Summary The settling velocities of a variety of shaped particles to simulate drilled cuttings were measured in both Newtonian and non-Newtonian fluids. The results showed that the particle drag coefficient is a function of the particle Reynolds number and, in the case of power-law-model fluids, of the flow behavior index. A new generalized model has been developed for predicting the setting velocities of particles of various shapes in both Newtonian and power-law fluids over a range of flow regimes. Introduction and Previous Investigations Drilling and fracturing fluids are generally classed as power-law-type fluids, and their viscosities vary with shear rate. The problems of drilled cuttings settling out from drilling fluids and of proppants from fracturing fluids are complicated by the shear-dependent characteristics of the fluids. To account for the non-Newtonian effect of drilling fluids on the settling velocity of drilled cuttings, Zeidler1 suggested use of the apparent viscosity at the wall and Moore2 adapted the effective viscosity for annular flow, as defined by Skelland.3 Note that the apparent viscosity or the effective viscosity represents the viscosity at a specific shear rate pertaining to that annular location in an annular flow situation, and does not necessarily represent the viscosity around the settling particles. When the fluid velocity approaches zero and the fluid becomes stagnant, both apparent and effective viscosities will approach infinity. For particles settling in fracturing fluids, several investigators suggested the use of an effective shear rate on a particle to calculate the equivalent Newtonian viscosity around the particle. Novotny4 suggested vp/ds and Daneshy5 suggested 3(vp/ds) to characterize the shear rate for stagnant fluids. When the fluid is in motion, Novotny4 claims that the effective shear rate on a particle is the vector sum of the shear rate caused by particle settling, vp/ds, and the shear rate imposed by fluid motion. On the basis of an apparent viscosity substitution, Shah6 established the correlation of [CD(2-n)NRem'2]½ vs. NRem' and found that this correlation was a function of the flow behavior index. Acharya7 considered the viscoelastic effect of some fracturing fluids and suggested use of a sophisticated drag-coefficient correlation for purely viscous non-Newtonian fluids and another correlation to account for the elastic effect. Other experimental work has been reported on investigations of the settling velocity of particles in drilling fiuids8–13 and fracturing fluids.14,15 In the previous investigations, however, no attempt was made to establish a drag-coefficient correlation for nonspherical particles settling in non-Newtonian fluids, such as disks or rectangular plates. These particles may be used to approximate the shape of drilled cuttings. Theory of Drag-Coefficient Correlation Assuming that the particles are separated sufficiently during settling so that they do not collide or interact with each other, the force causing a particle to settle may be expressed asEquation (1) The resistant force induced by the particle's motion consists of two components. One is the fluid viscous drag, which may be expressed asEquation (2) where Ap is the characteristic area of the particle parallel to the direction of motion. Anther component is the pressure drag, which may be expressed asEquation (3) where AN is the characteristic area of the particle normal to the direction of motion. The total resistant force, usually simply called the "drag force," is the sum of these two components and may be expressed asEquation (4) where CD is the drag coefficient and A is the characteristic area of the particle, which depends on the shape of the particle and its orientation during motion. For particles of different shapes, the distribution of the drag force between viscous drag and pressure drag may vary considerably. For a flat particle settling flatwise, pressure drag will predominate; for settling in an edgewise fashion, viscous drag will be dominant.

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A new physically based model is described in this paper which has been developed based on the analysis of forces acting on the cuttings and the associated dimensionless groups. The model can be used to predict the critical (minimum) flow rate (CFR) required to remove, or prevent the formation of, stationary cuttings beds on the low-side of the wellbore in deviated wells. The model has been validated with the experimental data obtained from an 8″ wellbore simulator located at BP Research Centre Sunbury, and with field data from 8-1/2″, 12-1/4″ and 17-1/2″ holes. By using the model, the effects of major drilling variables on cuttings transport are evaluated and compared with experimental and actual field data. The results show that the model accurately predicts the effects of the major variables such as: hole angle, penetration rate, mud properties and flow regime. In particular, the model provides a mechanism to support the use of both low and high viscosity pills, and mud low-shear enhancers - which have been widely used during drilling operations to improve hole cleaning. The model has been incorporated into a computer program which can predict the CFR under the specified drilling conditions. A series of simplified hole cleaning charts have also been developed which enable the optimum hole cleaning parameters to be determined at the rig-site.

