  We explore the motion of an axisymmetric gravity current in an anisotropic
porous medium in which the horizontal permeability is larger than the vertical
permeability. It is well known that the classical axisymmetric gravity current
supplied by a constant point source of fluid has an unphysical singularity near
the origin. We address this by considering a pressure-dominated region near the
origin which allows for vertical flow from the source, such that the current
remains of finite depth, whilst beyond this region the flow is
gravity-dominated. At early times the inner pressure-driven region controls the
spreading of the current, but at late times the inner region occupies a
progressively smaller fraction of the current such that the radius increases as
$\sim t^{3/7}$, while the depth near the origin increases approximately as
$\sim t^{1/7}$. The presence of anisotropy highlights this phenomenon, since
the vertical permeability maintains an effect on the flow at late times through
the pressure-driven flow near the origin. Using these results we provide some
quantitative insights into the dominant dynamics which control CO$_2$ migration
through permeable aquifers, as occurs in the context of carbon capture and
storage.
