A Hybrid Markov Chain Monte Carlo Method for Generating Permeability Fields Conditioned to Multiwell Pressure Data and Prior Information
Authors: Bonet-Cunha L. , Oliver D.S. , Redner R.A. , Reynolds A.C. 
Published: December 2007

Abstract
In order to properly evaluate the uncertainty in reservoir performance predictions, it is necessary to construct and sample the a posteriori probability density functions for the rock property fields. In this work, the a posteriori probability density function is constructed based on prior means and variograms (covariance function) for log-permeability and multiwell pressure data. Within the context of sampling the probability density function, we argue that the notion of equally probable realizations is the wrong paradigm for reservoir characterization. If the simulation of Gaussian random fields with a known variogram is the objective, it is shown that the variogram should not be incorporated directly into the objective function if simulated annealing is applied either to sample the a posteriori probability density function or to estimate a global minimum of the associated objective function. It is shown that the hybrid Markov chain Monte Carlo method provides a way to explore more fully the set of plausible log-permeability fields and does not suffer from the high rejection rates of more standard Markov chain Monte Carlo methods.

Introduction
To characterize the uncertainty in predicted reservoir performance, one wishes to simulate performance with a set of reservoir descriptions which honor all available data. Although simulated annealing represents a popular technique for generating reservoir descriptions which honor diverse data sets, our requirements are more strict. We do not simply wish to construct rock property fields which honor all data, we wish to generate a set of realizations of the rock property fields such that this set represents a correct sampling of the a posteriori probability density function for the rock property fields, i.e., we wish to derive the correct a posteriori probability density function conditioned to all data and then sample this probability distribution correctly. If one generates a set of N realizations which represent a proper sampling of the a posteriori probability density function, then one can characterize the uncertainty in performance predictions. To do this, we simply use each reservoir description (each realization) as input data for a reservoir simulation and generate the resulting reservoir performance. From these N flow simulations, one can compute the statistics (mean, median, variance, histogram) for each parameter or variable predicted by the flow simulations to provide a measure of the uncertainty or variability in predicted performance. Having characterized the uncertainty in predicted performance, one can make reservoir management decisions that account for our lack of complete knowledge of the true reservoir.
For the problem considered here, the objective is to determine realizations of the log-permeability field defined on reservoir simulator gridblocks. These realizations are obtained by sampling the probability density function conditioned to prior means, variograms (covariance function) and multiwell pressure data. Unlike other work, we use the pressure data directly instead of using pressure data to estimate an "average" permeability within some radius of investigation and then using this average permeability as a constraint when constructing realizations.
In earlier work, we estimated the most probable model (maximum a posteriori estimate) and then generated an approximate sampling of the a posteriori probability density function by generating realizations of the log-permeability field from the Cholesky decomposition of the a posteriori covariance matrix.
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