Scaling Formulae for the Wellbore Hydraulics Similitude with Drill Pipe Rotation and Eccentricity

Equivalent circulation density of the fluid circulation system in drilling rigs is determined by the frictional pressure losses in the wellbore annulus. Flow loop experiments are commonly used to simulate the annular wellbore hydraulics in the laboratory. However, proper scaling of the experiment design parameters including the drill pipe rotation and eccentricity has been a weak link in the literature. Our study uses the similarity laws and dimensional analysis to obtain a complete set of scaling formulae that would relate the pressure loss gradients of annular flows at the laboratory and wellbore scales while considering the effects of inner pipe rotation and eccentricity. Dimensional analysis is conducted for commonly encountered types of drilling fluid rheology, namely, Newtonian, power-law, and yield power-law. Appropriate dimensionless groups of the involved variables are developed to characterize fluid flow in an eccentric annulus with a rotating inner pipe. Characteristic shear strain rate at the pipe walls is obtained from the characteristic velocity and length scale of the considered annular flow. The relation between lab-scale and wellbore scale variables are obtained by imposing the geometric, kinematic, and dynamic similarities between the laboratory flow loop and wellbore annular flows. The outcomes of the considered scaling scheme is expressed in terms of closed-form formulae that would determine the flow rate and inner pipe rotation speed of the laboratory experiments in terms of the wellbore flow rate and drill pipe rotation speed, as well as other parameters of the problem, in such a way that the resulting Fanning friction factors of the laboratory and wellbore-scale annular flows become identical. Findings suggest that the appropriate value for lab flow rate and pipe rotation speed are linearly related to those of the field condition for all fluid types. The length ratio, density ratio, consistency index ratio, and power index determine the proportionality constant. Attaining complete similarity between the similitude and wellbore-scale annular flow may require the fluid rheology of the lab experiments to be different from the drilling fluid. The expressions of lab flow rate and rotational speed for the yield power-law fluid are identical to those of the power-law fluid case, provided that the yield stress of the lab fluid is constrained to a proper value.