Conditioning Permeability Fields to Pressure Data

Generating realizations of the permeability field drawn from a probability density function conditioned Oll inaccurate pressure or saturation data is difficult, even if the unconditional realizations are Gaussian ran-dom fields, because the problem is highly nonlinear. Inefficientmethods that generate large numbers of re-jected images, such as simulated annealing, must be ruled out as impractical because of the repeated need for reservoir flowsimulation. In this paper, we present a two-step Markov chain Monte Carlo method of proposing transitions in the Metropolis-Hastings algorithm such that the resulting state bas a high probability of acceptance. The first step is to propose an unconditional realization from a known probability distribution. This step could be carried out using any unconditional simulation tech-nique so it is not limited to simple stochastic modeIs. The secend part of the proposed transition involves "history matching " of the unconditional simulation to production data that has noise added. The decision to accept or reject the resulting "history matched" realization is made on the basis of the Metropolis-Hastings algorithm. Using this procedure on a simpie, but highly nonlinear, problem results in a suite of real-izations whose distribution is nearly indistinquishable from the analytical distribution. Because, however, calculation of the acceptance criterion for calibrated models is very difficult, we propose an approximate acceptance criterion in which al! proposed transitions are accepted. This method is shown to work wel! on a smal! nonlinear problem for which the approximate distribution can be compared to the desired distribu-tion