Linearisation of the Travel Time Functional in Porous Media Flows

TL;DR

This paper develops a method to accurately estimate the error in travel time calculations for fluid flow in porous media, crucial for safety assessments of radioactive waste storage, by analyzing the Gâteaux derivative of the travel time functional.

Contribution

It introduces a novel approach to derive goal-oriented error bounds for the travel time functional using its Gâteaux derivative in porous media flows.

Findings

The proposed error estimator performs well in numerical tests.

Application to a Sellafield-inspired case demonstrates practical relevance.

Abstract

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.