Page 313 - Air and Gas Drilling Manual

7-22 Air and Gas Drilling Manual
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1
37 .
f =− 2 log         D e i    + N 251 f      (7-70)
R
 
   
Equation 7-70 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
  2
 
 1 
f =   (7-71)
D 

 2 log. 
i
+ 1 14
   e   
 
Equation 7-71 is valid for values of Reynolds numbers greater than 4,000.
For follow-on calculations for flow in the drill string the absolute roughness for
commercial pipe, e p = 0.00015 ft, will be used for the inside surfaces of the drill
pipe and drill collars.
Equation 7-67 together with Equations 7-68 through 7-71 can be used in
sequential trial and error integration steps starting at the top of the inside of the drill
string (with the known exit pressure) and continuing for each subsequent change in
cross-sectional area inside the drill string until the pressure above the drill bit inside
the drill string is determined. These sequential change in drill string cross-sectional
area until the pressure at the bottom of the inside drill string is determined.
The mixture of incompressible fluid and the gas passing through the single
orifice has a high incompressible volume fraction and can, therefore, be assumed to
act physically as an incompressible mixture. Thus, borrowing from mud drilling
technology, the pressure change through the drill bit, ∆P b, can be approximated by
w ˙ 2
∆P = t (7-72)
b   2
π
2 g γ mixai C 2   D bi 4
4
The value of C represents the aerated fluid flow loss coefficient of the drill bit
orifice. Experiments have shown that aerated fluid flow is ver complex. The gas
and incompressible fluid components in the aerated mixture appear to alternate their
passage through the orifice. This means the aerated flow through the orifice is
inefficient. The value of C for aerated fluid flow should be taken as 0.70 to 0.85.
The pressure change obtained from Equation 7-72 is subtracted from the pressure
above the drill bit inside the drill string P ai obtained from Equation 7-67. The
annulus bottomhole pressure, P bh, is