Page 308 - Air and Gas Drilling Manual

f
b =
(7-49)
S 
i gD   R  2 w ˙ g 2 2 Chapter 7: Reverse Circulation Models 7-17
π
2  
i 4
  D i
4
In the above form, both sides of Equation 7-47 can be integrated. Using Equations
7-48 and 7-49 the solution to Equation 7-47 is
ln P ( 2 + b T av) = i h (7-50)
1 P ai a H
2
2 i T av 0
P at
Evaluating Equation 7-50 at the limits and rearranging the results gives
 P 2 + b T  2 a
2
ln  ai i av  = i H (7-51)
  P at 2 + b T   T av
2
i
av
Raising both sides of Equation 7-51 to the natural exponential exponent gives
2 aH
2 2 i
P ai + b T av T av
i
2 2 = e (7-52)
P at + b T av
i
Equation 7-52 can be rearranged and a solution obtained for P ai. This yields
.
 2 aH  05
i
−
2
T av
P =  P ( 2 + bT av) ebT 2  (7-53)
ai  at i i av 
   
The flow of gas to the well must be sufficient to carry solid rock cuttings from
the bottom of the annulus to the surface. Therefore, it is assumed that the flow
condition in both the inside of drill string and the annulus is turbulent. The
empirical von Karman relationship for determining the approximate Fanning friction
factor for the inside of the drill string is
  2
 
f =   1   (7-54)
  D  
i


+ 1 14
 2 log. 

e 
 