A New Generalized Model for Predicting Particle Settling Velocity in Viscoplastic Fluids
Authors: Okesanya Temitope , Kuru Ergun 
Published: September 2019

Abstract
An experimental study was conducted to measure the settling velocity of spherical particles in viscoplastic fluids. Using a mechanistic model based on the balance of the forces acting on the settling particle and detailed statistical analyses of the experimental results, a generalized model for predicting settling velocity of spherical particles in viscoplastic fluids was developed. The main objectives of the study were: i.) To measure the terminal settling velocity of particles in various viscoplastic fluids intending to expand the present database of experimental data ii.) To develop a new Drag coefficient-particle Reynolds number (CD-Rep) correlation that is applicable to both Newtonian and non-Newtonian viscoplastic fluids iii.) To present a general non-iterative approach for predicting settling velocities of particles in Newtonian and non-Newtonian viscoplastic fluids irrespective of their rheological models (Casson Model, Herschel Bulkley Model, and Bingham Model etc.).
The settling velocities of the spherical particles (Specific gravity ranging from 2.5 - 7.7; Diameters: ranging from 1.09 - 4.00 mm) in various Carbopol solutions were measured using Particle Image Shadowgraphy (PIS). The experimental results were combined with experimental data published in the literature to broaden the range and applicability of empirical analysis. Advanced statistical analysis programs (OriginPro 9.0 and MATLAB r2018b) were utilized together with extensive experimental data to develop a new CD-Rep correlation. In this study, a new modified shear Reynolds number (ReT*) was introduced, which physically quantifies the effects of non-Newtonian fluid rheological properties on the settling velocity. The newly developed CD-Rep correlation and the modified shear Reynolds number were incorporated into the Wilson et al. (2003) model to develop a generalized model that can be used for predicting particle settling velocity in viscoplastic fluids.
We have shown that presented new model predicts settling velocity better and yielded relatively more accurate results than existing models with the lowest approximate Mean Absolute Error (MAE) of 0.1 m/s for all data points. In addition to enhanced prediction accuracy, this new model occludes application constraints and offers prediction versatility that is lacking in current existing models by being valid for diverse rheological models of non-Newtonian viscoplastic fluids. The paper is concluded by presenting an illustrative and pragmatic example to calculate the terminal velocity of a spherical particle in a non-Newtonian viscoplastic fluid using the presented generalized model.
The knowledge of particle settling velocity in viscoplastic fluids is indispensable for the design, analysis, and optimization of a wide spectrum of industrial processes such as cuttings transport in oil and gas well drilling and proppant transport in hydraulic fracturing operations. By augmenting the current corpus of experimental data; we have provided much-needed particle settling velocity database that can be used for modeling of relevant transport processes (i.e. cuttings and/or proppants transport). Finally, by combining a mechanistic model describing the forces acting on the settling particles with the newly developed CD-Rep correlation, we have presented a new generalized predictive model of particle settling velocity in viscoplastic fluids that can be used for the optimization of particle transport in oil and gas well drilling and hydraulic fracturing operations.