Page 264 - Air and Gas Drilling Manual

Chapter 6: Direct Circulation Models 6-9


2



 


 
  T


1 + f     P P g   T av g    Q g + Q m      dh (6-25)

π
 2 ( h − )  ( D 2 − )  
gD
2
D
D
p
  4 h p  
      

Equation 6-25 contains only two independent variables, P and h. All of the
other terms in the equation are known constants. Separating variables in Equation 6-
25 and integrating from the surface to the bottom of the well yields
∫ P bh dP = ∫ H dh (6-26)
P
a
P e B () 0
2
where P e is the exit pressure at the top annulus (lb/ft , abs),
2
P bh is the bottomhole pressure at the bottom of the annulus (lb/ft , abs),
and
 
 
 w ˙ 
B () =  t 
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 + 2 ( f − )  π  

 gD h D p  ( D 2 − )  
2
D
  4 h p  
      

For this general derivation, the exit pressure, P e, is the pressure at the entrance to the
blooey line (in the case of air or gas drilling), or the pressure at the entrance to the
return flow line (in the case of aerated fluid drilling), or the back pressure upstream
of the control valve in the exit flow line (in the case of stable foam drilling).
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
Reynolds number, diameter (or hydraulic diameter), and absolute pipe roughness. In
general, the values for Reynolds number, diameter, and absolute pipe roughness are
known. The classic expression for the Reynolds number is