Page 163 - Air and gas Drilling Field Guide 3rd Edition

154 CHAPTER 6 Direct Circulation Models

6.2.3 Two-Phase Flow Through the Bit
There are two basic calculation techniques for determining the pressure change
through the constrictions of the drill bit orifices or nozzles.
The first technique assumes that the mixture of incompressible fluid and the
gas passing through the orifices has a high fraction of incompressible fluid vol-
ume. Under these conditions the mixture is assumed to act as an incompressible
fluid. Thus, borrowing from mud drilling technology, the pressure change
through the drill bit DP b can be approximated by [8, 9]
2
ð _ w g þ _ w m Þ
DP b ¼ ; (6-31)
C 2 p 2 D 4
2g g mixbh 4 e
2 2
where DP b is pressure change (lb/ft , N/m ), g mixbh is the specific weight of the
3 3
mixture at the bottom of the annulus (lb/ft , N/m ), C is the fluid flow loss coef-
ficient for drill bit orifices or nozzles (the value of this constant is dependent flow
conditions), and D e is the equivalent single orifice inside diameter (ft, m). For drill
bits with n equal diameter orifices (or nozzles), D e is
q ffiffiffiffiffiffiffiffiffiffiffi
2
D e ¼ nD ; (6-32)
n
where n is the number of equal diameter orifices (or nozzles) and D n is the orifice
(or nozzle) inside diameter (ft, m).
The pressure change obtained from Equation (6-31) is added to the bottom
hole annulus pressure P bh obtained from Equation (6-26). The pressure above
the drill bit inside the drill string P ai is
P ai ¼ P bh þ DP b ; (6-33)
2
where P ai is pressure above the drill bit inside the drill string (lb/ft abs, N/m 2
2
2
abs) and P bh is bottom hole pressure (lb/ft abs, N/m abs).
For fluid mixtures that are nearly all gas (with little incompressible fluid), the
pressure above the drill bit inside the drill string will depend on whether the crit-
ical flow conditions exist in the orifices or nozzle throats. The critical pressure
through the bit orifices or nozzles is [1]
k
k 1
P bh 2
¼ ; (6-34)
P ai c k þ 1
where k is the ratio of specific heats for the gas.
The right-hand side of Equation (6-34) is determined only by the value of
the specific heat ratio constant of a gas (e.g., for air k ¼ 1.4 and for natural gas
k ¼ 1.28). Thus, for air the critical pressure ratio is 0.528 and for natural gas
the critical ratio is 0.549. Therefore, if P ai is determined to be

P bh
for air P ai (6-35)
0:528
P bh
for natural gas P ai ; (6-36)
0:549