Consistent Formulation and Error Statistics for Reservoir History Matching

Abstract

Summary It is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the posterior probability density function of the uncertain static model parameters is proportional to the prior probability density of the parameters multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the data include, e.g., produced rates of oil, gas, and water from the wells. The reservoir prediction model is assumed to be perfect, and there are no errors besides those in the static parameters. The Bayesian formulation of this problem is given, e.g., in the recent paper by Evensen et al. (2019) , and serves as the fundamental description of the history-matching problem. However, this formulation is flawed. The historical rate data comes from the real production of the reservoir, and they contain errors. The conditioning methods usually take these errors into account, but we neglect them when we force the simulation model by the observed rates during the historical integration. Thus, in the history-matching problem, the model prediction depends on the same data that we condition on, which prevents the direct use of Bayes’ theorem. Here, we formulate Bayes’ theorem while taking into account the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the errors in the rate data used to force the reservoir simulation model. Also, we specify time-correlated rate errors that are consistent with the use of allocation tables to generate the rate measurements. The “red” errors lead to a stronger uncertainty increase for the simulation model and also reduces the impact of the rate measurements in the conditioning process (where the measurement error-covariance matrix becomes non-diagonal). We present results where the new subspace EnRML by Raanes et al. (2019) and Evensen et al. (2019) is used with a simple reservoir case. The result is a more consistent prediction model and a more realistic uncertainty estimate from the updated ensemble.