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

Abstract

Abstract The Kalman filter has 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. In this study, we propose an efficient, dimension-reduced Kalman filtering scheme based on Karhunen-Loeve and other orthogonal polynomial decompositions of the state vector. We consider flow in heterogeneous reservoirs with spatially variable permeability. The reservoir responses such as pressure head 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 Karhunen-Loeve (KL) decomposition, and reconstruction of this covariance from the KL decomposition can be done whenever needed. In each update, the forward problem is solved using the KL-based moment method, giving a set of functions from which the mean and covariance of the state vector 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 forward problem. The new approach is illustrated on a heteroge neous reservoir with dynamic measurements and its superior performance in terms of accuracy and efficiency relative to the EnKF method is discussed.