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190 Part III Underbalanced Drilling Systems

The superficial velocity of a given phase is defined as the volumetric
flow rate of the phase divided by the total pipe cross-sectional area for
flow. The third term on the right side of Eq. (9.23) represents a pressure
change due to a kinetic energy change, which is in most instances negligi-
ble in UBD operations.
Obviously, determining the value of liquid holdup y L is essential for
pressure calculations. The Hagedorn‒Brown correlation uses liquid
holdup from three charts based on the following dimensionless numbers:
Liquid velocity number, N vL :
r ffiffiffiffiffi
γ
4 L
N vL = F vL u SL (9.25)
σ
Gas velocity number, N vG :

γ
r ffiffiffiffiffi
4 L
N vG = F vG u SG (9.26)
σ
Pipe diameter number, N D :
γ
r ffiffiffiffiffi
L
N D = F D D H (9.27)
σ
Liquid viscosity number, N L :
r ffiffiffiffiffiffiffiffiffi
N L = F L μ 4 1 (9.28)
L
γ σ 3
L
In U.S. field units, the unit conversion factors are
F vL = 1:938
F vG = 1:938
F D = 120:872
F L = 0:15726

In SI units, the unit conversion factors are
F vL = 1:7964
F vG = 1:7964
F D = 31:664
F L = 0:55646
The first chart in Hagedorn and Brown’s (1965) work is used for deter-
mining the value of parameter (CN L ) based on N L . Guo and colleagues