Accounting for Model Discrepancy in Uncertainty Analysis by Combining Numerical Simulation and Bayesian Emulation Techniques

Summary Model discrepancy specifies unavoidable differences between a physical system and its corresponding computer model. Incomplete information, simplifications and lack of knowledge about the physical state originate model discrepancy. Misevaluation of model discrepancy exposes decision-makers to overconfident and biased forecasts, a risky situation. We describe a methodology to account for one type of model discrepancy in the Bayesian History Matching for Uncertainty Reduction (BHMUR), an approach that combines reservoir simulation and emulation techniques to find all reservoir scenarios consistent with observed data and uncertainties in the problem. Our methodology is an alternative and more rigorous tool to account for the model discrepancy caused by errors in target data while performing uncertainty analysis. Target data used in historical period contain observational errors that propagate through the simulator, causing one type of model discrepancy. We follow a systematic procedure for uncertainty reduction previously presented by the authors, expanding the step dedicated to the model discrepancy. Our methodology: (1) obtains a training set by evaluating model discrepancy in multiple scenarios of the search space, an expensive simulation-based process; (2) characterises the model discrepancy across the entire search space via Bayesian emulators; and (3) integrates the model discrepancy in the BHMUR via bias and covariance structures. The methodology is demonstrated in a case study: 27 valid emulators for model discrepancy were constructed and integrated into the implausibility analysis and uncertainty reduction process. Two perspectives showed the impact of this type of model discrepancy. Firstly, neglecting model discrepancy resulted in all the search space being implausible –an indicator to review the problem characterisation and uncertainties; by contrast, when considering the model discrepancy, the non-implausible region consists of 8% of the search space. Secondly, we demonstrated the uncertainty reduction in the historical and forecasting periods. A key finding is that the error in target data results in a substantial model discrepancy over many other simulation outputs, being both time and location dependent. We advance the applicability of BHMUR by proposing a statistically consistent tool to account for one type of model discrepancy in the uncertainty quantification process. We showed that errors in target data cause model discrepancy with a complex structure. Appropriate consideration of model discrepancy is vital to (a) identify the whole class of solutions consistent with historical data and uncertainties in the problem, (b) appropriately represent the physical system; (c) avoid making decisions based on over-confident and biased information while enabling more reliable production forecast.