Page 263 - Air and Gas Drilling Manual

6-8 Air and Gas Drilling Manual
Solving Equation 6-20 for Q yields
g
Q =   P   T av   Q (6-21)
 
 P   T  g
g
The three phase flow of gas, incompressible fluid, and rock cuttings up the
annulus can be described by a mixed specific weight term which is a function of its
position in the annulus. This mixed specific weight, γ mix, is
w ˙
γ = t (6-22)
 P g   T av 
mix
    Q + Q m
g
 P   T g 
In the derivation of Equation 6-22 the volume contribution of the solids (the rock
cuttings) is assumed to be small and negligible relative to the volumes of the gas
and the incompressible fluid in the mixture.
The velocity of this mixture changes as a function of its position in the annulus.
The velocity, V, of the three phase flow in the annulus is
Q + Q
V = π D ( 2 m p) (6-23)
2
4 h − D

where D h is the inside diameter of the outer surface of the annulus (the borehole) (ft),
D p is the outside diameter of the drill string (drill pipe and collars) (ft).
Substituting Equation 6-21 into Equation 6-23 yields
 P   T av 
g
 
  P   T   Q + Q m
g
V = g (6-24)
π D ( 2 − p)
2
4 h D
Substituting Equations 6-22 and 6-24 into Equation 6-1 yields
 
 
 w ˙ 
dP =  t 
  P   T av  
g
     Q + Q m 
g
   P   T   
g