GNGTS 2017 - 36° Convegno Nazionale

GNGTS 2017 S essione 3.1 529 in which the AVA responses of the top reservoir reflections are inverted. In particular, the implemented AVA-petrophysical inversion uses a previously defined linear rock-physics model (RPM) to rewrite the linear Aki and Richards (1980) equation for the P-P wave reflection coefficients in terms of contrasts in the petrophysical properties at the reflecting interface. This reformulation allows us to directly and analytically derive the posterior probability distribution of petrophysical properties conditioned upon the observed AVA response. To reliably represent the multimodal facies-dependent behavior of petrophysical properties, we assume a Gaussian mixture distribution for the a priori model, and we use the prior probability of facies as weights of the Gaussian components of the mixture. The implemented algorithm is based on sequential geostatistical simulations. The solution of the inverse problem is achieved by the explicit analytical expression of the posterior distribution of petrophysical properties. A sampling algorithm can be used to sequentially simulate several realizations of the estimated model. The implemented inversion method has been applied to on-shore field seismic data and validated with well log information. The method. The derivation of the linear forward modelling linking the P-P wave AVA response ( Rpp ) to the contrasts in the petrophysical properties at the reflecting interface, is detailed described in the companion paper “ Target-oriented, structurally constrained seismic- petrophysical inversion ” to which we refer the reader for more information. For the sake of readability, here we show the matrix formulation of this forward modelling: Rpp = FDm = Gm (1) where F and G are sparse matrices, the matrix D is the derivative matrix operator, while the vector m contains the petrophysical properties of interest (water saturation, Sw ; porosity φ; shaliness , Sh ) m = [ Sw , Sh , φ ] T . To reliably represent the multimodal behavior of petrophysical properties related to the different rock and fluid properties of different litho-fluid facies, we assume a Gaussian mixture distribution p ( m ) for the a priori model, and we use the prior probability of facies as weights of the Gaussian components of the mixture: (2) where Q is the total number of facies considered, λ h represents the weights of the components (i.e., the proportion of the different facies), whereas μ h m and C h m are the a priori mean vector and the a priori covariance matrix of petrophysical properties for each facies. If we assume that the forward model is linear as in equation 1, it results that at each spatial location i , the posterior distribution of petrophysical properties p ( m | d , m s ) is still a Gaussian mixture distribution: (3) where π h are the weights of the posterior distribution, and the vector m s represents the hard data constraints. The mean vector, the covariance matrix and the weights of the posterior distribution, can be computed as follows: (4.1) (4.2) (4.3)