Page 268 - Air and Gas Drilling Manual

wT
˙
=
P
(6-38)
ai
. 05

k+  1

k−  1

A n      gk S   g bh 2 . 05    Chapter 6: Direct Circulation Models 6-13

 
  R   k + 1  
 
where T bh is the temperature of the gas at the bottom of the well (˚R),
2
A n is the total cross-sectional area of the drill bit orifices (or nozzles) (ft ).
If the upstream pressure is less than the right hand side of either Equations 6-36 (or
6-37), the flow through the orifices or nozzles is subsonic and the upstream pressure
will be dependent on the pressure and temperature at the bottom of the borehole
annulus. The subsonic flow condition is a more complicated calculation situation
than the sonic flow condition. In this calculation situation and knowing the
bottomhole pressure and bottomhole temperature, the bottomhole specific weight
must be determined using Equation 4-11. Knowing the bottomhole pressure,
temperature, and specific weight, the upstream pressure can be obtained for the
subsonic condition. This equation is
k
 w  2 k  −1
  ˙ g 
   A n   
P = P  + 1  (6-39)
ai bh    
2 g  k  P γ bh 
bh
  k − 1  
 
3
where γ bh is the specific weight of the gas at the bottom of the annulus (lb/ft ). Note
that Equations 6-38 and 6-39 do not account for friction flow loss through the
orifices and nozzles.
6.2.4 Two-Phase Flow in the Drill String
The combined gas and incompressible fluid are injected into the top of the
inside drill string. Therefore, the flow of the fluids inside of the drill string must be
assumed to be two phase. The differential pressure, dP, in the downward flowing
two phase flow occurs over the incremental distance of dh at the depth h for a total
depth of well of H.
The differential pressure at a depth h can be approximated as
 fV 
2
dP = γ mix 1 −  dh (6-40)
  2 gD i  
where D i is the inside diameter of the drill pipe, or drill collars (ft).