A variational framework for inverse modeling: Case study in CO₂ sequestration

This study investigates probabilistic Bayesian inversion methods for CO₂ sequestration problems, focusing on the challenges of estimating subsurface properties in high-dimensional spaces, such as permeability and porosity, and accurately quantifying uncertainties. Traditional approaches, such as gradient-based optimization, Monte Carlo sampling, and Kalman filter-based methods, have notable limitations. Gradient-based methods are computationally efficient but fail to capture uncertainty and are prone to local minima. Monte Carlo methods, while effective in posterior estimation, become computationally infeasible in high-dimensional settings due to the curse of dimensionality. Kalman filter-based methods offer some uncertainty estimation but are limited by their reliance on Gaussian-shaped posterior and weakly nonlinear assumptions for the model-data relationships. To address these challenges, this study explores variational inversion (VI) techniques, recasting Bayesian inference as an optimization problem to efficiently approximate the posterior distribution. Specifically, we apply automatic differentiation variational inference (ADVI) and Stein variational gradient descent (SVGD) to 2D and 3D CO₂ sequestration inverse problems and compare their performance against Monte Carlo sampling and the ensemble smoother with multiple data assimilation (ES-MDA). Our results demonstrate that ADVI and SVGD offer significant computational advantages over traditional methods, while still be able to capture a reliable posterior estimation. ADVI provides more reasonable mean and uncertainty estimates than ES-MDA, even with smaller number of forward runs. SVGD delivers better mean and uncertainty estimates than both ADVI and ES-MDA, with the same computational cost as ES-MDA. Both ADVI and SVGD show better scalability than ES-MDA and MCMC, meaning they are less affected by the increasing dimensionality of the model parameters. These findings highlight the potential of ADVI and SVGD to offer a reliable alternative to traditional MCMC and ES-MDA methods.