Page 269 - Air and Gas Drilling Manual

6-14 Air and Gas Drilling Manual
The two-phase flow of gas and incompressible fluid down the inside of the drill
string can be described by a mixed specific weight term which is a function of
position in the drill string. This mixed specific weight term is
w ˙ g + w ˙ m
γ = (6-41)
mix
 P g   T av 
    Q + Q m
g
 P   T g 
The velocity of this mixture changes as a function of its position in the drill
string. The velocity of the two-phase flow in the drill string is
 P   T av 
g
 
  P   T   Q + Q m
g
V = π g (6-42)
D i 2
4
Equations 6-41 and 6-42 are functions of the pressure P at a depth of h.
Substituting Equations 6-41 and 6-42 into Equation 6-40 yields
 
 
 w ˙ + w ˙ 
dP =  g m 
  P   T av  
g
     Q + Q m 
g
   P   T   
g
   2 
    P g   T av Q + Q  
    P     g m  
  T


1 − f  π g   dh (6-43)

 2gD i  D 2  
  4 i  
      

Equation 6-43 contains only two independent variables, P and h. Separating
variables in Equation 6-43 and integrating from the bottom of the inside of the drill
string to the surface of the well yields
∫ P ai dP = ∫ H dh (6-44)
()
i
P in BP 0
where