An Accurate Slot-Flow Model for Non-Newtonian Fluid Flow Through Eccentric Annuli

This paper describes a new model for obtaining analytical solutions to the problem of non-Newtonian fluid flow through eccentric annuli. A discussion on non-Newtonian rheology is presented, followed by the development and solution of applicable differential equations using the Ostwald de Waele power-law model and a nonrectangular slot.Results indicate that velocity values are reduced greatly in the reduced region of the eccentric annulus. This is important in directional drilling where the drillpipe tends to lie against the hole. Design of mud flow for cuttings transport on the basis of the nominal average velocity could lead to serious problems associated with cuttings buildup in the low-velocity region of the annulus. Other practical applications of this work include the determination of velocity distribution in chemical processes involving fluid flow through eccentric annuli - e.g., heat exchangers and extruders - and more accurate velocity profiles inside journal bearings, particularly for small diameter ratios.The main advantage in the new approach is that iterative finite difference methods used by previous investigators are avoided. Previous work along present lines used a linearized model and resulted in velocity profiles of unacceptable accuracy. This study improves both the accuracy and the solution technique. Introduction In the petroleum industry, engineers routinely encounter Newtonian and non-Newtonian fluid flow through eccentric annuli during well drilling and, on a smaller scale, during through-casing production, gravel packing, and gas lifting. In analyzing the behavior of drilled cuttings in a wellbore annulus, previous investigators traditionally have assumed that the drillpipe and the hole or casing are concentric. As depicted in Fig. 1, the drillpipe usually is not concentric with the hole, especially during directional drilling when the pipe weight causes a strong tendency for the pipe to lie against the hole. Hence, a realistic prediction of cuttings behavior in an annulus necessarily includes an analysis of the velocity distribution of the transport fluid at various assumed levels of pipe/hole eccentricity.To ensure field applicability of the results, it is necessary to avoid complicated mathematical models that yield analytically intractable solutions. Since equations describing non-Newtonian flow through parallel plates are generally easier to manipulate than conventional annular-flow equations, the eccentric annulus is represented by a nonrectangular slot as shown in Figs. 2 and 3.The associated theory, results, and application are discussed in this paper. To permit use of the results in a wide variety of situations, results are presented in terms of dimensionless ratios. To set the stage for these discussions, several related publications are analyzed briefly. A more detailed literature review can be found in Ref. 1.One of the first studies on the subject was performed in 1955 by Tao and Donovan. They carried out both theoretical and experimental work on laminar and turbulent flow through narrow annuli and showed that the flow through an annulus with a rotating inner pipe could be treated as a higher flow velocity through an annulus of greater length with stationary walls.In 1959, Heyda carried out an analytical investigation of eccentric annulus velocity distribution. SPEJ P. 565^