Page 156 - Air and gas Drilling Field Guide 3rd Edition

6.2 General Derivation 147

6.2 GENERAL DERIVATION
The term P in represents the pressure of the injected drilling fluids into the top of
the drill string. The U-tube representation in Figure 6-1 shows the larger inside
diameter of the drill pipe at the top of the drill string where the drilling fluids
are injected. Below the drill pipe is shown the smaller inside diameter of the drill
collars, and below the drill collars is shown a schematic of the drill bit orifices (or
nozzles). The schematic shows the smaller annulus space between the outside of
the drill collars and the inside of the open borehole. Above is the annulus space
between the outside of the drill pipe and the inside of the open borehole. Then at
the top (in the annulus space) is the largest annulus space between the outside of
the drill pipe and the inside of the casing. At the top of the annulus the drilling
fluids 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 from the circulation system. Therefore, the derivation will
begin with the analysis of the flow of the gas and incompressible fluids in the
annulus and will continue through the circulation system in the upstream direc-
tion. Thus, this derivation will start with the annulus, continue through the drill
bit orifices, and then continue up the inside of the drill string to the surface. Fig-
ure 6-1 shows pressure P at any position in the annulus, which is referenced from
the surface to a depth by the term h. The total depth of the well is H. The differ-
ential pressure dP in the upward flowing three-phase flow occurs over an incre-
mental distance of dh. This differential pressure can be approximated as [1]
2
fV
dP ¼ g mix 1 þ dh; (6-1)
2gðD h D p Þ
2 2
where P is fluid pressure (lb/ft abs, N/m 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 other
3
3
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 h is the
inside diameter of the borehole (ft, m); D p is the outside diameter of the drill pipe
2
2
(ft, m); and g is the acceleration of gravity (32.2 ft/sec ,9.81 m/sec ).
The first term on the right side of Equation (6-1) represents the incremental
pressure change due to the hydrostatic weight of the column of fluid (with
entrained rock cuttings) in the annulus. The second term on the right side of
Equation (6-1) represents the incremental increase pressure change due to the
friction loss of the flowing fluid mixture.

6.2.1 Weight Rate of Flow of the Gas
In order to carry out derivation of the governing equations for direct 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