# Analysis of Different Statistical Models in Probabilistic Joint Estimation of Porosity and Litho-Fluid Facies from Acoustic Impedance Values

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Well Log Data Application

#### 3.2. Post-Stack Data Application

## 4. Discussion and Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Non-parametric and Gaussian-mixture joint $\mathrm{p}\left(\mathrm{m},\mathrm{d}|\mathrm{f}\right)$ distributions (parts

**a**, and

**b**, respectively) estimated from five out of seven available wells drilled through the reservoir interval. In (

**a**,

**b**) from left to right we represent the joint distributions pertaining to shale, brine sand, and gas sand. For visualization purposes, the color scales are different for each facies.

**Figure 2.**Inversion results for Well A for a non-parametric $\mathrm{p}\left(\mathrm{m},\mathrm{d}|\mathrm{f}\right)$. (

**a**) Logged acoustic impedance. (

**b**) Posterior porosity distribution (colour scale), maximum-a-posteriori (MAP) solution (white line), and logged porosity values (black line). (

**c**) Posterior distribution for litho-fluid facies. (

**d**) Actual facies profile derived from well log information. (

**e**) MAP solution for the facies classification. In (

**d**,

**e**) blue, green, and red code shale, brine sand, and gas sand, respectively.

**Figure 3.**As in Figure 2, but for the Gaussian-mixture $\mathrm{p}(\mathrm{m},\mathrm{d}|\mathrm{f})$ distribution.

**Figure 4.**Reconstruction rate and recognition rate for Well A associated to the non-parametric and Gaussian-mixture distributions (parts a and b, respectively). In (

**a**) and (

**b**) Sh, Bs, and Gs, refer to shale, brine, sand and gas sand, respectively.

**Figure 5.**Direct comparison between posterior porosity distributions (continuous curves) and actual porosity values (vertical lines) extracted for given depth positions. (

**a**,

**b**) refer to the non-parametric and Gaussian-mixture $\mathrm{p}\left(\mathrm{m},\mathrm{d}|\mathrm{f}\right)$ distributions, respectively. In (

**a**,

**b**), the same colour is used for the same depth position.

**Figure 6.**Inversion results for Well B for a non-parametric $\mathrm{p}\left(\mathrm{m},\mathrm{d}|\mathrm{f}\right)$. (

**a**) Logged acoustic impedance. (

**b**) Posterior porosity distribution (color scale), MAP solution (white line), and logged porosity values (black line). (

**c**) Posterior distribution for litho-fluid facies. (

**d**) Actual facies profile derived from well log information. (

**e**) MAP solution for the facies classification. In (

**d**,

**e**) blue, green, and red code shale, brine sand and gas sand, respectively.

**Figure 7.**As in Figure 6, but for the Gaussian-mixture $\mathrm{p}(\mathrm{m},\mathrm{d}|\mathrm{f})$ distribution.

**Figure 8.**As in Figure 4, but for Well B.

**Figure 9.**As in Figure 5, but for Well B.

**Figure 10.**Gaussian joint $\mathrm{p}\left(\mathrm{m},\mathrm{d}\right)$ distributions estimated from five out of seven available wells drilled through the reservoir interval.

**Figure 11.**Inversion results obtained for a simple Gaussian model. (

**a**,

**b**) refer to Well A and Well B, respectively. In both parts the left column represents the actual Ip values, whereas the right column depicts the posterior porosity distribution (color scale), the MAP solution (white line), and the logged porosity values (black line).

**Figure 12.**Direct comparison (for Well A) between posterior porosity distributions (continuous curves) and the actual porosity values (vertical lines) resulting from the Gaussian $\mathrm{p}\left(\mathrm{m},\mathrm{d}\right)$ model. The same colour is used for the same depth location.

