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Mud Hydraulics Fundamentals 29

Illustrative Example 2.1
A 10.5-ppg Newtonian fluid with a viscosity of 30 cp is circulating at 250 gpm
in an 8¾-in-diameter wellbore. Determine the flow regime inside a 4½-in OD,
16.60-lb/ft drill pipe (3.826-in ID), and in the drill pipe/hole annulus.
Solution
Inside the drill pipe:
v = 250 = 6:98 ft/s
2
2:448ð3:826Þ

N Re = 928 ð10:5Þð6:98Þð3:826Þ = 8,674
30
Since N Re > 4,000, turbulent flow exists inside the drill pipe.
In the annulus:
v = 250 = 1:82 ft/s
2 2
2:448ð8:75 − 4:5 Þ

N Re = 757 ð10:5Þð1:82Þð8:75 − 4:5Þ = 2,038
30
Since N Re < 2,100, laminar flow exists in the annular space.
Unfortunately, determining flow regimes is seldom this straightforward.
Laminar flow has been observed under controlled conditions for Reynolds
numbers as low as 1,200 and as high as 40,000 (Bourgoyne et al., 1986),
although we do not usually encounter such extremes in drilling operations.

Bingham Plastic Fluids
For Bingham plastic fluids, the equations for the Newtonian fluids need
to be modified by defining an apparent viscosity to account for the plastic
viscosity and yield point. For pipe flow, the definition is
6:66τ y d
μ = μ + (2.23)
a p
v
For the annular flow, the definition is
μ = μ + 5τ y ðd 2 − d 1 Þ (2.24)
a p
v
Equations (2.23) and (2.24) are valid for U.S. field units. When expressed in
SI units, the constant 6.66 becomes 0.1669, and the constant 5 becomes 0.1253.