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

TL;DR

The paper tackles hyperparameter uncertainty in geological carbon storage by introducing a hierarchical data assimilation framework that couples likelihood-free SMC-ABC for hyperparameters with ES-MDA for permeability, all accelerated by a 3D recurrent R-U-Net surrogate. This approach yields posterior hyperparameters and permeability realizations that closely match reference rejection-sampling results but with 1–2 orders of magnitude fewer forward evaluations. Across two synthetic true models, the method achieves substantial speedups (18× for true model 1 and 83× for true model 2) while maintaining accuracy in pressure and CO$_2$ saturation predictions at monitoring wells. The work demonstrates the value of treating hyperparameters hierarchically and using likelihood-free inference in subsurface history matching, with potential for extension to sequential decision-making and more advanced surrogate-assisted inference techniques.

Abstract

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.

Figures (19)

• Figure 1: Full simulation domain (left), and central storage aquifer with heterogeneous permeability field (right).
• Figure 2: Locations of the injection well and monitoring well in the storage aquifer.
• Figure 3: $\text{CO}_{2}$--brine two-phase flow functions. Curve in (b) is for porosity of 0.2 and permeability of 30 md.
• Figure 4: Pressure and saturation relative errors using the surrogate models trained with different number of training (ECLIPSE simulation) runs. Boxes show $\mathrm{P}_{90}$, $\mathrm{P}_{75}$, $\mathrm{P}_{50}$, $\mathrm{P}_{25}$ and $\mathrm{P}_{10}$ errors computed using Eqs. \ref{['error_p']} and \ref{['error_s']}.
• Figure 5: Comparison of pressure ensemble statistics from the surrogate model (red dashed curves) and simulation results (black solid curves) at grid blocks intersected by the monitoring well. The lower, middle and upper curves are $\mathrm{P}_{10}$, $\mathrm{P}_{50}$ and $\mathrm{P}_{90}$ results among the 500 test cases.