Page 164 - Air and gas Drilling Field Guide 3rd Edition

6.2 General Derivation 155

then the flow through the orifice or nozzle throat is sonic. Under these sonic
flow conditions, upstream pressure P ai does not depend on downstream pressure
P bh . For these sonic flow conditions, the upstream pressure P ai is
_ w g T 0:5
P ai ¼ bh ; (6-37)
" # 0:5
kþ1
ðÞ
2 k 1
gkS g
A n
k þ 1
R e

where T bh is the temperature of the gas at the bottom of the well ( R, K) and A n is
2 2
the total cross-sectional area of drill bit orifices or nozzles (ft ,m ).
If the upstream pressure is less than the right-hand side of either Equation
(6-35) or Equation (6-36), 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 com-
plicated calculation situation than the sonic flow condition. In this calculation sit-
uation and knowing the bottom hole pressure and bottom hole temperature, the
bottom hole specific weight must be determined using Equation (5-11). Knowing
the bottom hole pressure, temperature, and specific weight, the upstream pres-
sure can be obtained for the subsonic condition. This equation is
k
2 3
2
_ w g k 1
6 7
A n
6 7 ; (6-38)

k
P ai ¼ P bh6 þ 17
2g P bh g bh
4 5
k 1
3
where g bh is the specific weight of the gas at the bottom of the annulus (lb/ft ,
3
N/m ).
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. This is two-phase flow. 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.
Using Figure 6-1, the differential pressure at a depth h can be approximated as
2
fV
dP ¼ g mix 1 dh; (6-39)
2gD i
where D i is the inside diameter of the drill pipe, or drill collars (ft, m).
The two-phase flow of gas and incompressible fluid down the inside of the
drill string can be described by a mixed specific weight term, which is a function
of position in the drill string. This mixed specific weight term is

_ w g þ _ w m
: (6-40)
g mix ¼
P g T av
Q g þ Q m
P T g