Seismic reservoir characterization in offshore Nile Delta. Part II: probabilistic petrophysical-seismic inversion.

Reservoir characterization plays an essential role in integrated exploration and reservoir studies, as it provides an optimal understanding of the reservoir internal architecture and properties. In reservoir characterization studies seismic reflection data are often used to derive petrophysical rock properties (water saturation, porosity, shale content) from elastic parameters (seismic velocities, rock density or impedances). The rock-physics model is the link between elastic properties and such petrophysical parameters and it can be based on theoretical rock-physics equations or on empirical set of equations derived from available information (well-log or core data) and valid for the specific case of interest. The inverse problem of estimating petrophysical properties from seismic reflection data is multidimensional, ill-posed and it is strongly affected by noise and measurements errors. Therefore, it is not a surprise that the statistical approach to seismic reservoir characterization has become the most popular approach as it is able to take into account the uncertainties associated with the simplified rock-physics model, the error in the seismic data, and the natural variability of the petrophysical properties in the subsurface. The goal of this approach is to predict the probability of petrophysical variables when seismic velocities or impedances and density are assigned, and to capture the heterogeneity and complexity of the rocks and the uncertainty associated with the rock-physics model. For many examples of applications of this approach to reservoir characterization studies constrained by seismic and well-log data see for example Avseth et al. (2005). In this paper we apply a two-step procedure to seismic reservoir characterization. The first step is a Bayesian linearized amplitude versus angle inversion (AVA) in which, on the line of Buland and Omre (2003) and Chiappa and Mazzotti (2009), we derive the elastic properties of the subsurface and their associated uncertainties assuming Gaussian-distributed errors and Gaussian-distributed elastic characteristics. The second step is a petrophysical inversion that uses the outcomes of AVA inversion, the previously defined rock-physics model, their associated uncertainties and the prior distribution of the petrophysical variables, to derive the probability distributions of the petrophysical properties in the target zone. The derivation and the calibration of different rock-physics models is the topic of the companion paper titled “Seismic reservoir characterization in offshore Nile Delta. Part I: Comparing different methods to derive a reliable rock-physics model”. In that paper the empirical, linear, rock-physics model derived with a multilinear stepwise regression (named SR in the companion paper) and the theoretical rock-physics model (named TRPM in the companion paper) demonstrated to be the most reliable in predicting the elastic characteristics from the petrophysical properties. Then, these two rock-physics models are applied in the petrophysical inversion described here. In the context of petrophysical inversion the main difference of applying a linear or a non-linear rock-physics model lies in the fact that the former allows the joint distribution of petrophysical and elastic properties to be analytically computed, while the latter requires a Monte Carlo simulation to derive such joint distribution. We start with a brief theoretical description of the method and with a synthetic example based on actual well-log measurements. This test aims to demonstrate the applicability of the inversion method and to illustrate and compare the different results obtained by considering the empirical and the theoretical rock-physics models. Moreover, this synthetic test allows us to check the applicability and the reliability of the two rock-physics models in the specific case under examination. Then, a field case inversion is discussed. This inversion is performed for a single CMP location where well-control is available to validate the results