# 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

- Duijndam, A.J.W. Bayesian estimation in seismic inversion. Part I: Principles. Geophys. Prospect.
**1988**, 36, 878–898. [Google Scholar] [CrossRef] - Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation; Siam: Philadelphia, PA, USA, 2005. [Google Scholar]
- Avseth, P.; Mukerji, T.; Mavko, G. Quantitative Seismic Interpretation; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Doyen, P.M. Seismic Reservoir Characterization: An Earth Modelling Perspective; EAGE: Houten, The Netherlands, 2007. [Google Scholar]
- Bosch, M.; Mukerji, T.; Gonzalez, E.F. Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: A review. Geophysics
**2010**, 75, 75A165–75A176. [Google Scholar] [CrossRef] - Aleardi, M.; Ciabarri, F. Assessment of different approaches to rock-physics modeling: A case study from offshore Nile Delta. Geophysics
**2017**, 82, MR15–MR25. [Google Scholar] [CrossRef] - Mavko, G.; Mukerji, T. A rock physics strategy for quantifying uncertainty in common hydrocarbon indicators. Geophysics
**1998**, 63, 1997–2008. [Google Scholar] [CrossRef] - Bachrach, R. Joint estimation of porosity and saturation using stochastic rock-physics modeling. Geophysics
**2006**, 71, O53–O63. [Google Scholar] [CrossRef] - Gunning, J.; Glinsky, M.E. Detection of reservoir quality using Bayesian seismic inversion. Geophysics
**2007**, 72, R37–R49. [Google Scholar] [CrossRef] - Bosch, M.; Cara, L.; Rodrigues, J.; Navarro, A.; Díaz, M. A Monte Carlo approach to the joint estimation of reservoir and elastic parameters from seismic amplitudes. Geophysics
**2007**, 72, O29–O39. [Google Scholar] [CrossRef] - Rimstad, K.; Avseth, P.; Omre, H. Hierarchical Bayesian lithology/fluid prediction: A North Sea case study. Geophysics
**2012**, 77, B69–B85. [Google Scholar] [CrossRef] - Aleardi, M.; Ciabarri, F.; Mazzotti, A. Probabilistic estimation of reservoir properties by means of wide-angle AVA inversion and a petrophysical reformulation of the Zoeppritz equations. J. Appl. Geophys.
**2017**, 147, 28–41. [Google Scholar] [CrossRef] - Grana, D.; Bronston, M. Probabilistic formulation of AVO modeling and AVO-attribute-based facies classification using well logs. Geophysics
**2015**, 80, D343–D354. [Google Scholar] [CrossRef] - Mazzotti, A.; Zamboni, E. Petrophysical inversion of AVA data. Geophys. Prospect.
**2003**, 51, 517–530. [Google Scholar] [CrossRef] - Eidsvik, J.; Avseth, P.; Omre, H.; Mukerji, T.; Mavko, G. Stochastic reservoir characterization using prestack seismic data. Geophysics
**2004**, 69, 978–993. [Google Scholar] [CrossRef] [Green Version] - Kemper, M.; Gunning, J. Rock Physics Driven Joint Inversion to Facies and Reservoir Properties. ASEG Extended Abstr.
**2012**, 2012, 1–4. [Google Scholar] [CrossRef] [Green Version] - Gunning, J.S.; Kemper, M.; Pelham, A. Obstacles, challenges and strategies for facies estimation in AVO seismic inversion. In Proceedings of the 76th EAGE Conference and Exhibition, Amsterdam, The Netherlands, 16–19 June 2014. [Google Scholar] [CrossRef]
- De Figueiredo, L.P.; Grana, D.; Santos, M.; Figueiredo, W.; Roisenberg, M.; Neto, G.S. Bayesian seismic inversion based on rock-physics prior modeling for the joint estimation of acoustic impedance, porosity and lithofacies. J. Comput. Phys.
**2017**, 336, 128–142. [Google Scholar] [CrossRef] - Gunning, J.; Sams, M. Joint facies and rock properties Bayesian amplitude-versus-offset inversion using Markov random fields. Geophys. Prospect.
**2018**, 66, 904–919. [Google Scholar] [CrossRef] - Rimstad, K.; Omre, H. Impact of rock-physics depth trends and Markov random fields on hierarchical Bayesian lithology/fluid prediction. Geophysics
**2010**, 75, R93–R108. [Google Scholar] [CrossRef] - Grana, D.; Fjeldstad, T.; Omre, H. Bayesian Gaussian-mixture linear inversion for geophysical inverse problems. Math. Geosci.
**2017**, 49, 493–515. [Google Scholar] [CrossRef] - Aleardi, M.; Ciabarri, F.; Gukov, T. A two-step inversion approach for seismic-reservoir characterization and a comparison with a single-loop Markov-Chain Monte Carlo algorithm. Geophysics
**2018**, 83, R227–R244. [Google Scholar] [CrossRef] - Grana, D. Joint facies and reservoir properties inversion. Geophysics
**2018**, 83, M15–M24. [Google Scholar] [CrossRef] - Buland, A.; Omre, H. Bayesian linearized AVO inversion. Geophysics
**2003**, 68, 185–198. [Google Scholar] [CrossRef] - Azevedo, L.; Nunes, R.; Soares, A.; Mundin, E.C.; Neto, G.S. Integration of well data into geostatistical seismic amplitude variation with angle inversion for facies estimation. Geophysics
**2015**, 80, M113–M128. [Google Scholar] [CrossRef] - Sabeti, H.; Moradzadeh, A.; Ardejani, F.D.; Azevedo, L.; Soares, A.; Pereira, P.; Nunes, R. Geostatistical seismic inversion for non-stationary patterns using direct sequential simulation and co-simulation. Geophys. Prospect.
**2017**, 65, 25–48. [Google Scholar] [CrossRef] - Sammut, C.; Webb, G.I. Encyclopedia of Machine Learning; Springer Science & Business Media: Berlin, Germany, 2011. [Google Scholar]
- Çemen, I.; Fuchs, J.; Coffey, B.; Gertson, R.; Hager, C. Correlating Porosity with Acoustic Impedance in Sandstone Gas Reservoirs: Examples from the Atokan Sandstones of the Arkoma Basin, South Eastern Oklahoma. In Proceedings of the AAPG Annual Convention and Exhibition, Pittsburgh, PA, USA, 19–22 May 2013; p. 41255. [Google Scholar]
- Jafari, M.; Nikrouz, R.; Kadkhodaie, A. Estimation of acoustic-impedance model by using model-based seismic inversion on the Ghar Member of Asmari Formation in an oil field in southwestern Iran. Lead. Edge
**2017**, 36, 487–492. [Google Scholar] [CrossRef] - Das, B.; Chatterjee, R.; Singha, D.K.; Kumar, R. Post-stack seismic inversion and attribute analysis in shallow offshore of Krishna-Godavari basin, India. J. Geol. Soc. India
**2017**, 90, 32–40. [Google Scholar] [CrossRef] - Larsen, A.L.; Ulvmoen, M.; Omre, H.; Buland, A. Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model. Geophysics
**2006**, 71, R69–R78. [Google Scholar] [CrossRef] - Avseth, P.; Flesche, H.; Van Wijngaarden, A.J. AVO classification of lithology and pore fluids constrained by rock physics depth trends. Lead. Edge
**2003**, 22, 1004–1011. [Google Scholar] [CrossRef] - Aleardi, M.; Ciabarri, F.; Calabrò, R. Two- and single-stage seismic-petrophysical inversions applied in Nile Delta. Lead. Edge
**2018**, 37, 510–518. [Google Scholar] [CrossRef]

**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