GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.1 21 The method. In the following discussion we will use m to indicate the elastic properties, typically P and S-impedance ( Ip and Is , respectively) and density, R to indicate the petrophysical properties, such as water saturation ( Sw ), porosity ( φ ) and shaliness ( Sh ), whereas d obs indicates the observed seismic data (typically the measured AVA responses). The method we use is a two-step procedure: a Bayesian linearized AVA inversion followed by a probabilistic petrophysical inversion. This petrophysical inversion makes use of the a-priori distribution of the petrophysical properties p(R) derived on the basis of well-log data, of the previously defined rock-physics model and of the results of AVA inversion to derive the probabilistic distribution of the petrophysical properties in the subsurface. The first step of the petrophysical-seismic inversion is the Bayesian AVA inversion that jointly estimates the posterior distributions of the elastic properties in the subsurface by making use of a reformulation of the linear approximation of the Zoeppritz equation derived by Aki and Richards (1980). In particular, we parameterize the inversion in terms of P and S-impedance ( Ip and Is , respectively) and density. In terms of impedances, the P-wave reflection coefficient Rpp as a function of the reflection angle ( θ ) can be written as follows: (1) where Ip— , Is— and ρ— are, respectively, the averages of impedances and density at the reflecting interface, whereas ΔIp , ΔIs and Δρ are the corresponding contrasts. However, density estimates are not used in the petrophysical inversion since they are obviously correlated with the impedances ones and because the linear AVA inversion cannot retrieve reliable information about density with realistic noise levels (Buland and Omre, 2003). Following Stolt and Weglein (1985), the single-interface reflection coefficient in Eq. 1 can be easily extended to a time- continuous reflectivity function. The elastic properties estimated by Bayesian AVA inversion are delivered according to the following posterior probability distribution: (2) where: G indicates the Gaussian distribution where the posterior expectation and the covariance are equal to μ m|dobs and Σ m|dobs , respectively. The a-priori distributions and the vertical correlation of the elastic properties, needed to derive the posterior distribution in Eq. 2, can be determined from available well-log data. For full details about the Bayesian linearized AVA inversion see Buland and Omre (2003). For what concerns the petrophysical inversion, we apply the method proposed by Grana and Della Rossa (2010) and briefly summarized in the following. Considering all variables as random vectors, we can write the rock-physics model ( f RPM ) as: (3) where: ε is the random error that describes the accuracy of the rock-physics model and can be determined by comparing the available well-log data with the predicted data. For the prior distribution of the petrophysical properties p(R) we assume a multivariate Gaussian mixture (GM) that is a linear combination of Gaussian distributions: (4) where: Nc indicate the number of components of the mixture and a k are the weights associated with each component (with ). Generally, each component is a specific litho-fluid class previously determined from available log data and from the geological knowledge of the investigated area. In this work, we consider three litho-fluid classes that are gas-sand, brine- sand and shale. The technique we adopt, to estimate the parameters of the Gaussian components