Dynamic Reservoir Data Assimilation With an Efficient, Dimension-Reduced Kalman Filter

Summary Kalman filter-based methods have been widely applied for assimilating new measurements to continuously update the estimate of state variables, such as reservoir properties and responses. The standard Kalman filtering scheme requires computing and storing the covariance matrix of state variables, which is computationally expensive for large-scale problems with millions of gridblocks. In the ensemble Kalman filter (EnKF), this problem is alleviated with sampling from a limited number of realizations and computing the required subset of the covariance matrix at each update. However, the goodness of the (ensemble) covariance approximated from the limited ensemble depends on the number of realizations used and the representativity of a given ensemble. In this study, we propose an efficient, dimension-reduced Kalman filtering scheme based on Karhunen-Loeve (KL) and other orthogonal polynomial decompositions of the state variables. We consider flow in heterogeneous reservoirs with spatially variable permeability. The reservoir responses such as pressure are measured at some locations at various time intervals. The aim is to dynamically characterize the reservoir properties and to predict the reservoir performance and its uncertainty at future times. In our scheme, the covariance of the reservoir properties is approximated by a small set of eigenvalues and eigenfunctions using the KL decomposition and the reconstruction of the covariance from the KL decomposition can be done whenever needed. In each update, the forecast step is solved using the KL-based moment method, giving a set of functions from which the mean and covariance of the state variables can be constructed, when needed. The statistics of both the reservoir properties and the reservoir responses are then updated with the available measurements at this time using the auto- and cross-covariances obtained from the forecast step. The new approach is illustrated on a heterogeneous reservoir with dynamic measurements and the results are compared with those from the EnKF method, in terms of accuracy and efficiency. Introduction Owing to the high cost associated with direct measurements of reservoir properties, for instance permeability and porosity, the number of direct observations is always limited. However, the reservoir exhibits a high degree of spatial variability at all length scales resulted from its intrinsically complicated nature. This combination of significant spatial heterogeneity with a relatively small number of direct observations leads to uncertainty in characterizing reservoir properties, which in turn results in uncertainty in estimating or predicting the corresponding reservoir responses.