Page 284 - Air and Gas Drilling Manual

g (
f)
2
w ˙
∆ P =
(6-93)
b
π
2 g γ mixbh + C 2 w ˙   2 D e 4 Chapter 6: Direct Circulation Models 6-29
 
4
The magnitude of C represents the loss coefficient of the drill bit nozzles for the
aerated fluid mixture of the stable foam components. The value of C for this aerated
fluid mixture is assumed to be 0.70 to 0.85. For drill bits with n equal diameter
orifices (or nozzles), D e becomes
D = n D 2 n (6-94)
e
The pressure change obtained from Equation 6-93 is added to the bottomhole
pressure P bh obtained from Equation 6-88. The pressure above the drill bit inside the
drill string, P ai, is
P ai = P bh + ∆ P b (6-95)
For performed stable foam flowing through the orifices (or nozzles), the pressure
above the drill bit inside the drill string can be approximated by
 2 
w ˙ g (  + w ˙ f)   
P = P +    1 − 1  (6-96)
ai bh 2 γ
 gA n    γ mixbh mixai  
 
Equation 6-96 must be solved by trial and error techniques since γ mixai depends on
the pressure P ai. Note that this equation does not account for friction flow loss
through the orifices and nozzles.
For fluid mixtures that are nearly all gas (with little incompressible fluid), the
pressure above the drill bit inside the drill string will depend upon whether the
critical flow conditions exist in the orifices or nozzle throats. The critical pressure
through the bit orifices or nozzles is
k
 P bh   2 k  −1
  =   (6-97)
 P  c  k + 1 
ai
The right hand side of Equation 6-97 is determined only by the value of the
specific heat ratio constant of a gas (e.g., for air k = 1.4 and for natural gas k =
1.28). Thus, for air the critical pressure ratio is 0.528 and for natural gas the critical
ratio is 0.549. Therefore, if P ai is determined to be