The Characterization of Non-Newtonian Systems by A Dual Differentiation-Integration Method
Authors: Savins J.G. 
Published: December 2007

Abstract
Analytical procedures are described for optimizing the selection of a rheological model when it is desired to express the functional relationship between true shear rate and shearing stress in analytical form. The procedures are extended to two principal model categories - "Generalized Newtonian" and "Viscoplastic" models. Graphical examples are presented to illustrate the utility of certain characteristic derivative functions in distinguishing between these categories and in determining how well a particular model reflects the flow behavior of rheologically anomalous materials in simple shearing flows.

INTRODUCTION
Modern drilling fluids and fracturing fluids, being chemically complex, are frequently non-Newtonian - a peculiarity which complicates the solution of problems such as designing flowline and process equipment, and predicting laminar-turbulent behavior, solid - suspending ability and cutting- transport efficiency. Therefore, techniques of data analysis which accurately characterize the behavior of rheologically complex materials are needed for the accurate formulation of solutions of these and similar problems. It is the purpose of this paper to describe the application of a sensitive method of rheological data analysis - the dual differentiation-integration method1 of optimizing the selection of the functional relation between shear rate and shearing stress - to a variety of materials of interest in drilling and production practices.

BASIC CONCEPTS

THE DIFFERENTIATION METHOD
To a first approximation, the viscosity function for a non-Newtonian system can be expressed in a form analogous to Newton's law of viscosity by the following generalization2,3 between the stress tensor r (i.e., the viscous part of the pressure tensor) and D the rate of deformation tensor.Equation 1
in which the viscosity function is some function, say, of the three invariants I1, I2 and I3 of D, as follows.Equations 2–4
On introducing the restrictions that the system is incompressible and the motion is rectilinear, then I1 = 0 and I3 = 0; and with the further restriction that the flow is slip-free, the viscosity (an invariant) is then determined solely by I2. With this approach one can set up a suite of integral equations relating observed kinematical and dynamical parameters for certain types of flow experiments involving simple shearing motion, e.g., flow in a cylindrical tube, flow between rotating concentric cylinders, and flow between fixed parallel surfaces of large aspect ratio. Taken in the order cited, these integral equations are as follows.

THE DIFFERENTIATION METHOD
To a first approximation, the viscosity function for a non-Newtonian system can be expressed in a form analogous to Newton's law of viscosity by the following generalization2,3 between the stress tensor r (i.e., the viscous part of the pressure tensor) and D the rate of deformation tensor.Equation 1
in which the viscosity function is some function, say, of the three invariants I1, I2 and I3 of D, as follows.Equations 2–4
On introducing the restrictions that the system is incompressible and the motion is rectilinear, then I1 = 0 and I3 = 0; and with the further restriction that the flow is slip-free, the viscosity (an invariant) is then determined solely by I2. With this approach one can set up a suite of integral equations relating observed kinematical and dynamical parameters for certain types of flow experiments involving simple shearing motion, e.g., flow in a cylindrical tube, flow between rotating concentric cylinders, and flow between fixed parallel surfaces of large aspect ratio. Taken in the order cited, these integral equations are as follows.