Page 314 - Air and Gas Drilling Manual

Chapter 7: Reverse Circulation Models 7-23
− ∆
=
P
P
P
ai
bh
b
For fluid mixtures that have a high gas volume fraction, the pressure above the
drill bit inside the drill string can be approximated by (7-73)
 w ˙ 2   1 1 
P bh = P ai −  t   −  (7-74)
2
  gA   γ mixbh γ mixai  
 
bi
Equation 7-74 must be solved by trial and error techniques since γ mixai depends on
the pressure P bh. Note that this equation does not account for friction flow loss
through the orifices and nozzles.
The above equation will yield calculations that show that for most practical
parameters the bottomhole pressure, P bh, will differ very little from the pressure
above the drill bit opening inside the drill string, P ai. Therefore, it can usually be
assumed that
P ≈ P (7-75)
bh ai
The flow condition in the annulus is two phase (gas and fluid) flow. Equation
7-75 becomes
∫ P bh dP = ∫ H dh (7-76)
P
a
P in B () 0
where
 
 
 w ˙ + w ˙ 
B () =  g m 
P
a
  P   T av  
g
     Q + Q m 
g
   P   T   
g
   2 
    P g   T av  
    P   g   Q g + Q m   
  T

1 − f  π  

(
 2gD h − D )  ( D 2 − D )  
2
p
  4 h p  
      

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