Page 266 - Air and Gas Drilling Manual

Chapter 6: Direct Circulation Models 6-11
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 6-26 together with Equations 6-27 through 6-30 can be used in
sequential integration steps starting at the top of the annulus (with the known exit
pressure) and continuing for each subsequent change in annulus cross-sectional area
until the bottomhole pressure is determined. These sequential calculation steps
require trial and error solutions. The trial and error process requires the selection of
the upper limit of the pressure in each integral on the right side of Equation 6-26.
This upper limit pressure selection must give a right side integral solution that is
equal to the left side integral solution.
6.2.3 Two-Phase Flow Through the Bit
There are three 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 incompressible fluid volume fraction. 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,
∆P b, can be approximated by [6, 7]
g ( m) 2
w ˙ + w ˙
∆ P = 2 (6-31)
b
π
2 g γ C 2   D 4
mixbh   e
4
2
where ∆P b is pressure change (lb/ft ),
3
γ mixbh is the mixture specific weight at the bottom of the annulus (lb/ft ),
C is the fluid flow loss coefficient for drill bit orifices or nozzles (the
value of this constant is dependent on the type of gas or aerated flow),
D e is the equivalent single orifice inside diameter (ft).
For drill bits with n equal diameter orifices (or nozzles), D e becomes
D = n D 2 n (6-32)
e
where n is the number of equal diameter orifices (or nozzles),
D n is the orifice (or nozzle) inside diameter (ft).
The pressure change obtained from Equation 6-31 is added to the bottomhole
pressure P bh obtained from Equation 6-26. The pressure above the drill bit inside the
drill string, P ai, is

P ai = P bh + ∆ P b (6-33)