3 - Concentric, Rotating, Annular Flow

Abstract

This chapter describes the analytical solutions for concentric, rotating, and annular flow. Analytical solutions for the nonlinearly coupled axial and circumferential velocities, their deformation, stress and pressure fields, are obtained for the annular flow in an inclined borehole with a centered, rotating drillstring or casing. The closed form solutions are used to derive formulas for volume flow rate, maximum borehole wall stress, apparent viscosity, and other quantities. The result for Newtonian flow, an exact solution to the Navier–Stokes equations, is considered first, without geometric approximation. An approximate solution for pseudoplastic and dilatant power law fluids is developed. It is found that for Newtonian flows, the stress is linearly proportional to the shear rate and the proportionality constant is the laminar fluid viscosity. In studies on borehole erosion, annular velocity plays an important role, since drilling mud carries abrasive cuttings. The magnitude of fluid shear stress may also be important in unconsolidated sands where tangential surface forces assist in wall erosion. The individual components can be obtained by evaluating equations for power law fluids, and equations Newtonian fluids.