Page 305 - Air and Gas Drilling Manual

7-14 Air and Gas Drilling Manual

2


 

 
  T

1 − f     P P g   T av g    Q g + Q m         

π
 2 ( h − )  ( D 2 − )  
gD
2
D
D
p
  4 h p  
      

2
where P in is the injection pressure into the top of the annulus space (lb/ft , abs).
The Fanning friction factor f given in the above equation is determined by the
standard fluid mechanics empirical expressions relating the friction factor to the
Reynolds number, diameter (or hydraulic diameter), and absolute pipe roughness. In
general, the values for Reynolds number, diameter, and absolute pipe roughness are
known. The classic expression for the Reynolds number is
( D h − ) V
D
p
N = (7-41)
R
ν
where D h D p is the hydraulic diameter for the annulus (ft).
There are three flow conditions that can exist in the annulus. These are laminar,
transitional, and turbulent flow conditions.
The empirical expression for the Fanning friction factor for laminar flow
conditions is
f = 64 (7-42)
N R
This equation can be solved directly once the Reynolds number is known. Equation
7-42 is valid for values for Reynolds numbers from 0 to 2,000.
The empirical expression for the Fanning friction factor for transitional flow
conditions is known as the Colebrook equation. The Fanning friction factor cannot
be determined directly and must be solved by trial and error. This empirical
expression is
   
  e  
.
1 =− 2 log    D − D p  + 251   (7-43)
h
f  37 . N R f 
 
   
where e is the absolute roughness of the annulus surface (ft). Equation 7-43 is valid
for values of Reynolds numbers from 2,000 to 4,000.
The empirical expression for the Fanning friction factor for turbulent flow
conditions is known as the von Karman equation. This empirical expression is