We present an analytical solution for buoyancy-driven "filling box" flows in axisymmetric porous media having closed bottom and side boundaries. The flow consists first and foremost of a descending, point source plume. When plume fluid reaches the (horizontal) bottom boundary, it begins to flow radially outward in the form of an axisymmetric gravity current. The leading edge of the gravity current advances with time as until it reaches the vertical sidewalls. At this point, the flow is characterized by a vertically ascending "first front" that steadily advects towards the plume source. We assume the plume to be in a Darcy regime, i.e. , with , where Re and Pe are, respectively, the Reynolds and P,clet numbers, and derive a similarity solution for the plume by applying a boundary layer approximation. Formulas are therefore obtained for the vertical variation of the plume volume flux and area-averaged concentration. The former result shows important qualitative differences with the analogue equation derived in the limit . In particular, the plume volume flux is now predicted to explicitly depend on the reservoir permeability, plume buoyancy flux and fluid viscosity. The gravity current problem is likewise solved using a self-similar solution, this time adapted from the work of Lyle et al. (J Fluid Mech 543:293-302, 2005) but connected to the outflow conditions of the plume. Finally, in solving for the motion of the first front, we apply a volume flux balance equation and therefore estimate the time scale required for the first front to advect from the bottom of the control volume to the source elevation. By synthesizing the above results, we can estimate the total volume of source fluid and mass of solute that can be injected into an axisymmetric reservoir without overflow. Predictions can also be made for the time-variable mean concentration of this contaminated fluid layer, which must obviously be less than the source concentration.