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

Abstract

Abstract. 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 non-sorbing 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 Hpdiv, Ωq-conforming finite elements. The proposed a posteriori error bound is derived based on a variant of the standard Dual-WeightedResidual 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 UK.Key words. Travel time functional, groundwater flows, adaptivity, goal-oriented a posteriori error estimation, mixed finite element methods 1. Introduction. In recent decades the use of numerical simulations in hydrogeological studies has become commonplace across a range of applications. Amongst these, modelling the post-closure safety performance of deep geological storage of radioactive waste is of particular interest for a posteriori error estimation. Efficient and reliable simulations are required in order to assess the suitability of a specific location for siting a waste repository. Furthermore, there is a critical need to verify any computational results with rigorous error bounds as the effects of an inaccurate simulation could be extremely costly. In the safety assessment of radioactive waste facilities, one of the key quantities of interest is the time taken for a non-sorbing radioactive solute, transported by groundwater, to travel from a potential site deep underground to the biosphere [28,41,44]. Additionally, accurate computation of the travel time has applications in streamline methods for modelling other subsurface flows; for instance in oil and gas reservoir management [36].The suitability of finite element methods has been demonstrated for many of the complex geometries and physical effects that are associated with the numerical approximation of groundwater flow and contaminant transport problems [16,17,20,21,37,41,42]. There are, however, problems associated with the use of nodal-based elements, such as lack of mass conservation at an elemental level and unphysical streamlines, as noted in [21,24,42,43]. These problems are not observed when using a (conforming) mixed finite element method, or finite difference method, in which pressure and velocity are computed simultaneously. Indeed, when employing the latter class of methods on triangular meshes, the tracing of streamlines is straightforward and has been shown to yield physical results, even o...

Linearisation of the Travel Time Functional in Porous Media Flows

“…Three numerical experiments are conducted in Section 4: two simple, academic-style examples aim to build confidence in the proposed a posteriori error estimate, while the last one adaptively simulates the leakage of radioactive waste within a domain inspired by the (albeit greatly simplified) Sellafield site, located in Cumbria, in the UK. This final, physically motivated example, matches the experiment conducted in [25] but uses the new linearisation result instead. Lastly, some concluding remarks are discussed in Section 5.…”

“…To each of the sub-domains we assign a fixed, constant, permeability and porosity (cf. Example III), given by the dataset used in [25].…”

“…In this paper, our motivation is in the post-closure safety assessment of facilities intended for use as deep geological storage of high-level radioactive waste [25,3,47,43].…”

“…Here, we are solely interested in the time-of-flight for a non-sorbing solute (which has leaked from the repository) to make its way to the surface, or boundary, of the domain; this time is represented by the (nonlinear) travel time functional. Previously, work undertaken in [25] employed goal-oriented a posteriori error estimation for this functional, relying on a finite-difference approximation of its Gâteaux derivative.…”

“…Related Literature. Groundwater flow, governed by Darcy's equations, represents a viable simplified model for the fluid flow [43,25] and will be exploited within this paper. It is assumed that whilst the surrounding rocks may not be saturated while the repository is being built, that they will eventually become saturated in its operational lifetime; thus, it is sufficient that in a post-closure assessment we can consider saturated conditions, and therefore use the time independent Darcy's equations as our model, rather than the usual Richards equations for capillary flow [26, p. 3].…”