Scaling Equations for Annular Flow of Yield-Power Law Fluid and Drill Cuttings

Estimation of frictional pressure loss along the wellbore annulus is the key to determining the wellbore equivalent circulating density. Flow loop experiments are commonly used at smaller scales of flow to measure this quantity. However, a complete set of scaling equations between the measured pressure drop gradient along the annulus of a flow loop device and the one occurring in the wellbore is not reported in the literature. This study applies dimensional analysis to make such a connection while accounting for drillpipe rotation and eccentricity, as well as solid cuttings load, in the annular flow of yield-power law drilling fluids. Simultaneous application of geometric, kinematic, dynamic, and rheological similarities allows for developing direct relations between the quantities that govern the laboratory and wellbore scales of flow. For this purpose, a set of dimensionless groups is identified and held equal between the two flow scales. Dimensional analysis shows that scaling the two-phase flow of drilling fluid and drill cuttings entails 10 dimensionless groups. The solution of the obtained scaling equations determines the required volumetric rates of fluid and solid particles, the inner pipe rotation speed, and the fluid rheology in the flow loop device to establish a complete similitude with the wellbore annular flow. The required similarity between the Reynolds numbers of solid particles or drill cuttings at the two scales of flow introduces a constraint on the rheology of the fluid to be used in the laboratory flow loop. This finding indicates that attaining a complete similitude requires the two fluids to be dissimilar. Once all scaling requirements of the considered similitude are applied, the pressure gradient along the wellbore annulus can be obtained directly in terms of the measured pressure drop in the laboratory flow loop.