Fast Kalman filters for real-time CO2 storage monitoring and reservoir characterization [electronic resource]
Keywords: Civil and Environmental Engineering.

The operation of most engineered hydrogeophysical systems relies on simulating physical processes using numerical models with uncertain parameters and initial conditions. The model predictions can be greatly improved using data assimilation techniques. Kalman-type techniques sequentially assimilate monitoring data in real-time to correct model predictions and quantify the prediction uncertainties. Essentially, this update constitutes a nonlinear optimization, which is solved by linearizing an objective function around the model forecast and applying a linear correction to the predictions. However, direct application of Kalman-type data assimilation techniques to geologic CO2 storage (GCS) monitoring is challenging. KF is computational intractable for the size of the GCS problem, typically at the scale of millions. Such large state dimension is required to capture potential CO2 leakage at small scale, where a fine resolution of the estimated field is important. The popular Monte-Carlo based ensemble Kalman filters (EnKFs) are computationally feasible for such large scale system, but give unreliable uncertainty estimates. Moreover, as the model parameters are highly uncertain and the system dynamics is governed by nonlinear multi-phase flow, the optimization problem becomes strongly nonlinear and a linear Kalman correction may yield nonphysical results. Conventional Kalman filters have 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 matrix-vector products. Quadratic scaling indicates that for a realistic reservoir size, KF will take a few months to assimilate a single data set, while for a linear-scaling filter it will only take a few minutes. By adopting a reduced-rank approximation of the covariance, low-rank KFs such as the EnKF can provide significant computational speedup compared to the conventional full-rank KFs provided that the rank N is small compared to the number of unknowns, i.e., N < < m. A small rank leads to linear computational scaling and minimizes the number of reservoir simulations. However, the efficiency of such low-rank KFs is determined by the accuracy of the low rank approximation. A large approximation error may lead to spurious correlations, underestimation of uncertainty, etc., which prompts the filter to make nonphysical corrections or fail to capture small-scale variations in the state variables. Recognizing the trade-offs between computational cost and estimation accuracy in conventional Kalman filters, this work investigates a few common hydrogeophysical data assimilation scenarios and derives Kalman filter variants that exploit the formulation of the state space models as well as the covariance structures to achieve computational efficiency while maintaining the quality of uncertainty quantification. In this dissertation, three fast Kalman filter algorithms have been developed for combined characterization and monitoring of large-scale non-linear systems. The proposed methods address the current lack of algorithms for reliable and cost effective real time estimation. The developed algorithms are tested, validated and also packaged in a user-friendly software tool that can be used for learning purposes, but most importantly, that can be easily extended to perform data assimilation in a wide range of applications, extending far beyond hydrogeophysical applications.