Page 181 - Air and gas Drilling Field Guide 3rd Edition

172 CHAPTER 7 Reverse Circulation Models

where D i is the inside diameter of the drill string (ft).
Substituting Equation (7-21) into Equation (7-23) yields

P g T av
Q g þ Q m
P T g
V ¼ : (7-24)
D
p 2
4 i
Substituting Equations (7-22) and (7-24) into Equation (7-1) yields
2 3
6 _ w t 7
6 7
dP ¼
4 5
P g T av
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 þ 6 7 dh: (7-25)
2gD i 4 p 2 5
> D >
> i >
4
> >
: ;
Equation (7-25) contains only two independent variables: P and h. All of the
other terms in the equation are known constants. Separating variables in
Equation (7-25) and integrating from the exit (at the surface) to the bottom of
the inside of the drill string yields
dP
ð ð H
P ai
¼ dh; (7-26)
B i PðÞ
P e 0
2
where P e is the exit pressure at the top of the inside of the drill string (lb/ft abs,
2
N/m abs), P ai is the pressure above the bit inside the bottom of the drill string
2 2
(lb/ft abs, N/m abs), and
2 3
_ w t
6 7
6 7
B i PðÞ ¼
4 5
P g T av
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 þ 6 7 :
4 p 2 5
2gD i
> D >
> >
4
> i >
: ;
For this general derivation, exit pressure, P e , is atmospheric pressure at the end of
the blooey line from the top of the inside of the drill string (in the case of air or
gas drilling) and at the end of the return flow line from the top of the inside of the
drill string (in the case of aerated fluid drilling or stable foam drilling).
The Darcy–Weisbach friction factor f given in the aforementioned equation is
usually determined by standard fluid mechanics empirical correlations relating
the friction factor to the Reynolds number, diameter, and absolute pipe roughness.