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Mud Hydraulics Fundamentals 21

different types of fluids that are often encountered in the industry.
Curve a characterizes the fluids that are the most common in nature.
The shear stress is proportional to the shear rate, meaning that flow
resistance increases linearly with flow deformation. Water and oil are
examples of fluids in this category. These fluids are called Newtonian
fluids.
Curve b shows a linear relationship between the shear rate and the
shear stress, except in the low-shear-rate region. The shear stress takes a
nonzero value at zero shear rate. This nonzero shear stress is called gel
strength. It means that an initial force is required to deform and mobilize
the fluid. Because of the plastic behavior, this type of fluid is called plastic
fluid, or Bingham plastic fluid. Plastic fluids can be obtained by adding
claylike solid particles to Newtonian fluids.
Curve c shows a nonlinear relationship between the shear rate and the
shear stress. The flow resistance increases less than linearly with deforma-
tion. Fluid of this type is called pseudo plastic fluid, or Power Law fluid.
Polymer solutions usually fall in this category.
Curve d shows a nonlinear relationship between the shear rate and the
shear stress with a nonzero shear stress value at zero shear rate. An initial
force is required to deform and mobilize the fluid. The flow resistance
increases less than linearly with deformation. The behavior of this fluid
was first modeled by Herschel and Bulkley (1926) and is called Herschel-
Bulkley fluid.
Curve e also shows a nonlinear relationship between the shear rate
and the shear stress. The flow resistance increases greater than linearly
with deformation. This type of fluid is called dilatant fluid, which can be
obtained by adding starchlike materials to Newtonian fluids.

2.2.2 Rheological Models
Different rheological models are used to describe the flow behavior of
fluids. Newtonian fluids are described by the Newtonian model expressed
as

τ = μ _γ (2.1)

where
2
τ = shear stress, lb/100 ft or Pa
μ = viscosity, cp or Pa-s
_ γ = shear rate, s -1