Scaling Equations for Benchtop Laboratory Simulator of Wellbore Hydraulics

Determining frictional pressure losses along a wellbore annulus is the key to estimation of the wellbore equivalent circulating density. Flow-loop experiments are often used at smaller scales of flow to measure the frictional pressure losses. However, a complete set of scaling equations between the measured pressure drop in a flow loop device and the one occurring in the wellbore has not been reported in the literature. This study applies dimensional analysis to make such connection while accounting for drill pipe rotation, eccentricity, and cuttings load in the annular flow of power-law drilling fluids. Simultaneous application of geometric, kinematic, dynamic, and rheological similarities allows for developing direct relations between the operational and flow quantities at the laboratory and wellbore scales of flow. For this purpose, the pertinent dimensionless groups are identified and set equal between the two flow scales. Results indicate that scaling the two-phase flow of drilling fluid and cuttings entails nine (9) dimensionless groups. The obtained scaling equations provide the required volumetric rate of fluid and particles, the inner pipe rotation speed, as well as the fluid rheology and other design parameters of the flow-loop device to establish the full similitude with the corresponding wellbore hydraulics. In particular, the Reynolds number of cuttings necessitates introducing a constraint on the rheology of fluid to be used in the laboratory flow loop. 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.