Page 238 - Air and gas Drilling Field Guide 3rd Edition

9.4 Nonfriction and Friction Illustrative Examples 229

multiphase oil and gas flow tubing production theory to aerated drilling annulus
problems. This production theory application includes only major friction losses
and was not applicable to complicated borehole geometry. New additions to this
methodology, which do not come from production literature, have included major
and minor losses and can be applied to complicated borehole geometry. Also
included in this second methodology is the inclusion of liquid holdup correlations,
which were first examined in the production literature [16, 17].
Chapter 6 derived the basic aerated fluid drilling governing equations and pre-
sented their auxiliary friction factor and nozzle flow equations. These equations
form the foundation for both methodologies as discussed in this treatise.

9.4.1 Nonfriction Approximation
The simple nonfriction methodology allows straightforward deterministic approx-
imate solutions of aerated drilling problems. However, the practical applicability
of these nonfriction solutions is limited to shallow (generally less than 3000 ft
of depth) wells with simple geometric profiles. The nonfriction solution will be
applied to a deep well example only as a demonstration and ultimate comparison
of the results to those obtained from the friction solution.
In what follows, the basic equations in Chapter 6 for aerated drilling are used
to derive the nonfriction governing equation. Letting f ! 0 in Equation (6-48)
yields

ð ð H
dP
P bh
3 ¼ dh;
2
0
P e
6 7
6 _ w t 7
6 7
6 7

P g T av
4 5
Q g þ Q f
P T g
2
where P bh is the pressure at the bottom of the annulus (lb/ft , abs) and P e is the
2
pressure at the exit from the annulus (lb/ft , abs). This equation can be rear-
ranged and integrated to yield

P bh
H

T av Q g Q f
¼ h :
P g In P þ P

T g _ w t _ w t
0

P e
Evaluating the aforementioned equation at the limits, rearranging the result, and
solving for gas volumetric flow rate, Q g , yields

T av P bh
ð
P g Q g ln þ Q f P bh P s Þ ¼ _ w t H: (9-7)
T s P s
The nonfriction solution is usually not applied to a deep well like our example.
This is only done as a demonstration. It is instructive to compare the nonfriction
solution results to the full friction solution results obtained later in this chapter.
The approximate values for the volumetric flow rates of the incompressible fluid