Bayesian Inference of Poroelastic Properties Using an Energy-based Poromechanics Model

Fluid injection into deep underground formations related to CO2 sequestration, wastewater injection, etc., is increasingly relevant for the energy sector. Injected fluids in porous deformable elastic media increase pore pressure, reduce normal effective stress, and change the available friction along fractures and faults. Consequently, slip can occur, causing seismic events. Understanding this mechanism and identifying the stress and pressure fields around the injection wellbores play a central role in assessing the seismic hazard. One of the crucial steps is inferring the unknown model parameters (i.e., poroelastic properties) from the noisy data of injection sites. Due to the indirect relation between the uncertain parameters and the empirical observation (i.e., number of earthquakes and stress drop variations in injection sites) and the high dimension of the parameters’ domain, the inverse problem is computationally expensive. 

This thesis presents a variational energy-based continuum mechanics framework to model large-deformation poroelasticity to characterize the evolution of stress, pore pressure, and other mechanical quantities. A numerical approach based on finite elements is applied to analyze the behavior of saturated and unsaturated porous media. The proposed model is amenable to the use of arbitrary energy density functions for both the fluid and solid phases. Consequently, this model can be effectively applied to multiphase porous systems, such as CO2 sequestration in deep saline sedimentary formations. Using the variational energy-based framework, we can also predict the complicated behavior of CO2 and its phase transition under high pressure close to the injection wells. 

Furthermore, the proposed variational approach can potentially have advantages for numerical methods as well as for combining with data-driven models in a Bayesian framework. Consequently, we adopt a Bayesian inference framework to integrate the partial differential equations (PDEs) of the forward mechanical model with models of uncertainty for observations and parameters. Using this approach, we can project approximate hazard assessments by exploring scenario ensembles generated from posterior knowledge. 

The Bayesian framework provides a probabilistic characterization of the unknown parameters of the physics-based model by updating the prior knowledge of these parameters based on the noisy measurements of injection sites. Maximizing the updated probability distribution or the posterior distribution provides the solution to the high-dimension inverse problem. To quantify the uncertainty and predictability of the Bayesian method’s solution, we investigate sampling algorithms and their challenges to explore high-dimension parameter spaces. We discuss the accelerated Markov Chain Monte Carlo (MCMC) algorithms using the local gradient and Hessian (of the posterior) information to get samples from the posterior distribution. 

We also investigate the variational Bayesian inference methods for quantifying the uncertainty of posterior distribution and their challenges to explore nonlinear distributions. We discuss a particle-based variational inference method. This approach allows us to approximate the posterior distribution with a set of samples or particles. We compare the performance of this method with sampling algorithms for exploring the posterior distribution.