Spatially correlated Markov chain Monte Carlo method for petrophysical inversion

Abstract

Stochastic methods for seismic inversion problems for the estimation of rock and fluid properties are commonly adopted in reservoir characterization studies. Among the numerous algorithms, Markov chain Monte Carlo (McMC) methods represent a family of algorithms for the estimation of the posterior distribution of the variables of interest. In seismic reservoir characterization studies, McMC methods can be used to estimate the posterior distribution of elastic and petrophysical properties from seismic data, which allows predicting the most likely model and its uncertainty. However, McMC methods require large computational time to exhaustively sample the model space. We propose an efficient implementation of a McMC method based on the Metropolis algorithm for petrophysical inversion. The inverse problem is solved based on an efficient 1- dimensional (single trace) approach with lateral constraints imposed by a pre-conditioned prior model, i.e., a prior model that is re-calculated based on previously inverted traces. The application of the proposed method to 2- and 3-dimensional problems is based on a sequential path where the seismic traces are sequentially visited according to a random path and where the parameters of the prior model are computed using kriging with locally variable mean. The resulting McMC algorithm efficiently provides spatially correlated realizations of petrophysical properties that are used to estimate the posterior distribution of the model variables. The methodology is demonstrated on 1- and 2-dimensional applications in a clastic reservoir with synthetic seismic data.