We propose a semi-analytical closure approximation to the stratified flow of two Herschel-Bulkley fluids flowing along a circular pipe. The closure proposed appears to approximate the volumetric fluxes within 10-15% over a broad range of rheological parameters and interface heights. Accuracy is diminished at more extreme viscosity ratios when the less viscous fluid is in a thin layer. We demonstrate that the closure is able to give a good prediction of displacement front velocities in the viscous regime, comparing well with experimental values, and can be used for approximating other design features of a displacement process, e.g. the maximal static wall layer (in the case of a yield stress fluid) and the minimal flow rate to prevent backflow. (C) 2012 Elsevier B.V. All rights reserved.