Fast parameter and state estimation with the Spectral Kalman Filter: an application for CO2 injection in heterogeneous domains

Abstract:

The Kalman Filter has been widely used for dynamic monitoring in reservoir engineering, and has recently gained popularity in hydrogeologic applications. A common characteristic of such applications is that the physical processes of interest are greatly affected by preferential flow (e.g., contaminant spreading, CO₂ leakage), which can only be delineated if the problem is finely discretized into a large number of unknowns. However, for problems with large numbers of unknowns (e.g., larger than 10,000), the Kalman Filter has prohibitively expensive computation and storage costs. The EnKF, which is typically used to reduce the cost of computing the covariance in such cases converges slowly to the best estimate, and for a reasonable number of realizations, the estimate may not be accurate, especially for strongly heterogeneous systems. We present the Spectral Kalman Filter, a new Kalman Filter implementation that has a dramatically reduced computational cost compared to the full Kalman Filter, with comparable or higher accuracy than the EnKF for the same computational cost. Our algorithm’s computational efficiency is achieved by a recurrence that updates small cross-covariance matrices instead of large covariance matrices, in combination with a low-rank approximation of the noise covariance matrix. In addition, instead of computing the expensive Jacobian matrix, a matrix-free method is used to obtain sensitivities. Finally, the error of our method can be explicitly controlled by reducing the time between matrix updates. The frequency of these updates is controlled independently from the data assimilation steps. We demonstrate the performance of the Spectral Kalman Filter for the joint estimation of domain properties and state evolution by assimilation of quasi-continuous data during a hypothetical CO₂ injection in a heterogeneous domain.