Parameterization Techniques to Improve Mass Conservation and Data Assimilation for Ensemble Kalman Filter

Abstract

Abstract The ensemble Kalman filter (EnKF) is an attractive method for assisted or automatic history matching. In particular, it has been shown to be possible to efficiently match many types of data by adjusting many types of reservoir model variables, including gridblock permeabilities and porosities, facies locations, relative permeability curves, fault transmissibilities, and initial fluid contacts. At each analysis step of EnKF, model and state variables are updated to honor the production history and prior knowledge of the reservoir description using a correction that is constructed from a linear combination of the forward models. Because the update in the ensemble Kalman filter is linear and variables may be adjusted beyond their plausible range in which case the updated values are sometimes truncated, mass conservation may be violated in the updated models. Proper selection of the model and state variables to be updated in EnKF is critically important for achieving good data match, retaining geological realism of the updated ensemble and satisfying nonlinear constraints that may apply. In this paper, we discuss parameterization techniques for these three different purposes. We revisit the updating formulation of EnKF and investigate the constraint on the weighting coefficients at the analysis step. We show that by reparameterization of the state vector, nonlinear constraints such as conservation of mass or volume, can be satisfied without complicating the updating process. The ensemble approximation of the covariance tends to be strongly corrupted by noise when the true correlation is small. Localization can regularize the ensemble-based estimate and reduce the detrimental effects of the spurious correlation, but also eliminate the effect of true correlations whose magnitude in individual gridblocks is small, but whose cumulative effect may be large. Appropriate parameterization in addition to localization could better constrain the solution of the inverse problem. Finally, the choice of model variables may also depend on observations that are available. Including model variables that are highly sensitive to certain types of data provides improved data match and more realistic updates. A large scale reservoir model is used to illustrate these parameterization techniques. By using these techniques multiple realistic history matched models were obtained that provide reliable basis for accessing uncertainty and for design of reservoir management strategy.