Data Assimilation Using the Constrained Ensemble Kalman Filter

Abstract

Abstract When the ensemble Kalman filter (EnKF) is used for history matching, the resulting updates to reservoir properties sometimes exceed physical plausible bounds, especially when the problem is highly nonlinear. Problems of this type are often encountered during history matching compositional models using EnKF. In this paper, we illustrate the problem using an example in which the updated molar density of CO2 in some regions is observed to take negative values while molar densities of the remaining components are increased. Standard truncation schemes avoid negative values of molar densities, but do not address the problem of increased molar densities of other components. The results can include a spurious increase in reservoir pressure with a subsequent inability to maintain injection. In this paper, we present a method for constrained EnKF which takes into account the physical constraints on the plausible values of state variables during the data assimilation. The proposed method can be implemented in two different approaches, both of which convert inequality constraints to a small number of equality constraints. The first approach uses Lagrange multipliers to apply the active constraints. In the second approach, the constraints are used as virtual observations for calibrating the model parameters within plausible ranges. The constrained EnKF method is tested on a 2D compositional model and on a highly heterogeneous 3-phase flow reservoir model. The effect of the constraints on mass conservation are illustrated using a 1D Buckley-Leverett flow example. Results show that the constrained EnKF is able to enforce the nonnegativity constraints on molar densities and bound constraints on saturations (all phase saturations must be between 0 and 1), and achieve a better estimation of reservoir properties than is obtained using only truncation with EnKF.