Abstract
In recent years, it is shown that the inverse problem theory based on Bayesian estimation provides a powerful methodology not only to generate rock property fields conditioned to both static and dynamic data, but also to assess the uncertainty in performance predictions. To date, standard applications of inverse problem theory given in the literature assume that rock property fields obey multinormal distribution and are second order stationary. In this work, we extend the inverse problem theory to cases where rock property fields (only porosity and permeability fields are considered) can be characterized by fractal distributions. Fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) are considered. To the authors' knowledge, there exists no study in the literature considering generation of fractal rock property fields conditioned to dynamic data; particularly to well-test pressure data.
We show that available Bayesian estimation techniques based on the assumption of normal/second-order stationary distributions can be directly used to generate conditional fGn rock property fields to both hard and pressure data because fGn distributions are normal/second-order stationary. On the other hand, we show that because fBm is not second-order stationary, these Bayesian estimation techniques can only be used with implementation of a pseudo-covariance (generalized covariance) approach to generate conditional fBm fields to static and well-test pressure data.
Using synthetic examples generated from two and three-dimensional single-phase flow simulators, we show that the inverse problem methodology can be applied to generate realizations of conditional fBm/fGn porosity and permeability fields to well-test pressure data. We conclude by showing how one can then assess the uncertainty in reservoir performance predictions appropriately using these realizations.