**Figure 13.**Inversion results for the synthetic post-stack seismic experiment on Well A for a non-parametric statistical model. (

**a**) Comparison between the observed stack trace (black line) and the predicted trace by the post-stack inversion (red line). (

**b**) Post-stack inversion results. The blue line illustrates the true Ip values (interpolated to the seismic sampling interval), the red line represents the MAP solution (${\mathsf{\mu}}_{d|s}$), whereas the green lines delimit the 95% confidence interval. (

**c**) Posterior porosity distribution (colour scale), MAP solution (white line), and logged porosity values interpolated to the seismic sampling interval (black line). (

**d**) Posterior distribution for the litho-fluid facies. (

**e**) Actual facies profile derived from well log information. (

**f**) MAP solution for the facies classification. In (

**d**,

**e**) blue, green, and red code shale, brine sand, and gas sand, respectively.

**Figure 14.**As in Figure 13, but for the Gaussian-mixture assumption.

**Figure 15.**Contingency analysis results for Well B and pertaining to the non-parametric and Gaussian-mixture distributions (parts

**a**and

**b**, respectively). In (

**a**,

**b**) Sh, Bs, and Gs, refer to shale, brine, sand, and gas sand, respectively.

**Figure 16.**As in Figure 13, but for Well B.

**Figure 17.**As in Figure 14, but for Well B.

**Figure 18.**Inversion results for the synthetic post-stack seismic experiments on Well A when a simple Gaussian model is considered. (

**a**) Comparison between the observed stack trace (black line) and the predicted trace by the post-stack inversion (red line). (

**b**) Post-stack inversion results. The blue line illustrates the true Ip values (interpolated to the seismic sampling interval), the red line represents the MAP solution (${\mathsf{\mu}}_{\mathrm{d}|\mathrm{s}}$), whereas the green lines delimit the 95% confidence interval. (

**c**) Posterior porosity distribution (colour scale), MAP solution (white line), and logged porosity values interpolated to the seismic sampling interval (black line).

Non-Parametric $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | Gaussian-Mixture $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | |
---|---|---|

Well A | 0.9296 | 0.8884 |

Well B | 0.9533 | 0.9195 |

**Table 2.**Linear correlation coefficients between the actual porosity profile and the MAP solutions provided by the non-parametric and the Gaussian-mixture models.

Non-Parametric $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | Gaussian-Mixture $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | |
---|---|---|

Well A | 0.9264 | 0.9024 |

Well B | 0.9012 | 0.8825 |

Well A | Well B |
---|---|

0.7788 | 0.8332 |

**Table 4.**Linear correlation coefficients between the actual porosity profile and the MAP solutions provided by the Gaussian model.

Well A | Well B |
---|---|

0.8454 | 0.8241 |

Non-Parametric $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | Gaussian-Mixture $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | |
---|---|---|

Well A | 0.7687 | 0.6331 |

Well B | 0.7388 | 0.7178 |

**Table 6.**Linear correlation coefficients between the actual porosity profile and the MAP solutions yielded by the non-parametric and the Gaussian-mixture models.

Non-Parametric $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | Gaussian-Mixture $\mathbf{p}\left(\mathbf{m},\mathbf{d}|\mathbf{f}\right)$ | |
---|---|---|

Well A | 0.8247 | 0.8011 |

Well B | 0.8436 | 0.8201 |

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**MDPI and ACS Style**

Aleardi, M. Analysis of Different Statistical Models in Probabilistic Joint Estimation of Porosity and Litho-Fluid Facies from Acoustic Impedance Values. *Geosciences* **2018**, *8*, 388.
https://doi.org/10.3390/geosciences8110388

**AMA Style**

Aleardi M. Analysis of Different Statistical Models in Probabilistic Joint Estimation of Porosity and Litho-Fluid Facies from Acoustic Impedance Values. *Geosciences*. 2018; 8(11):388.
https://doi.org/10.3390/geosciences8110388

**Chicago/Turabian Style**

Aleardi, Mattia. 2018. "Analysis of Different Statistical Models in Probabilistic Joint Estimation of Porosity and Litho-Fluid Facies from Acoustic Impedance Values" *Geosciences* 8, no. 11: 388.
https://doi.org/10.3390/geosciences8110388