Page 188 - Air and gas Drilling Field Guide 3rd Edition

7.5 Air and Gas Drilling Model 179

The exit pressure in reverse circulation air (or gas) drilling operations is atmo-
spheric pressure, P at , at the top of the inside of the drill string. Separating vari-
ables in Equation (7-43) yields
ð ð H
P ai
dP
¼ dh; (7-44)
B i PðÞ 0
P at
where
8 9
32
2 3 2
P g
> T av >
> >
> Q g >
< f P =
6 _ w t 7 6 T g 7
6 7 1 þ 6 7 :
B i PðÞ ¼
P g T av > 2gD i D >
4 5 4 p 2 5
> >
> 4 i >
P T g Q g : ;
Using Equations (7-6), (7-7), and (7-13), Equation (7-44) can be rearranged to give
ð ð H
P ai PdP a i
¼ dh; (7-45)

2
P þ b i T 2 T av
P at av 0
where

S _ w s
a i ¼ 1 þ (7-46)
_ w g
R e
2 2
f _ w g
b i ¼ R e : (7-47)
p 2
S
2gD i 4
D
4 i
In this form, both sides of Equation (7-45) can be integrated. Using Equations
(7-46) and (7-47), the solution to Equation (7-45) is
P
ai
1 2 a i H
ln ðP þ b i T Þ ¼ jhj : (7-48)
2
2 T av
av 0
P at
Evaluating Equation (7-48) at the limits and rearranging the results give
2 2
P þ b i T 2a i
ln ai av ¼ H: (7-49)
2
P þ b i T 2 T av
at av
Raising both sides of Equation (7-49) to the natural exponential exponent gives
2a i H
2
2
P þ b i T av ¼ e T av : (7-50)
ai
2
P þ b i T 2
at av
Equation (7-50) can be rearranged and a solution obtained for P ai . This will yield
2 3 0:5
2a i H

P þ b i T e (7-51)
2 7
6 2 2 T av b i T 5 :
P ai ¼ 4 at av av
The von Karman empirical correlation for wholly turbulent flow conditions can
be used to determine the friction factor in Equation (7-47)[1]. This correlation is