Hydraulics - Deformation of a Fluid by Simple Shear - Power Law Model

Power Law Model

This model is defined by the relationship:

Shear Stress = Consistency Factor x Shear Rate flow behavior index

It describes the thickness or pumpability of the fluid, and is somewhat analogous to the apparent viscosity. The flow behavior index (n) indicates the degree of non-Newtonian characteristics of the fluid. As the fluid becomes more viscous, the consistency factors (k) increases; as a fluid becomes more shear thinning “n” decreases. When “n” is 1 the fluid is Newtonian. If “n” is greater than 1, the fluid is classed as Dilatant (the apparent viscosity increases as the shear rate increases). If “n” is between zero and 1 the fluid is classified as Pseudoplastic, exhibiting shear-thinning; (i.e., the apparent viscosity decreases as the shear rate increases). For drilling fluids, this is a desirable property and most drilling fluids are pseudoplastics.

While the Power Law Model is more accurate then the Bingham Model at low shear rates, it does not include a yield stress. This results in poor results at extremely low shear rates.

A modification to the Power Law Model, the OXY Model, was proposed for use in oil-based muds. The major difference is the viscometer readings used to determine the “k” and “n” values. Power Law uses the 300 and 600 rpm readings, the OXY Model uses the 6 and 100 rpm readings. In addition, other models have been proposed that tend to exhibit behavior between the Bingham and Power Law models at low shear rates.

Non-Newtonian fluids may show a degree of time-dependent behavior. (For example, the apparent viscosity for a fixed shear rate does not remain constant, but varies to some maximum or minimum with the duration of shear.) If the apparent viscosity decreases with flow time, the fluid is termed “Thixotropic”. Once flow has ceased, a thixotropic fluid will show an increase in apparent viscosity. When apparent viscosity increases with flow time, the fluid is “Rheopectic”. The shear stress developed in most drilling fluids is dependent upon the duration of shear. A time lag exists between an adjustment of shear rate and the stabilization of shear stress. This is due to the breaking up of clay

particles at high shear rates and the aggregation of clay particles when shear rate is decreased, both occurrences take a noticeable length of time.

“Gel strength” is used to measure this time dependent behavior. This gel strength measures the attractive forces of a fluid while under static conditions. If the gel strength increases steadily with time, the gel strength is classed strong or progressive. If it increases slowly with time, it is classed as weak or fragile.

When strong gels occur, excessive pressures may be required to break circulation.