  When a fluid is injected into a porous medium saturated with an ambient fluid
of a greater density, the injected fluid forms a plume that rises upwards due
to buoyancy. In the near-field of the injection point the plume adjusts its
speed to match the buoyancy velocity of the porous medium, either thinning or
thickening to conserve mass. These adjustments are the dominant controls on the
near-field plume shape, rather than mixing with the ambient fluid, which occurs
over larger vertical distances. In this study, we focus on the plume behaviour
in the near-field, demonstrating that for moderate injection rates the plume
will reach a steady state, whereby it matches the buoyancy velocity over a few
plume width-scales from the injection point. However, for very small injection
rates an instability occurs in which the steady plume breaks apart due to the
insurmountable density contrast with the surrounding fluid. The steady shape of
the plume in the near-field only depends on a single dimensionless parameter,
which is the ratio between the inlet velocity and the buoyancy velocity. A
linear stability analysis is performed, indicating that for small velocity
ratios an infinitesimal perturbation can be constructed that becomes unstable,
whilst for moderate velocity ratios the shape is shown to be stable. Finally,
we comment on the application of such flows to the context of CO$_2$
sequestration in porous geological reservoirs.
