The compressed state Kalman filter for nonlinear state estimation: Application to large-scale reservoir monitoring

Abstract

Reservoir monitoring aims to provide snapshots of reservoir conditions and their uncertainties to assist operation management and risk analysis. These snapshots may contain millions of state variables, e.g., pressures and saturations, which can be estimated by assimilating data in real time using the Kalman filter (KF). However, the KF has a computational cost that scales quadratically with the number of unknowns, m, due to the cost of computing and storing the covariance and Jacobian matrices, along with their products. The compressed state Kalman filter (CSKF) adapts the KF for solving large-scale monitoring problems. The CSKF uses N preselected orthogonal bases to compute an accurate rank-N approximation of the covariance that is close to the optimal spectral approximation given by SVD. The CSKF has a computational cost that scales linearly in m and uses an efficient matrix-free approach that propagates uncertainties using N + 1 forward model evaluations, where N 蠐 m. Here we present a generalized CSKF algorithm for nonlinear state estimation problems such as CO2 monitoring. For simultaneous estimation of multiple types of state variables, the algorithm allows selecting bases that represent the variability of each state type. Through synthetic numerical experiments of CO2 monitoring, we show that the CSKF can reproduce the Kalman gain accurately even for large compression ratios (m/N). For a given computational cost, the CSKF uses a robust and flexible compression scheme that gives more reliable uncertainty estimates than the ensemble Kalman filter, which may display loss of ensemble variability leading to suboptimal uncertainty estimates.