Novel Monte Carlo Approaches for Uncertainty Quantification in Porous Media Flow


Rob Scheichl, U of Bath

Abstract: The quantification of uncertainty in groundwater flow plays a central role in the safety assessment of radioactive waste disposal and of CO2 capture and storage underground. Stochastic modelling of data uncertainties in the rock permeabilities lead to elliptic PDEs with random coefficients. A typical computational goal is the estimation of the expected value or higher order moments of some relevant quantities of interest, such as the effective permeability or the breakthrough time of a plume of radionuclides. Because of the typically large variances and short correlation lengths in groundwater flow applications, methods based on truncated Karhunen-Loeve expansions are only of limited use and Monte Carlo type methods are still most commonly used in practice. To overcome the notoriously slow convergence of conventional Monte Carlo, we formulate and implement novel methods based on (i) deterministic rules to cover probability space (Quasi-Monte Carlo) and (ii) hierarchies of spatial grids (multilevel Monte Carlo). It has been proven theoretically for both of these approaches that for certain classes of problems they have the potential to significantly outperform conventional Monte Carlo. A full theoretical justification that the porous media flow applications discussed here belong to those problem classes has not yet been achieved, but experimentally our numerical results show that both methods do indeed always clearly outperform conventional Monte Carlo even within this more complicated setting.