  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.
