We consider the displacement flow of a weakly compressible waxy crude oil from a pipeline, in the case that the displacing fluid is incompressible and less viscous. We show that fluid compressibility only has a significant effect on the timescale over which all residual fluid is drained from the pipeline, but no noticeable effect on the initial breakthrough of new fluid. We derive analytic estimates for this drainage time, for the cases where either the pressure drop or the displacement rate is fixed. In the case of the fixed displacement rate, the drainage time may be infinite since it is possible for there to be a residual static layer of waxy crude left behind on the walls of the pipe, see also [M. Allouche, I.A. Frigaard, G. Sona, Static wall layers in the displacement of two visco-plastic fluids in a plane channel, J. Fluid Mech. 424 (2000) 243-277]. We provide estimates for the maximal residual wax fraction, which is attained by our simplified displacement model. However, fully two-dimensional displacement simulations typically produce a significantly thinner static layer, due to the occurrence of fully two-dimensional velocity and stress fields close to the displacement front. (C) 2007 Elsevier B.V. All rights reserved.