Page 165 - Air and gas Drilling Field Guide 3rd Edition

156 CHAPTER 6 Direct Circulation Models

The velocity of this mixture changes as a function of its position in the drill string.
The velocity of the two-phase flow in the drill string is

P g T av
Q g þ Q m
P T g
V ¼ : (6-41)
p D 2 i
4
Equations (6-40) and (6-41) are functions of the pressure P at a depth of h. Substi-
tuting Equations (6-40) and (6-41) into Equation (6-39) yields
8 9
32
2 3 2
P g T av
> >
> >
Q g þ Q m
> >
< =
6 _ w g þ _ w m 7 f 6 P T g 7
6 7 1 6 7 dh: (6-42)
dP ¼
P g T av > 2gD i p D >
4 5 4 2 5
> i >
> 4 >
P T g Q g þ Q m : ;
Equation (6-42) contains only two independent variables: P and h. Separating vari-
ables in Equation (6-42) and integrating from the bottom of the inside of the drill
string to the surface of the well yields
ð ð H
P ai
dP
¼ dh; (6-43)
B i ðPÞ
P in 0
where
8 9
32
2 3 2
P g T av
> >
> >
> Q g þ Q m >
< f P =
6 _ w g þ _ w m 7 6 T g 7
6 7 1 6 7
B i ðPÞ¼
P g T av > 2gD i p D >
4 5 4 2 5
> i >
> 4 >
P T g ;
Q g þ Q m :
2
and P in is the injection pressure into the inside of the drill string (lb/ft abs,
2
N/m abs).
The friction factor f given in the aforementioned equation is determined by
the standard fluid mechanics empirical correlations relating the friction factor
to the Reynolds number, diameter, and absolute pipe roughness. In general,
values for Reynolds number, diameter, and absolute pipe roughness are known.
The classic correlation for the Reynolds number is
D i V
N R ¼ ; (6-44)
n
where D i is the inside diameter for the drill string (ft, m).
Three flow conditions can exist in the inside of the drill string. These are lam-
inar, transitional, and turbulent.
The empirical correlation for the friction factor for laminar flow conditions is
64
f ¼ N R : (6-45)
This equation can be solved directly once the Reynolds number is known. In gen-
eral, Equation (6-45) is valid for values for Reynolds numbers from 0 to 2000.