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.