Methods

#### 2.4. Polynomial Chaos Expansion Method

#### 2.5. Probabilistic Collocation Method

## 3. Results and Discussion

#### 3.1. Establishment of the Reference Model and Alternative Model Set

#### 3.2. Construction of PCM-Based Response Surface Model in BMA Multi-Model Analysis

**Y**random realizations need to be generated in order to ensure the convergence of MC simulation results [50]. Here, after the convergence test introduced in [50], it was found that it required at least 10,000 realizations for each model to ensure the MC result convergence. Thus, it requires at least 60,000 groundwater simulation models to be run in order to obtain a reasonably converged result for the BMA multi-model analysis with six postulated alternative models. Even for a simple groundwater model such as the two-dimensional synthetic model here, it is tedious to obtain the results. Therefore, it is necessary to construct a response surface model to improve the efficiency of the BMA multi-model analysis.

**Y**measurements, we need to use the conditional case. As mentioned in Section 2, the KLE to generate the conditional

**Y**realizations is based on eigenvalues and eigenfunctions in the polynomial expansion. The generated conditional

**Y**realization is mainly determined by the terms associated with the large eigenvalues. The computed eigenvalues for three calibrated variogram models were ranked from largest to smallest, and are depicted in Figure 6.

**Y**field, we need to select the proper polynomial order in PCE for hydraulic head. In this synthetic case, it can be shown that second order PCE (i.e., P = 2) was sufficient to approximate the hydraulic head. To obtain the collocations points, the roots of third-order univariate Hermite polynomial were used:

#### 3.3. Comparison Results

^{th}grid node. The comparison results are listed in Table 2. It can be found that the multi-model mean can be computed with the highest accuracy by using the PCM-based response surface model. This can be attributed to the fact that mean is a first-order statistical moment. For higher-order statistical moments, between-model variance had the largest error. However, the magnitude of the between-model variance was small relative to the within-model variance, and thus it did not significantly affect the total variance. The relative error was approximately 1%, which is acceptable. Overall, it is beneficial to use PCM-based response surface method to conduct BMA multi-model analysis.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Reference log hydraulic conductivity field and hydraulic head field distribution: (

**a**) the reference log hydraulic conductivity field; (

**b**) the reference hydraulic head field at the 20th time step. The black rectangle represents the location of the recharge window.

**Figure 4.**Sampling locations of log hydraulic conductivity and hydraulic head. Y denotes the log hydraulic conductivity sampling locations and H represents the hydraulic head monitoring well location.

**Figure 5.**Results of variogram model calibrations against sample variogram. Solid lines denote the calibrated variogram model curves, and black dots represent the sample variogram.

**Figure 10.**Comparison of model averaged within-model variance based on: (

**a**) MC method; (

**b**) PCM method.

**Figure 11.**Comparison of model averaged between-model variance based on: (

**a**) MC method; (

**b**) PCM method.

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

Discretization | |

Row | 40 |

Column | 40 |

Grid spacing | 1 |

Stress period | 1 |

Time step | 20 |

Reference geostatistical model | |

Type | TpvG |

A | 0.1 |

H | 0.25 |

λ_{u} | 25 |

Reference flow condition | |

Prescribed head on left boundary | 10 |

Prescribed head on right boundary | 5 |

Impervious upper and bottom boundaries | 0 |

Pumping rate | 5 |

Recharge rate | 0.01 |

Storage coefficient | 0.05 |

Porosity | 0.15 |

Sampling information | |

Number of lnK measurements | 10 |

Number of head measurements | 20 |

Measurement error | 1% of the observed head value |

Setting of multi-model analysis | |

Number of postulated models | 6 |

Exp0 | |

Exp1 | |

Labels of the postulated models | Gau0 |

Gau1 | |

Sph0 | |

Sph1 |

Statistics | MC-Based | PCM-Based | Relative Error |
---|---|---|---|

Multi-model mean | 7.3546 | 7.3536 | 0.013% |

Within-model variance | 0.1560 | 0.1585 | 1.586% |

Between-model variance | 0.0066 | 0.0061 | 8.274% |

Total variance | 0.1626 | 0.1645 | 1.182% |

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**MDPI and ACS Style**

Xue, L.; Dai, C.; Wu, Y.; Wang, L.
Towards Improving the Efficiency of Bayesian Model Averaging Analysis for Flow in Porous Media via the Probabilistic Collocation Method. *Water* **2018**, *10*, 412.
https://doi.org/10.3390/w10040412

**AMA Style**

Xue L, Dai C, Wu Y, Wang L.
Towards Improving the Efficiency of Bayesian Model Averaging Analysis for Flow in Porous Media via the Probabilistic Collocation Method. *Water*. 2018; 10(4):412.
https://doi.org/10.3390/w10040412

**Chicago/Turabian Style**

Xue, Liang, Cheng Dai, Yujuan Wu, and Lei Wang.
2018. "Towards Improving the Efficiency of Bayesian Model Averaging Analysis for Flow in Porous Media via the Probabilistic Collocation Method" *Water* 10, no. 4: 412.
https://doi.org/10.3390/w10040412