Linearisation of the Travel Time Functional in Porous Media Flows

The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the post-closure safety assessment of deep geological storage facilities for radioactive waste where leaked, non-sorbing, solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented $a$ $posteriori$ error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its G\^ateaux derivative. A mixed finite element method is implemented to approximate Darcy's equations and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, in the UK.