GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.1 555 EFFECT OF DIFFERENT STATISTICAL MODELS IN PROBABILISTIC JOINT ESTIMATION OF POROSITY AND LITHO-FLUID FACIES FROM ACOUSTIC IMPEDANCE VALUES M. Aleardi University of Pisa, Earth Sciences Department, Pisa, Italy Introduction. The estimation of petrophysical reservoir properties (i.e. porosity, shale content, fluid saturation) and litho-fluid facies around the target area is a common, highly ill- conditioned problem that is often casted into a Bayesian framework. Independently from the inversion approach adopted (analytical or numerical), the correct choice of the underlying statistical model always plays a crucial role in any geophysical Bayesian inversion. For what concerns the reservoir characterization problem, many authors have demonstrated that such statistical model should be able to correctly capture the facies-dependency of petrophysical and/or elastic properties related to the different lithologic and fluid-saturation conditions. In particular, it has been demonstrated that the accounting for such facies-dependency often provides more accurate descriptions of the uncertainties affecting the sought parameters. However, as the author is aware an in-depth discussion and a comparison of the results provided by different statistical models is still lacking for reservoir characterization studies. Focusing on this peculiar aspect, I use an inversion approach for the joint estimation of porosity and litho-fluid facies from logged and post-stack inverted acoustic impedance ( Ip ) values. The inversion approach I employ is a modification of the method proposed by Grana (2018) that is adapted to consider Gaussian-mixture and Gaussian distributions, and to jointly invert porosity and logged or inverted Ip values. This work is mainly aimed at analysing and comparing the results provided by three different statistical assumptions about the underlying joint distribution of the petrophysical model relating porosity and Ip values. To this end, I consider a simple Gaussian assumption that neglects the facies dependency of porosity and acoustic impedance values, whereas an analytical Gaussian-mixture distribution and a non- parametric mixture distribution relate each component of the mixture to a specific litho-fluid facies. In particular, the Gaussian or Gaussian-mixture models are often employed in seismic inversions because they allow for an analytical computation of the posterior model and make it possible to easily include additional constraints (i.e. geostatistical constraints) into the inversion kernel. Differently, a non-parametric distribution is not restricted by any statistical assumption about the underlying statistical model, but it impedes an analytical derivation of the posterior model and also complicates the inclusion of additional regularization operators or geostatistical constraints into the inversion framework. This work focuses the attention on well log data pertaining to a clastic gas-saturated reservoir. All the three considered statistical models are directly estimated from 5 out of 7 available wells drilled trough the reservoir zone. The kernel density technique is used to derive the non- parametric distribution. One of the two remaining wells is here used as blind test to validate the inversion results, whereas the analysis of the maximum-a-posteriori (MAP) solutions, and the coverage ratio are used to more quantitatively assess the final predictions. The method. The method used in this work is a modification of that proposed by Grana (2018). If we consider a Bayesian setting, the goal of the inversion is to estimate the probability of the petrophysical properties ( m) and the litho-fluid-facies ( f) given the logged or inverted elastic properties ( d) . In this context, the sought probability distributions can be computed as: (1) where, in our case p ( m , d  | f ) is the joint distribution of the porosity and Ip values within each facies, which can be estimated from available well log data. The probability p ( f  | d ) represents the conditional distribution of facies given the observed data that can be computed as: