Page 176 - Air and gas Drilling Field Guide 3rd Edition

7.2 General Derivation 167

7.2 GENERAL DERIVATION
The term P in represents the pressure of the injected drilling fluid into the top of
the annulus. The U-tube representation in Figure 7-1 shows the largest annulus
space between the outside of the drill pipe and the inside of the casing. Next is
the annulus space between the outside of the drill pipe and the inside of the open
hole. Then at the bottom of the annulus is the space between the outside of the
drill collars and the inside of the open hole. At the bottom of the drill string is the
single large opening in the drill bit that allows the drilling fluids with entrained
rock cuttings to pass into the inside of the drill string. The schematic shows
the smaller inside diameter of the drill collars. Above the drill collars is the larger
inside diameter of the drill pipe. At the top of the drill pipe the drilling fluid with
the entrained cuttings exit the circulation system at pressure P e .
As in all compressible flow problems, the process of solution must commence
with a known pressure and temperature and, in this case, the pressure and tem-
perature at the exit. Therefore, the derivation will begin with the analysis of the
inside of the drill string. Figure 7-1 shows pressure P at any position in the inside
of the drill string, which is referenced from the surface to a depth h. The total
depth of the well is H. The differential pressure dP in the upward flowing
three-phase flow occurs over an incremental distance of dh. This differential pres-
sure can be approximated as [1]
2
fV
dP ¼ g mix 1 þ dh; (7-1)
2gD i
2
2
where P is fluid pressure (lb/ft abs, N/cm abs); h is the reference depth (ft, m);
H is the total depth (ft, m); g mix is the specific weight of the mixture of air (or
3
3
other gas), incompressible fluid, and rock cuttings (lb/ft , N/m ); f is the
Darcy–Weisbach friction factor; V is the average velocity in the annulus (ft/sec,
m/sec); D i is the inside diameter of the drill string (ft, m); and g is the acceleration
2
2
of gravity (32.2 ft/sec , 9.81 m/sec ).
The first term on the right side of Equation (7-1) represents the incremental
pressure change due to the hydrostatic weight of the column of fluids (with
entrained rock cuttings) inside the drill string. The second term on the right side
of Equation (7-1) represents the incremental pressure change due to the friction
resistance to the flowing fluids in the drill string.

7.2.1 Weight Rate of Flow of the Gas
In order to carry out derivation of the governing equations for reverse circulation,
the weight rate of flow of air (or gas) to the well must be determined. Assuming
that the compressed air is provided by a compressor(s), the weight rate of flow
through the circulating system is determined from the atmospheric pressure and
temperature of the air at the compressor location on the surface of the earth,
and the characteristics of the compressor(s). For air, the atmospheric pressure