Page 184 - Air and gas Drilling Field Guide 3rd Edition

7.2 General Derivation 175

7.2.4 Two-Phase Flow in the Annulus
The downward flow condition in the annulus is two phase. Differential pressure,
dP, occurs over the incremental distance of dh. This differential pressure can be
approximated as
" #
fV 2
dP ¼ g mix 1 dh; (7-34)
2gD h D p
where D h is the inside diameter of the annulus (ft, m)(the borehole) and D p is the
outside diameter of the drill string (pipe and collars) (ft, m).
The two-phase flow of gas and incompressible fluid down the annulus space to
the bottom of the well can be described by a mixed specific weight term, which is a
function of position in the annulus. This mixed specific weight term is
_ w g þ _ w m
g mix ¼ : (7-35)
P g T av
Q g þ Q m
P T g
The velocity of this mixture changes as a function of its position in the annu-
lus. The velocity of the two phase flow in the annulus is

P g T av
Q g þ Q m
P T g
V ¼ : (7-36)
p 2 2
D D
4 h p
Substituting Equations (7-35) and (7-36) into Equation (7-34) yields
2 3
6 _ w g þ _ w m 7
6 7
dP ¼
P g T av
4 5
Q g þ Q m
P T g
8 9
32
2
P g T av
> >
> >
> Q g þ Q m >
f 6 P T g 7
< =
1 6 7 dh: (7-37)
4 p 2 2 5
> 2gD h D p D D >
> >
4
> h p >
: ;
Equation (7-37) contains only two independent variables: P and h. Separating
variables in Equation (7-37) and integrating from the injection (at the surface)
to the bottom of the annulus yields
dP
P bh
ð ð H
¼ dh; (7-38)
B a PðÞ
P in 0