Summary. The laminar eccentric annular flow of non-Newtonian fluids is analyzed with a new method where an eccentric annulus is represented by an infinite number of concentric annuli with variable outer radii. The analytical solutions for the shear stress, shear rate, velocity, and volumetric flow rate/pressure gradient are obtained for both power-law and Bingham-plastic fluids. This method is shown to provide more accurate approximations for various profiles and good predictions of the volumetric flow rate/pressure gradient in eccentric annular flow. In addition, turbulent eccentric annular flow is discussed. Introduction Fluid flow through an annular space is an often-encountered engineering problem that has been under investigation for many decades. If the annular space is concentric, the flow can currently be analyzed without much difficulty. But if the annular space is eccentric, - i.e., the axes of the inner and outer tubes do not coincide with each other - a great deal of effort is required. Unfortunately, the latter case represents the majority of realistic situations. For example, in drilling operations, the drillpipe is usually positioned eccentrically in the wellbore, especially in a deviated wellbore where the drillpipe has a strong tendency to offset toward the low side because of gravitational effects. A number of studies have focused on the eccentric annular flow problem. Initially, mathematicians studied Newtonian fluids as a hydrodynamic problem, using the bipolar-coordinate system to transform the eccentric annular geometry into a rectangular region. Redberger and Charles used a similar approach to evaluate the velocity profile and the volumetric flow rate for Newtonian fluids numerically. Subsequently, Mitsuishi and Aoyagi extended the approach to non-Newtonian fluids, and Guckes presented procedures for calculating the volumetric flow rate for power-law and Bingham-plastic fluids. Although the bipolar-coordinate method may theoretically give exact solutions, the procedures are extremely tedious and involve laborious computations. To find simple approximations, Tao and Donovans treated an eccentric annulus as a variable-height slot and developed the analytical solutions of the velocity profile and the volumetric flow rate for Newtonian fluids. Vaughn later extended this method to power-law fluids, and Iyoho and Azar modified the model to calculate the slot height. Tosun and liner et al. recently extended Iyoho and Azar's slot-height model to approximate volumetric flow rates for both Newtonian and non-Newtonian fluids. Note that the slot model, because it is in essence a modified model for flow between parallel plates, will result in unrealistic symmetric profiles of the shear-stress/shear-rate magnitudes and the velocity. In this analysis, an eccentric annulus is treated as being composed of an infinite number of concentric annuli with variable outer radii. Using this method, we develop analytical solutions for the shear stress, shear rate, velocity, and volumetric flow rate/pressure gradient for both power-law and Bingham-plastic fluids. Concentric Annular Flow The power-law and Bingham-plastic models are the most commonly used models for describing the rheological behavior of viscous shear-thinning fluids, which represent the majority of non-Newtonian fluids. The power-law model is usually expressed as (1) and the Bingham-plastic model can be written as (2) where "+" is for -Y less than 0 and "-" is for 'Y >0. Fredrickson and Bird and Laird first analyzed the flow o power-law and Bingham-plastic fluids through concentric annuli. Based on their analyses, for the steady-state and isothermal flow of incompressible fluids through concentric annuli, the equation of motion can be integrated in cylindrical coordinates to yield (3) where c is an integration constant and gp is the pressure gradient defined by(4) Power-Law Fluids. For power-law fluids, applying the boundary condition rzr = 0 at r = ro, we can establish the expression for the shear-stress profile as(4) The expression for the shear-rate profile can be found by combining Eqs. 1 and 4:(5) where "+" is for r less than ro, "-" for r >ro, and s = 1/n. The velocity profile is obtained by integrating Eq. 5: (6a) (6b) From Eqs. 4 through 6, we can see that the velocity profiles and the magnitudes of shear stress/shear rate are not symmetric about the radial position of r=ro, where the shear stress is zero and the velocity is maximum. The determining equation for ro may be found by combining Eqs. 6a and 6b and setting them equal for r=ro:(7) Fig. 1 shows ro/r2 vs. n and r1/r2. ro is rather insensitive to the variation of the flow-behaviour index, n. In fact, within an error of about 3 %, Eq. 7 may be approximated by setting n = 1 for the cases where n>0.5 and r1/r2>0.3; i.e., (8) Hanks and Larsen found that the volumetric flow rate in the concentric annular flow of power-law fluids may be expressed as (9) where qaD, the dimensionless volumetric flow rate, may be defined as(10) SPEPE P. 91^

The eccentric annular flow of power-law and Bingham plastic fluids is analysed in this paper using a new method in which an eccentric annulus is represented by infinite concentric annuli with variable outer radii. For the case of power-law fluids, ana1ytical solutions of the shear stress and velocity profiles have been obtained which are valid over the entire eccentric annulus. For Bingham plastic fluids, the profiles have been obtained at the minimum and maximum dimension positions of the eccentric annulus, which are -considered to be the important positions for the analysis of cuttings transport in an inclined eccentric drilling annulus. The results show that the local velocity and shear stress have greater magnitude in the enlarged region of an eccentric annulus than in the reduced region. In addition, like the case of concentric annular flow, the present analysis indicates that the profiles of the velocity and the magnitude of the shear stress in an eccentric annulus are not symmetric in the radial direction. Previous studies gave either unrealistic symmetric profiles or very complicated solutions. Based on the above analysis, the effect of the eccentricity of a drilling annulus on cuttings transport is evaluated. The results of this analysis will aid in the design of drilling programmes, particularly for deviated wells, to ensure efficient transport of drilled cuttings. Introduction Drilling engineers deal routinely with the flow of drilling fluids which are non-Newtonian in rheological behaviour, in the annular space between the drillpipe and the casing or wellbore. Traditionally, the assumption is made that the drillpipe is placed concentrically in the casing or wellbore when the annular flow and the behaviour of the transported drilled cuttings is analysed. In reality, however, the drillpipe is usually positioned eccentrically, especially in deviated wells where the drillpipe has a strong tendency to lie against the low side of the hole. A number of studies have been reported for the flow of non-Newtonian fluids through eccentric annuli. Heyda, Redberger and Charles, Mitsuishi and Aoyagi, and Guckes approached the problem by using bipolar co-ordinates to define the eccentric annular geometry and developed methods for the calculation of the velocity profile which involve extensive numerical iterative computation. Vaughn and Iyoho and Azar treated an eccentric annulus as a slot with variable height and developed the analytical models of the velocity profile for power-law fluids. However, because their models are in essence the modified model for flow between parallel plates, unrealistic symmetric velocity and linear shear stress profiles result. In this paper, the annular flow of drilling fluids is analysed based on the eccentric annular geometry using a new method which treats an eccentric annulus as an infinite concentric annuli with variable outer radii. The two most commonly used rheological models i.e. power-law and Bingham plastic models, are used in the analysis to define the rheological behaviour of the drilling fluids. Then, based on the above analysis, the effect of the pipe/hole eccentricity on cuttings transport is evaluated.