Adaptive modelling of variably saturated seepage problems

TL;DR

This paper introduces an adaptive finite element method for modeling variably saturated seepage in porous media, effectively reducing error in water flux predictions at seepage faces through goal-oriented adaptivity.

Contribution

The work develops a novel goal-oriented adaptive finite element approach for nonlinear seepage problems formulated as variational inequalities, focusing on error reduction in specific quantities of interest.

Findings

Rapid error reduction demonstrated in numerical examples

Method effectively handles nonlinear free boundary problems

Accurate prediction of water flux across seepage faces

Abstract

In this article we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy-Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.