An Improved Mathematical Model for Relating Shear Stress to Shear Rate in Drilling Fluids and Cement Slurries
Authors: Robertson R.E. , Stiff H.A. 
Published: December 2007

Abstract
The Newtonian, Bingham, and power law models previously have been used to approximate the previously have been used to approximate the rheology of drilling fluids and cements. The proposed yield-pseudoplastic model provides more consistently accurate descriptions of the rheology of such fluids. Simple explicit relationships between the wall shear rate and the volumetric flow rate in both pipe and annular flow have been derived from this model for use in engineering calculations.

Introduction
Two mathematical models have been widely used with drilling fluids and cement slurries for relating shear stress to shear rate. The most popular is that of Bingham,.T = Ty + ny, .............................(1)
which describes this relationship as linear after an initial yield. Very few, if any, drilling fluids or cement slurries conform to this model, and no explicit relationship can be derived between the shear rate and the volumetric flow rate in a pipe or an annulus.
In recent years, the Ostwald-de Waele or "power law" model,.T = K yn,...................................(2)
has gained popularity. Eq. 2 describes a fluid with no yield stress and a constant ratio between the logarithms of the shear stress and the shear rate over a workable range. Simple explicit relationships between the shear rate and the volumetric flow rate in a pipe and an annulus can be derived from the equation, but the model often does not fit actual shear stress and shear rate data.
Actual shear stress/shear rate data for many fluids place them in the category of yield-pseudoplastics, fluids that exhibit a yield stress as well as a nonlinear relationship between shear stress and shear rate once flow is initiated. A three-parameter model for such fluids, proposed by Herschel and Bulkley, combines the characteristics of the Bingham and power law models:.T = Ty + K yn ..............................(3)
Eq. 3 describes the behavior of yield-pseudoplastics reasonably well, but again, no explicit relationship can be derived between the shear rate and the volumetric flow rate in a pipe or an annulus.
Thus, the need exists for a model that will adequately describe yield-pseudoplastics, such as drilling fluids and cement slurries, and that has the analytical utility of the power law model for engineering calculations.

PROPOSED MODEL PROPOSED MODEL
The proposed model takes the form.T = A (y + C)B,.............................(4)
It adequately describes the relationship between shear rate and shear stress for most drilling fluids and cement slurries. A simple explicit equation replacing shear rate to the volumetric flow rate in a pipe or annulus can be derived from Eq. 4. As an pipe or annulus can be derived from Eq. 4. As an added feature, the values of the constants characterize the fluid. Thus, it can be seen that when B = 1.0 and C = 0, Eq. 4 becomes.T = A y, ...................................(5)
which describes the flow properties of a Newtonian fluid. When B = 1.0 and C 0, the fluid is a Bingham plastic, as described in Eq. 1. When B 1.0 and plastic, as described in Eq. 1. When B 1.0 and C = 0, the fluid follows the power law model, as shown in Eq. 2.
The parameters A and B can be considered similarly to the parameters of the power law model. However, the third parameter, C, has a somewhat different connotation than the yield stress of the Bingham model.
SPEJ
P. 31