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Mud Hydraulics Fundamentals 33

In the annulus:
2 3
n

ρν ð2−nÞ d 2 − d 1
6 7
N Re = 4ð2n + 1Þ 6 2 n 7 (2.38)
n 6 d 2 − d 1 n 2ð2n + 1Þ 5
7
4
τ y + K
2ν nC
a
where the constants C c and C are expressed respectively as follows:

a

C c = 1 − 1 τ y n (2.39)
2n + 1 " ð3n + 1Þq #
τ y + K
3
nπðd/2Þ
and

C = 1 − 1 τ y ) (2.40)

a ( n
n + 1
τ y + K 2qð2n + 1Þ
2 2
nπ½ðd 2 /2Þ − ðd 1 /2ފ½ðd 2 /2Þ − ðd 1 /2Þ Š
The critical Reynolds number N Rec inside the drill pipe and in the
annulus can be estimated respectively as

1

4ð3n + 1Þ 1−z
N Rec = (2.41)
ny
and
1

8ð2n + 1Þ 1−z
N Rec = (2.42)
ny
where
logðnÞ + 3:93
y = (2.43)
50

z = 1:75 − logðnÞ (2.44)
7

The critical Reynolds number (N Rec ) is the criterion for determining the
flow regime. The flow becomes turbulent once it is over the critical
Reynolds number; otherwise it is laminar flow.