Page 172 - Air and gas Drilling Field Guide 3rd Edition

6.5 Air and Gas Drilling Model 163

ð ð H
P ai dP
¼ dh; (6-68)
B i ðPÞ
P in 0
where
8 9
32
2 3 2
P g T av
> >
> >
> Q g >
< =
_ w g T g
6 7 f 6 P 7
6 7 1 6 7 :
B i ðPÞ¼ 2
P g T av > 2gD i i >
4 5 4 p D 5
> 4 >
> >
P T g Q g : ;
Using Equations (6-6), (6-7), and (6-13), Equation (6-68) can be rearranged to give
PdP a i
ð ð H
P ai
¼ dh; (6-69)
2
2
ðP b i T Þ T av
P in av 0
where
S g
a i ¼ R e (6-70)
2 2
f _ w g
b i ¼ R e : (6-71)
p 2

2gD i S g 4
D
4 i
In the form just described, both sides of Equation (6-69) can be integrated. Using
the constants in Equations (6-70) and (6-71), the solution to Equation (6-69) is
P
ai
1 2 a i H
lnðP b i T Þ ¼ jhj : (6-72)
2
2 av T av 0
P in
Evaluating Equation (6-72) at the limits and rearranging the results give
2 2
P b i T 2a i
ln ai av ¼ H: (6-73)
2
2
P b i T av T av
in
Raising both sides of Equation (6-73) to the natural exponent gives
2
2
P b i T av ¼ e T av : (6-74)
2a i H
ai
2
2
P b i T av
in
Equation (6-74) can be rearranged and a solution obtained for P in . This is
30:5

2 2a i H
2
P þ b i T 2 e T av 1
ai av
P in ¼ 4 5 : (6-75)
2a i H
e Tav
The von Karman empirical correlation can be used to determine the friction fac-
tor in Equation (6-71)[1]. This empirical expression is
2 3 2
1
6 7
f ¼ 6 7 : (6-76)
4 5
D i
2 log þ 1:14
e