Likelihood-Free Inference and Hierarchical Data Assimilation for Geological Carbon Storage

Data assimilation will be essential for the management and expansion of geological carbon storage operations. In traditional data assimilation approaches a fixed set of geological hyperparameters, such as mean and standard deviation of log-permeability, is often assumed. Such hyperparameters, however, may be highly uncertain in practical CO2 storage applications where measurements are scarce. In this study, we develop a hierarchical data assimilation framework for carbon storage that treats hyperparameters as uncertain variables characterized by hyperprior distributions. To deal with the computationally intractable likelihood function in hyperparameter estimation, we apply a likelihood-free (or simulation-based) inference algorithm, specifically sequential Monte Carlo-based approximate Bayesian computation (SMC-ABC), to draw posterior samples of hyperparameters given dynamic monitoring well data. In the second step we use an ensemble smoother with multiple data assimilation (ESMDA) procedure to provide posterior realizations of grid-block permeability. To reduce computational costs, a 3D recurrent R-U-Net deep learning-based surrogate model is applied for forward function evaluations. A rejection sampling (RS) procedure for data assimilation is applied to provide reference posterior results. Detailed posterior results from SMC-ABC-ESMDA are compared to those from the reference RS method. Close agreement is achieved with 'converged' RS results, for two synthetic true models, in all quantities considered. Importantly, the SMC-ABC-ESMDA procedure provides speedup of 1-2 orders of magnitude relative to RS for the two cases. A modified standalone ESMDA procedure is introduced for comparison purposes. For the same number of function evaluations, the hierarchical approach is shown to provide superior results for posterior hyperparameter distributions and monitoring well pressure predictions.