Mixing in the downward displacement of a turbulent wash by a laminar spacer or cement slurry

When drilling an oil well, the well is lined by sinking a steel casing, or liner, into the drilling mud (see Figure 1). The inside of the liner forms a central pipe, and leaves an annular gap between the outside of the liner and the surrounding rock. The inside and outside of the liner are initially filled with drilling mud, which is displaced by pumping a sequence of fluids down inside the liner from the surface. Typically, a chemical wash is pumped down first, followed by a spacer, and finally a cement slurry. The wash, which is usually water based, is less dense than the spacer, a water-based suspension, which is itself less dense than the cement slurry. The scenario of a light wash being forced down a pipe by a denser spacer fluid is liable to instabilities of Rayleigh–Taylor type. The two fluids are miscible, so mixing between the wash and the spacer is likely to take place. This may impair the intended efficiency of the wash in displacing mud from the walls of the annulus. Schlumberger Dowell asked the study group to estimate the amount of mixing between the wash and the spacer, and its dependence on parameters such as the pumping rate, diameter of the pipe, viscosity of the fluids etc. In particular, they asked us to consider the following questions: