The Bayesian framework allows one to integrate production data and static data into a posteriori probability density function for reservoir variables (model parameters). The problem of generating realizations of the reservoir variables for the assessment of uncertainty in reservoir description or predicted reservoir performance then becomes a problem of sampling this posteriori pdf to obtain a suite of realizations. Generation of a realization by the randomized maximum likelihood method requires the minimization of an objective function that includes production data misfit terms and a model misfit term that arises from static data. Minimization of this objective function with an optimization algorithm is equivalent to the automatic history matching of production data with a prior model constructed from static data providing regularization. Because of the computational cost of computing sensitivity coefficients and the need to solve matrix problems involving the covariance matrix for the prior model, this approach has not been applied to problems where the number of data and number of reservoir model parameters are both large and the forward problem is solved by a conventional finite difference simulator. In this work, we illustrate that computational efficiency problems can be overcome by using a scaled limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm to minimize the objective function, and using approximate computational stencils to approximate the multiplication of a vector by the prior covariance matrix or its inverse. Implementation of the limited memory BFGS method requires only the gradient of the objective function which can be obtained from a single solution of the adjoint problem; individual sensitivity coefficients are not needed. We apply the overall process to two examples. The first is a true field example in which a realization of log-permeabilities at 26,019 grid blocks is generated by the automatic history matching of pressure data and the second is a pseudo-field example that provides a very rough approximation to a North Sea reservoir in which a realization of log-permeabilities at 9750 gridblocks is computed by the automatic history matching of GOR and pressure data. Introduction Bayes theorem provides a general framework for updating a probability density function as new data or information on the model becomes available. The Bayesian setting offers a distinct advantage. If one can generate a suite of realizations which represent a correct sampling of the posteriori pdf, then the suite of samples provide an assessment of the uncertainty in reservoir variables. Moreover, by predicting future reservoir performance under proposed operating conditions for each realization, one can characterize the uncertainty in future performance predictions by constructing statistics for the set of outcomes1,2. Ning and Oliver3 have recently presented a comparison of methods for sampling the posteriori pdf. Their results indicate that the randomized maximum likelihood method is adequate for evaluating uncertainty with a relatively limited number of samples. In this work, we consider the case where a prior geostatistical model constructed from static data is available and is represented by a multivariate Gaussian pdf. Then the posteriori pdf conditional to production data is such that calculation of the maximum a posteriori estimate or generation of a realization by the randomized maximum likelihood method is equivalent to the minimization of an appropriate objective function4,5,6,2.

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