GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.1 13 transfer function. The weights associated with each node are computed such that the value at the output layer is equal to the training value in the least-squares sense. The NN optimization is essentially a steepest descent algorithm iteratively adjusting initial random weights using a technique called back-propagation (Haykin, 1999). For a review of ������ �������� ��� ����� neural networks and their geophysical applications, see van der Baan and Jutten (2000). Genetic algorithms (GA) optimization. Genetic algorithms are a stochastic optimization method based on the mechanics of natural selection and evolution according to the Darwinian principle of “survival of the fittest” (Holland, 1975). In a GAoptimization procedure a population of randomly generated individuals, which represent candidate solutions to an optimization problem, is evolved toward better solutions applying three main genetic operators, which are selection, cross-over and mutation. For more details see Mitchell (1996). In this work, we apply a GA optimization in which a population of 100 individuals evolves into 50 iterations. In this GA optimization, the equations describing the RPM are: (2) where: EP represents a generic elastic property (e.g. P-wave velocity, density…), PP is a generic petrophysical property (e.g. porosity, water saturation…) and N is the number of petrophysical properties considered in the regression process. The weight of each input variable is given by the coefficient a, the exponent b is used to reproduce the effects of variations in the petrophysical properties on the elastic property under consideration , whereas k is the intercept of the final equation. The coefficients ( k ; a 1 , a 2 ,..., a N ; b 1 , b 2 ,..., b N ) are contained in each individual that is evolved during the GA optimization in which the L 2 norm between observed and predicted elastic properties defines the error function to be minimized. ���� ���� ��� � ���������� � Note that Eq. 2 represents a generalization of classical depth trends (Banchs et al., 2001). Theoretical rock-physics model (TRPM). With theoretical rock-physics model we refer to one or more theoretical equations that establish a relationship between elastic attributes and petrophysical rock properties. To this end, several models exist (e.g. granular media models and inclusion models). The reader can find an extensive discussion of TRPM in Avseth et al. (2005) and Mavko et al. (2009). In this work, following Avseth et al. (2005), we use the Hertz-Mindlin theory to define the shale and sand dry elastic properties at critical porosity and hydrostatic pressure. To simulate the compaction effect we used the Hashin-Strikmann lower bound, whereas the Gassmann equation defines the saturated elastic properties. Taking into account the depth interval considered in this study, characterized by a mechanical compaction regime, we assume a shale totally formed by smectite mineral and a not-cemented sand totally formed by quartz grains. The shale and sand critical porosities are fixed to 70% and 40%, respectively. Results. In this chapter, we analyze and comment the RPMs resulting from theoretical (TRPM) approach and from linear (SR) and non-linear (NN and GA) empirical approaches. The well-log data used to estimate the RPMs pertain to four exploration wells drilled through the reservoir zone (sand) and the encasing non-reservoir rocks (shale). The petrophysical ( Sh, Sw, φ ) and elastic ( Vp, Vs, density ) properties, that we consider, are all derived from appropriate formation evaluation analysis of actual well-log measurements and have been subjected to an accurate outlier removing procedure. We know that non-linear relations often relate petrophysical properties and elastic characteristics. For example, non-linear relations link the shale content to Vp and Vs and the water saturation to Vp if considered in their full range from 0% to 100% (see Avseth et al., 2005 for more details). However, in order to investigate the capability of a linear method in deriving a reliable rock-physics model, in this specific case, we consider the entire shale content and water saturation ranges (from 0% to 100%). Differently, the depth interval is limited to the target sands and the encasing shales and ranges from 2400 to 3000 m, approximately.