Bayesian Style History Matching: Another Way to Under-Estimate Forecast Uncertainty

摘要

Current theoretical formulations of assisted history matching (AHM) problems within the Bayesian framework, e.g., ensemble Kalman filter (EnKF) and randomized maximum likelihood (RML), are typically based on the assumption that simulation models can accurately reproduce field data within the measurement error. However, this assumption does not hold for AHM problems of real assets. This paper critically investigates the impact of using realistic, inaccurate simulation models. In particular it demonstrates the risk of underestimating uncertainty, when conditioning real-life models to large numbers of field data. Even though it is well-known, that model error and under-modeling impacts Bayesian methods, the practical effect that uncertainty may be severely underestimated, simply by using all available data is not well appreciated. Besides highlighting this effect, also a mitigation strategy to counteract this problem will be proposed and shown to be effective for the analytical toy model as well as for the real field case used as tests in this paper. After briefly reviewing the Bayesian method and its underlying assumptions, limitations of AHM approaches within the Bayesian framework are analyzed using a simple analytical model in which forecast uncertainty can be computed both with and without constraints due to historic data. In particular the model can be used to illustrate the impact of using an inaccurate, or incomplete, simulation model. The observations from this analytical work can then be generalized to real-life workflows that are currently implemented in many commercial and proprietary tools. To mitigate the observed problem, a fairly simple but effective modification of the AHM workflow is proposed and tested on the analytical test case. The same mitigation procedure is then also applied to improve uncertainty quantification of production forecasts using a real asset model. In order to see if the proposed workflow indeed leads to a more credible uncertainty assessment for forecast results, a specific realization of the asset model is used to generate synthetic production data. The model used for history matching and uncertainty quantification uses a different geological realization and hence can never reproduce the production results (which are assumed to have negligible noise). Also in this realistic setting, it is shown that forecasts easily can be underestimated when large numbers of data are used to constrain forecast uncertainty in an imperfect model with accurate data. This undesired effect comes out as the flip-side of the attractive property of Bayesian methods that model parameters can be inferred with increased accuracy if the number of data is increased in a perfect model with noisy data.