Sequential design strategy for kriging and cokriging-based machine learning in the context of reservoir history-matching

Numerical models representing geological reservoirs can be used to forecast production and help engineers to design optimal development plans. These models should be as representative as possible of the true dynamic behavior and reproduce available static and dynamic data. However, identifying models constrained to production data can be very challenging and time consuming. Machine learning techniques can be considered to mimic and replace the fluid flow simulator in the process. However, the benefit of these approaches strongly depends on the simulation time required to train reliable predictors. Previous studies highlighted the potential of the multi-fidelity approach rooted in cokriging to efficiently provide accurate estimations of fluid flow simulator outputs. This technique consists in combining simulation results obtained on several levels of resolution for the reservoir model to predict the output properties on the finest level (the most accurate one). The degraded levels can correspond for instance to a coarser discretization in space or time, or to less complex physics. The idea behind is to take advantage of the coarse level low-cost information to limit the total simulation time required to train the meta-models. In this paper, we propose a new sequential design strategy for iteratively and automatically training (kriging and) cokriging based meta-models. As highlighted on two synthetic cases, this approach makes it possible to identify training sets leading to accurate estimations for the error between measured and simulated production data (objective function) while requiring limited simulation times.