A probabilistic collocation Eulerian-Lagrangian localized adjoint method on sparse grids for assessing CO₂ leakage through wells in randomly heterogeneous porous media

We develop a probabilistic collocation Eulerian-Lagrangian localized adjoint method on sparse grids for assessing CO₂ leakage through wells in randomly heterogeneous porous media, by utilizing the intrinsic mathematical, numerical, and physical properties of the mathematical model. We model the process in which CO₂ is injected into a deep aquifer, spreads within the aquifer and, upon reaching a leaky well, rises up to a shallower aquifer, to quantify the leakage rate, which depends on the pressure build-up in the aquifer due to injection and the buoyancy of CO₂. The underlying Eulerian-Lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in CO₂ sequestration. The sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre-existing Eulerian-Lagrangian methods in a nonintrusive manner. It also provides a scalable framework to consider uncertainty in a straightforward parallel manner. Preliminary numerical experiments show the feasibility and potential of the method.