Settling Velocity of Variously Shaped Particles in Drilling and Fracturing Fluids

Abstract

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.

Sedimentation of a cylindrical particle in a Carreau fluid

“…Torrest [26] claimed that Stokes's law is capable of describing the settling of gravel chips in viscous non-Newtonian hydroxyethyl cellulose polymer solutions. Peden and Luo [27] reported the settling velocity of cylindrical and disk-shaped particles in drilling and fracturing fluids. Reynolds and Jones [28] measured the drag forces on various types of particles in non-Newtonian fluids.…”

New model for single spherical particle settling velocity in power law (visco-inelastic) fluids

“…It was also noted that the data could have been influenced by the confining walls but no wall correction was applied. Peden and Luo (1987) have also reported experimental data of spheres falling in aqueous solutions of CMC and HEC. The two constants in the relationship between drag coefficient and particle Reynolds number were reported to be functions of the power law flow behavior index but the dependence was found to be irregular.…”

“…In his investigation, unfortunately Chhabra (1990) did not acknowledge and include the relevant work reported in the petroleum engineering literature, except the study of Peden and Luo (1987). Novotny (1977) studied particle transport using vertical fractures.…”

Numerical Modelling of Cuttings Transport with Foam in Inclined Wells

“…The determination of this coefficient for Newtonian fluids has been a subject of numerous investigations in the past [86][87][88]. Several drag coefficient models for non-Newtonian fluids have also been proposed [89][90][91][92][93][94][95]. Chhabra [96] carried out a detailed comparison of the different correlations developed by the different authors for power law fluids and found out that correlations by Acharya et al [93] and Matijasic and Glasnovic [95] gave the best result followed by that of Darby [94].…”