Page 274 - Air and Gas Drilling Manual

Chapter 6: Direct Circulation Models 6-19
pipe and drill collars, and inside surface of the casing. The openhole surfaces of
boreholes can be approximated with an absolute roughness, e oh = 0.01 ft (i.e., this
example value is the same as concrete pipe which approximates borehole surfaces in
limestone and dolomite sedimentary rocks, or in similar competent igneous and
metamorphic rocks, see Table 8-1).
Equation 6-57 together with Equations 6-52, 6-53 and 6-58 can be used in
sequential calculation steps starting at the top of the annulus and continuing for each
subsequent change in cross-sectional area in the annulus until the bottomhole
pressure is determined.
The flow condition through the drill bit orifices or nozzles is single phase (air or
gas) flow. The character (sonic or subsonic) of the gas flow through the drill bit
orifices or nozzles is determined by the critical pressure ratio equation. The critical
pressure ratio equation for bottomhole conditions is
k
 P bh   2 k  −1
  =   (6-59)
 P   k + 1 
ai
c
Using k = 1.4 for air and k = 1.28 for natural gas, then from Equation 6-59 if P ai is
determined to be
P bh
for air P ai ≥ (6-60)
0 528
.
P bh
for natural gas P ai ≥ (6-61)
0 549
.
the flow through the orifice or nozzle throat is sonic. Under these sonic flow
conditions, the upstream pressure, P ai, does not depend on downstream pressure, P bh.
For sonic flow conditions the upstream pressure is

wT bh . 05
˙
g
P ai = (6-62)
  k+  1  . 05
  gk S   2   k−  1 
A n      
  R   k + 1  
 
If the upstream pressure is less than the right hand side of either Equation 6-60
(or 6-61), the flow through the orifices or nozzles is subsonic and the upstream
pressure will be dependent upon the pressure and temperature at the bottom of the
well (downstream). For these subsonic conditions the upstream pressure is