Quantification of prediction uncertainty using imperfect subsurface models with model error estimation

Subsurface reservoirs are far more heterogeneous and complex than the simulation models in terms of scale, assumptions and description. In this work, we address the issue of prediction reliability while calibrating imperfect/low-fidelity reservoir models. The main goal is to avoid over-confident and inaccurate predictions by including a model for the bias terms (i.e. error-model of a predefined form) during the history matching process. Our aim is to obtain unbiased posterior distributions of the physical model parameters and thus improving the prediction capacity of the calibrated low-fidelity reservoir models. We formulate the parameter estimation problem as a joint estimation of the imperfect model parameters and the error-model parameters. The structure of the error-model and the prior distributions of the error-model parameters are evaluated before calibration through analysis of leading sources of the modeling errors. We adopt a Bayesian framework for solving the inverse problem, where we utilize the ensemble smoother with multiple data assimilation (ES-MDA) as a practical history matching algorithm. We provide two test cases, where the impact of typical model errors originating from grid coarsening/upscaling and from utilizing an imperfect geological model description is investigated. For both cases results from the ES-MDA update with and without accounting for model error are compared in terms of estimated physical model parameters, quality of match to historical data and forecasting ability compared to held out data. The test results show that calibration of the imperfect physical model without accounting for model errors results in extreme values of the calibrated model parameters and a biased posterior distribution. With accounting for modeling errors the posterior distribution of the model parameters is less biased (i.e. nearly unbiased) and improved forecasting skills with higher prediction accuracy/reliability is observed. Moreover, the consistency between the different runs of the ES-MDA is improved by including the modeling error component. Although the examples in the paper consider the oil-water system with permeabilities being parameters of the physical model, the developed methodology is general and can be applied to typical ground water hydrology models.