# Application of Spectral Clustering Algorithm to ES-MDA with DCT for History Matching of Gas Channel Reservoirs

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. ES-MDA

#### 2.2. DCT

#### 2.3. PFR

#### 2.4. SCA

#### 2.5. Overall Workflow for History Matching

## 3. Results and Discussion

#### 3.1. Facies Distribution

#### 3.2. Gas and Water Productions

^{4}for comparison. The average of the objective functions of the PFR case is more than double that of the SCA case. Although the 13 realizations had more than 106 errors in Figure 12a, there were only two realizations in Figure 12b. This explains the reason why the average value of the PFR case is larger than that of the SCA case. From these results, we can verify quantitatively that SCA achieved better history matching performance than the PFR.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) One realization example of the discrete cosine transform (DCT) principle and application to history matching procedure and (

**b**) sensitivity analysis of the DCT coefficients.

**Figure 2.**One realization example of the preservation of facies ratio (PFR) principle and application to an updated facies index field.

**Figure 5.**Facies distributions of each cluster and cluster combination with average errors of dynamic data.

**Figure 10.**Comparison between final updated facies distribution using PFR and SCA: Averages of one hundred realizations of (

**a**) initial models, updated models by (

**b**) PFR and (

**c**) SCA in the first row, and four example realizations below the first row.

**Figure 11.**Gas and water well production rates of initial ensemble and updated ensembles using PFR and SCA.

Clusters 1–7 | Average Index Values | Assigned Class |
---|---|---|

6 | 1.9419 | Cluster A |

7 | 1.7291 | Cluster B |

1 | 1.5202 | Cluster C |

3 | 1.3482 | Cluster D |

5 | 1.1916 | Cluster E |

2 | 1.0533 | Cluster F |

4 | 0.9495 | Cluster G |

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

Reservoir grid system | 75 by 75 by 1 |

Well locations, grid coordinate (from the upper left) Well numbering P1–P16 | (14, 14), (30, 14), (46, 14), (62, 14), (14, 30), (30, 30), (46, 30), (62, 30), (14, 46), (30, 46), (46, 46), (62, 46), (14, 62), (30, 62), (46, 62), (62, 62) |

Observed data types | Well gas production rate, Well bottomhole pressure |

Porosity, fraction | 0.2 (sandstone), 0.1 (shale) |

Permeability, md | 100 (sandstone), 1 (shale) |

Relative permeability, modified Brooks-Corey relation | Connate water saturation = 0.25 Connate gas saturation = 0.2 Sandstone (nw = 3, ng = 2) Shale (nw = 5, ng = 4) |

Initial water saturation, fraction | 0.25 |

Initial reservoir pressure, psia | 3000 |

Bottomhole pressure limit, psia | 1000 |

The number of DCT coefficients used in a reservoir | 465 (8% of total) |

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**MDPI and ACS Style**

Kim, S.; Lee, K.
Application of Spectral Clustering Algorithm to ES-MDA with DCT for History Matching of Gas Channel Reservoirs. *Energies* **2019**, *12*, 4394.
https://doi.org/10.3390/en12224394

**AMA Style**

Kim S, Lee K.
Application of Spectral Clustering Algorithm to ES-MDA with DCT for History Matching of Gas Channel Reservoirs. *Energies*. 2019; 12(22):4394.
https://doi.org/10.3390/en12224394

**Chicago/Turabian Style**

Kim, Sungil, and Kyungbook Lee.
2019. "Application of Spectral Clustering Algorithm to ES-MDA with DCT for History Matching of Gas Channel Reservoirs" *Energies* 12, no. 22: 4394.
https://doi.org/10.3390/en12224394