Page 304 - Air and Gas Drilling Manual

The velocity of this mixture changes as a function of its position in the annulus.
The velocity of the two phase flow in the annulus is
 P   T av  Chapter 7: Reverse Circulation Models 7-13
g
  P   T   Q + Q m
 
g
V = g (7-38)
π D ( 2 − p)
2
4 h D
Substituting Equations 7-37 and 7-38 into Equation 7-36 yields
 
 
 w ˙ g + w ˙ m 
dP =  
  P   T av  
g
     Q + Q m 
g
   P   T   
g
   2 
    P g   T av  
    P     Q g + Q m  
  T


1 − f  π g   dh (7-39)

 2 ( h − )  ( D 2 − )  
D
gD
2
D
p
  4 h p  
      

Equation 7-39 contains only two independent variables, P and h. Separating
variables in Equation 7-39 and integrating from the injection (at the surface) to
bottom of the annulus yields
∫ P bh dP = ∫ H dh (7-40)
P
a
P in B () 0
where
 
 
 w ˙ g + w ˙ m 
B () =  
P
a
  P   T av  
g
     Q + Q m 
g
   P   T   
g