Geology Reference
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6.3.2 Non-Newtonian fluids.
When the flow is non-Newtonian, few published models for very low
frequency waves are available. In order to obtain rough attenuation estimates,
we adopt the following procedure. Non-Newtonian drilling fluids are typically
modeled by a simple power law rheology model with “n” and “K” coefficients.
The apparent viscosity is variable throughout the pipe cross-section of the pipe
and depends on the radius R and the volume flow rate Q. The volume flow rate
Q in steady flow is exactly
Q = {SR 3 /(3 + 1/n)} {R/(2K)} 1/n (dp/dz) 1/n (6.3c)
from which we can calculate the pressure gradient dp/dz. What is the effective
Newtonian viscosity P effective that will give the same flow rate Q for identical
parameters n, K and R? For this, we use the well known pipe flow Hagen-
Poiseuille formula rewritten in the form
P effective = {SR 4 /(8Q)} dp/dz (6.3d)
and substitute the pressure gradient known from the first calculation. This
viscosity value can then be used with the calculator of Figures 6.3a and 6.3b.
Again, this procedure gives an approximate “engineering solution” that is
roughly correct and useful for MWD job planning purposes.
Figure 6.3a . Newtonian model input form.
Figure 6.3b . Newtonian attenuation solutions.
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