  We consider an energy-based boundary condition to impose an equilibrium
wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on
voxel-set-type computational domains. These domains typically stem from the
micro-CT imaging of porous rock and approximate a (on {\mu}m scale) smooth
domain with a certain resolution. Planar surfaces that are perpendicular to the
main axes are naturally approximated by a layer of voxels. However, planar
surfaces in any other directions and curved surfaces yield a jagged/rough
surface approximation by voxels. For the standard Cahn-Hilliard formulation,
where the contact angle between the diffuse interface and the domain boundary
(fluid-solid interface/wall) is 90 degrees, jagged surfaces have no impact on
the contact angle. However, a prescribed contact angle smaller or larger than
90 degrees on jagged voxel surfaces is amplified in either direction. As a
remedy, we propose the introduction of surface energy correction factors for
each fluid-solid voxel face that counterbalance the difference of the voxel-set
surface area with the underlying smooth one. The discretization of the model
equations is performed with the discontinuous Galerkin method, however, the
presented semi-analytical approach of correcting the surface energy is equally
applicable to other direct numerical methods such as finite elements, finite
volumes, or finite differences, since the correction factors appear in the
strong formulation of the model.
