When a near-horizontal well is sealed by means of a cement plug, a slow buoyancy-driven slumping flow can result along the axis of the well. The heavy plug slumps downwards along the bottom of the well, displacing the lighter wellbore fluids towards the top. Thus, a lower tail can slump down from the plug, towards the bottom of the well, while an upper tail can slump upwards, towards the top of the well. We derive a bound for the maximal length which the cement may slump, in each tail, before the motion is arrested by the yield stress of the cement slurry. We also predict the profile of the interface between fluids when the slumping motion stops. Our initial analysis is carried out for the geometrically simpler situation of a flow between parallel plates. Later we generalise our approach to flows in a pipe geometry, divided by a stratified interface. (C) 2004 Elsevier B.V. All rights reserved.