GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.1 557 actual logged Ip values to infer porosity and facies. Second, I exploit the well log information to compute synthetic seismic traces that, in a first inversion step are converted into Ip values and associated uncertainties that become the input for the following inversion step aimed at estimating porosity and litho-fluid facies. In this synthetic application I employ a convolutional forward operator to derive the post-stack seismic trace, whereas a simple analytical least- square Bayesian inversion is adopted to estimate the Ip values and the associated uncertainty from post-stack traces. In the seismic examples the Chapman-Kolmogorov equation is used to correctly propagate the uncertainty affecting the estimated Ip values into the uncertainties associated to the final porosity and facies predictions. In addition, a 1D Markov Chain prior model is employed to vertically constrain the predicted facies profile. Well log data application. I first describe the three joint p ( m , d  | f ) distributions derived from 5 out of 7 available wells that reached the reservoir (Fig. 1). The first is the non-parametric distribution estimated through the kernel density technique; the second is the analytical Gaussian- mixture that assumes Gaussian distributed porosity and Ip values within each facies. The third is the simple Gaussian distribution. The two multimodal distributions (non-parametric and Gaussian-mixture) derived for the shale seem very similar, whereas their differences are more prominent for the brine and gas sand facies. In particular, the Gaussian-mixture assumption completely masks the multimodality of the p ( m , d  | f = brine sand ) distribution that is instead correctly modelled by the non-parametric model. This multimodality could be related to sands with different mineralogic or textural characteristics. In Fig. 1c we observe that the Gaussian prior is not able to reliably model the underlying relation linking porosity and Ip values. In other words, the Gaussian model constitutes an oversimplification of the actual, underlying petrophysical model. Fig. 2 represents the final results. We observe five significant decreases of the acoustic impedance value that mark the main sand layers. The target, gas saturated reservoir is located between 1400-1420 m. In Fig. 2a I show the results for the non-parametric distribution and we observe that the MAP solution for the porosity closely matches the actual porosity values and correctly captures the fine-layered structure of the reservoir. The outcomes of the facies classification show a satisfactory match with the true facies profile derived from borehole information. In particular, note the high probability that a gas saturated layer occurs at the target depth (1400-1420 m). Fig. 2b shows the results obtained for the same well but employing the Gaussian-mixture p ( m , d  | f ) distribution. We observe that the MAP solution for the porosity is now characterized by a poorer match with the logged porosity values than that obtained with the non-parametric p ( m , d  | f ) model. The facies prediction still shows a satisfactory match with the actual facies profile and, more importantly, the main gas saturated layer is still correctly identified. The MAP solutions derived from the Gaussian model (Fig. 2c), seem still able to capture the vertical porosity variability but the oversimplified statistical p ( m , d ) model translates into higher posterior uncertainties (i.e. wider posterior distributions) compared to the Gaussian-mixture and the non-parametric p ( m , d  | f ) distributions. In other terms, the suboptimal underlying statistical model results in more inaccurate prediction intervals compared to the previous tests. The 90% coverage probability ratios are equal to 0.92, 0.8884 and 0.7788 for the non-parametric, Gaussian-mixture and Gaussian model, respectively. These values confirm that the non-parametric approach outperforms the other two models, although the Gaussian-mixture distribution provides quite accurate predictions. A direct comparison of posterior distributions, of linear correlation coefficients, and the contingency analysis tools have also been employed to more quantitatively assess the final predictions. However, these analyses are not shown here for the lack of space. Post-stack data application. I now extend the inversion tests on post-stack seismic data. For confidentiality reasons, I limit the attention to synthetic data computed on the basis of actual well log information and adopting a 1D convolutional forward modelling with a 45-Hz Ricker wavelet as the source signature and 0.002 s as the sampling interval. To better simulate