Page 300 - Air and Gas Drilling Manual

Chapter 7: Reverse Circulation Models 7-9


2


av

 


 
  T


1 + f     P P g   T π g    Q g + Q m      dh (7-25)

 2gD i  D 2  
  4 i  
      

Equation 7-25 contains only two independent variables, P and h. All of the
other terms in the equation are known constants. Separating variables in Equation 7-
25 and integrating from the exit (at the surface) to the bottom of the inside of the
drill string yields
∫ P ai dP = ∫ H dh (7-26)
()
i
P e BP 0
2
where P e is the exit pressure at the top of the inside of the drill string (lb/ft , abs),
2
P ai is the pressure above the bit inside the drill string (lb/ft , abs),
and
 
 
 w ˙ 
BP() =  t 
i P  
  g T av  
     Q + Q m 
g
   P   T   
g
   2 
    P g   T av  
    P   g   Q g + Q m   
  T

1 + f  π  

 2gD i  D 2  
  4 i  
      

For this general derivation, the exit pressure, P e, is atmospheric pressure at the end
of the blooey line from the annulus (in the case of air or gas drilling) and at the end
of the flow line from the annulus (in the case of aerated fluid 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, 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