GNGTS 2017 - 36° Convegno Nazionale

530 GNGTS 2017 S essione 3.1 where and . The linear operators B i and A are selector operators that extract the values of the petrophysical properties at the current location ( i ) and at the locations previously visited. In particular, B i can be written as: B i = [0,...,0,1,0,...,0] (5) with the one positioned at the i- th column. If the sub-vector m s has size X and m has size Y, the operator A is given by: (6) where A has dimensions X × Y and the ones are in the i 1 , i 2 , …, i x columns. In the inversion, the geostatistical information about the spatial correlation of petrophysical properties, and about their mutual correlation is expressed by the matrix that can be obtained by a double Kronecker product between a stationary covariance matrix (expressing the mutual correlation of petrophysical properties within a given facies), and the two spatial correlation functions that define the lateral variability of petrophysical properties along the two horizontal directions. In particular, the two spatial correlation functions follow a first-order exponential function, which can be generically written as: (7) where q is a given spatial direction, and k is the parameter that defines the spatial dependency. To run the implemented inversion algorithm along a simple 1D spatial profile, we first choose a random searching path. We then visit each spatial location i , and compute at each point the posterior distribution p ( m | d ) of petrophysical properties m conditioned by seismic data d . If previously estimated values and/or hard data constrains (i.e. well log constraints; m s ) are available in a given searching neighborhood, the posterior distribution (Eqs. 4.1 and 4.2) is also conditioned by m s ( p ( m | d, m s ). If needed, we can draw a random value m i at the current position i according to the estimated posterior distribution. We iterate this procedure until all the locations have been visited. To extend the inversion to 2D or 3D dataset we need to introduce additional for loops for the additional spatial directions. At each location, the posterior distribution is conditioned by the seismic data (AVA response) observed at the current location, the hard data constraints, and the previously estimated values in neighboring positions. Despite the analytical expression of the posterior distribution, this methodology is quite computer demanding. For this reason, we use the forward-backward algorithm based on the Cholesky factorization to efficiently compute the matrix inverse. Field data application . The reservoir considered in this work is located in a shale-sand sequence and is constituted by gas-bearing sands at the depth range of 900 -1000 m. For the description of the multilinear rock-physics model derived for the investigated area, we again refer the reader to the companion paper “ Target-oriented, structurally constrained seismic- petrophysical inversion ” in which the same exploration field and seismic data are considered. Fig. 1 represents the estimated petrophysical properties along the interpreted top of the reservoir when a laterally unconstrained Bayesian inversion (LUBI) is applied. In this case, each AVA response extracted from each data gather is inverted separately and a simple a priori Gaussian distribution for the petrophysical properties is assumed. Note the scattering visible in the estimated 2D maps of reservoir properties due to noise contaminating the observed data. Fig. 2 shows the results achieved by the implemented inversion approach that incorporates geostatistical and hard data constraints. By comparing Fig. 2 with Fig. 3, we can appreciate that the implemented algorithm produces low-noise images with more laterally continuous