Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 2: Steady laminar inertial flows

We study steady inertial flows of a Bingham fluid through 2D “washout” geometries, that represent longitudinal sections of an oil and gas well under construction, thus extending our previous work (Roustaei et al., 2015) into the inertial regime. The washout geometry is characterised by dimensionless depth h and length δ−1. The other dimensionless parameters of problem are the Reynolds number Reand Bingham number B. The effects of increasing Re were studied for fixed B and for fixed Hedström number, He=ReB. The latter represents an increase in flow rate for fixed geometry and fluid properties. In both cases we observed that the variation in flowing area of the washout (i.e. that area that is mobilised) was non-monotone. Increasing Reresulted in a straightening of the streamlines passing through the washout region, in the main part of the channel. For fixed He, beyond a first critical value of Re the flowing area was observed to decrease, i.e. increasing the flow rate results in larger parts of the washout being static, as is quite counterintuitive and contrary to the industrial perception that pumping faster will circulate/condition the mud better. This trend persists for a significant range of Re. On passing a second critical value of Re, we observe the onset of zones of recirculation within the washout. The flowing area thus increases but the area of washout from which fluid is displaced during conditioning continues to decrease. Finally, we considered self-selection of the flow geometry, as observed in Roustaei et al. (2015) for Re=0. We conclude that the same phenomenon holds in inertial flows for a given geometry, i.e. at large hthe flowing areas of washouts with static regions become similar in shape.