Goal-Oriented A Posteriori Error Estimation For The Travel Time Functional In Porous Media Flows

In this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows. The key application of this work is in the safety assessment of radioactive waste facilities; in this setting, the travel time functional measures the time taken for a nonsorbing radioactive solute, transported by groundwater, to travel from a potential site deep underground to the biosphere. To ensure the computability of the travel time functional, we employ a mixed formulation of Darcy's law and conservation of mass, together with Raviart--Thomas $H(\mathrm{div} , \Omega)$-conforming finite elements. The proposed a posteriori error bound is derived based on a variant of the standard dual-weighted-residual approximation, which takes into account the lack of smoothness of the underlying functional of interest. The proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the geological units found at the Sellafield site in the U.K.