Page 257 - Air and Gas Drilling Manual

6-2 Air and Gas Drilling Manual
used are treated fresh water, treated salt water (formation water), water based drilling
muds, diesel oil, oil based drilling muds, and crude oil (formation oil).
It is assumed that compressible gases can be approximated by the perfect gas
law. Further, it is assumed that the mixture of compressed gas and incompressible
fluid will be uniform and homogeneous. When the solid rock cuttings are added to
the mixture of compressible gas and incompressible fluid, the solid rock particles are
assumed to be uniform in size and density and will be distributed uniformly in the
mixture of gas and fluid. Also, it is assumed that the rock particles move with the
same velocity as circulating gas and fluid and that the resulting uniform mixtures
can be approximated by known basic fluid mechanics relationships [1].
The assumption of uniformity of the two or three phases in the mixtures is an
important issue in light of the technology developed for gas lift assisted oil
production [2, 3]. The aeration of oil (or other formation produced fluids) from the
bottom of a well with the flow of gas from the surface (down the annulus between
the casing and the production tubing) is somewhat similar to the aeration of fluid
and rock cuttings from the bottom of a well with a flow of gas and fluid from the
surface (down the inside of the drill string). However, in most oil production
situations the two phase flow takes place inside of the tubing. In the drilling
situation, the gas and fluid are injected together into the top of the drill string and
move together down the inside of the drill string, through the bit orifices or nozzles,
and then the resulting three phase flow (gas, fluid, and rock cuttings) moves up the
annulus to the surface. Thus, the geometry of flow for the two operations is quite
different and probably not comparable [4].
6.2 General Derivation
The term, P in, represents the pressure of the injected drilling fluids into the top
of 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 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 a 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 temperature
at the exit from the circulation system. Therefore, the derivation will begin with the
analysis of the flow of the gas in the annulus and will continue through the
circulation system in the upstream direction. Thus, this derivation will start with
the annulus, continues through the drill bit orifices, and then continue up the inside
of the drill string to the surface. Figure 6-1 shows the 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 differential pressure, dP, in the upward flowing
three phase flow occurs over an incremental distance of dh. This differential pressure
can be approximated as [1]