Abstract
Efficient history matching (model updating) of geologically complex reservoirs is important in many applications, but it is central in closed-loop reservoir modeling, in which real-time model updating is required. Within the context of closed-loop modeling, one history matching approach receiving the most attention to date is the ensemble Kalman filter (EnKF). Although the EnKF has many advantages such as ease of implementation and efficient uncertainty quantification, it is technically appropriate only for random fields (e.g., permeability) characterized by two-point geostatistics (multi-Gaussian random fields). Realistic systems however are much better described by multi-point geostatistics, which is capable of representing key geological structures such as channels. History matching algorithms that are able to reproduce realistic geology provide enhanced predictive capacity and can therefore lead to better reservoir management and optimization.
In this work, we propose and formulate a generalized EnKF using kernels, capable of representing non-Gaussian random fields characterized by multi-point geostatistics. The main drawback of the standard EnKF is that the Kalman update essentially results in a linear combination of the forecasted ensemble, and the EnKF only uses the covariance and cross-covariance between the random fields (to be updated) and observations, thereby only preserving two-point statistics. Kernel methods allow the creation of nonlinear generalizations of linear algorithms that can be exclusively written in terms of dot products. By deriving the EnKF in a high-dimensional feature space implicitly defined using kernels, both the Kalman gain and update equations are nonlinearized, thus providing a completely general nonlinear set of EnKF equations, the nonlinearity being controlled by the kernel. By choosing high order polynomial kernels, multi-point statistics and therefore geological realism of the updated random fields can be preserved. The procedure is applied to two example cases where permeability is updated using production data as obserations, and is shown to better reproduce complex geology compared to the standard EnKF, while providing reasonable match to the production data.
