Page 245 - Air and gas Drilling Field Guide 3rd Edition

236 CHAPTER 9 Aerated Fluids Drilling

9.4.3 Major and Minor Losses and the Effect of Fluid Holdup
The mathematical models described by Equation (6-26) for the fluid flow in the
annulus was formulated on the assumption of homogeneous multiphase flow
where all the fluids are assumed to flow at the same velocity. In reality, in vertical
upward multiphase flow streams, the liquid and solid phases always flow slower
(due to gravity) than the gas phase, resulting in liquid holdup. In vertical down-
ward multiphase flow streams, the liquid phase (due to gravity) flows faster than
the gas phase, which results in gas holdup.
In upward flow (in the annulus), differences in phase velocities result in annu-
lus volume fractions of liquids and solids that are different from the volume frac-
tions at the injection point (surface). To be specific, the amount of pipe occupied
by a phase is often different from its proportion of the total volumetric flow rate
at the injection point. This allows the annulus to essentially “store” more liquid
in it relative to the homogeneous model discussed in the previous subsection.
The most important empirical model for this type of flow is the generalized
Hagedorn–Brown model [5].
Empirical models similar to those just described for downward multiphase
flow (inside the drill pipe) and for multiphase flow through the drill bit nozzles
do not exist for petroleum applications.

Liquid Holdup
The term liquid holdup is defined mathematically as
V L
y L ¼ ; (9-9)
V
where y L is the incompressible (liquid) fluid fraction, V L is the liquid phase
3
3
3
3
volume (ft ,m ), and V is the total volume in the annulus (ft ,m ).
Liquid holdup depends on flow regime, fluid properties, and pipe size and
configuration. Its value can only be determined quantitatively through experi-
mental measurements. A direct application of the liquid holdup is to use it for
estimating mixture specific weight in two-phase flow. This mixed specific weight
was used in Equation (6-26). This new mixed specific weight is
¼ y L g þ 1 y L Þg ; (9-10)
ð
g mix L G
3
3
where g L is the liquid-specific weight (lb/ft , N/m ) and g G is the gas-specific
3 3
weight (lb/ft , N/m ).
The generalized Hagedorn–Brown correlation is widely used in petroleum pro-
duction engineering for multiphase flow performance calculations [5]. Most
applications pertain to multiphase production fluids flowing up the inside of
the production tubing. This discussion applies the generalized Hagedorn–Brown
correlation to multiphase drilling fluids up the annulus.
The generalized Hagedorn–Brown correlation determines liquid holdup values
from three empirical charts [5]. All of these charts are accessed by using the fol-
lowing additional dimensionless numbers: