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176 Part III Underbalanced Drilling Systems

On the contrary, gravity causes the dense phase to slip down in a
downward flow—that is, the lighter phase moves downward slower than
the denser phase. Because of the gravity effect, in an upward flow the in
situ volume fraction of the denser phase will be greater than the input
volume fraction of the denser phase—that is, the denser phase is “held
up” in the conduit relative to the lighter phase. The terms liquid holdup
and gas holdup are used to describe the in situ volume fraction of the
liquid and gas phase in upward and downward flows, respectively. The
liquid holdup has been well documented in the petroleum production
literature, while the gas holdup has not been thoroughly studied. Liquid
holdup is mathematically defined as

y L = V L (9.1)
V
where

y L = liquid holdup, fraction
3
3
V L = volume of liquid phase in the pipe segment (ft ,m )
3 3
V = volume of the pipe segment (ft ,m )
Liquid holdup depends on flow regime, fluid properties, and conduit size
and configuration. Its value can only be quantitatively determined through
experimental measurements. A direct application of the liquid holdup is to
use it for estimating mixture specific weight in a two-phase flow:
γ = y L γ + ð1 − y L Þγ (9.2)
mix L G
where
3 3
γ = liquid specific weight (lb/ft , N/m )
L
3
3
γ = in situ gas specific weight (lb/ft , N/m )
G
Because the in situ gas specific weight is much less than the liquid specific
weight, the former is usually neglected in most engineering analyses.
9.2.3 Multiphase Flow Models

Mathematical models used for describing multiphase flow fall into two
categories: homogeneous flow models and separated flow models. Liquid
holdup is not considered in homogeneous flow models but is considered
in separated flow models.
Bubbly flow regimes can be approximately described by homoge-
neous flow models. Lage and Time’s (2000) work indicates that a bubbly