Randomized Maximum Likelihood with Permeability Samples Generated by a Predictor Corrector Technique

Abstract

Abstract To assess the uncertainty in reservoir descriptions and performance predictions, the a posteriori probability density function (pdf) for the unknown model parameters is needed. The only practical way to evaluate the posterior pdf is by sampling. Permeability samples for performance forecasting should be consistent with all available information. Markov chain Monte Carlo (McMC) samples correctly from the posterior distribution, but is computationally extremely intensive. Randomized maximum likelihood (RML) is a good approximation to McMC that involves history matching of each permeability sample. Although RML is computationally less intensive than McMC, the computational effort to generate a sufficiently large number of samples is therefore huge for all but very small reservoir models. In this work, we have investigated if the sampling procedure can be made more efficient by using a predictor-corrector approach in the history matching step of RML. The predictor applies sequential history matching to obtain an estimate with few degrees of freedom, utilizing only part of the available information. The corrector downscales the predictor estimate in a two-step procedure involving all available information, including the estimate obtained with the predictor. The first corrector step is a variant of Kriging. The second corrector step is a history match, again involving few degrees of freedom, with basis functions derived from the results of the predictor. The method is demonstrated and compared to regular RML through numerical examples, indicating that predictor-corrector RML is computationally more efficient than regular RML.