Page 160 - Air and gas Drilling Field Guide 3rd Edition

6.2 General Derivation 151

where T r is the reference atmospheric temperature ( R, K). Once the reference
temperature is changed to absolute, no other changes need to be made in Equa-
tion (6-16).
The absolute average temperature T av over the first depth interval below the
surface is
T r þ T h1
T av1 ¼ : (6-17)
2
The T av for follow-on intervals will be the average of the absolute temperature at
the top and the absolute temperature at the bottom of the interval. Follow-on
average temperatures will be

T h1 þ T h2
T av2 ¼ .. . ; (6-18)
2
where T h1 is the temperature at the bottom of the first interval ( R, K) and T h2 is

the temperature at the bottom of the second interval ( R, K). Follow-on T av inter-

val temperatures are determined in sequence in a similar method as above.
The relationship between the weight rate of flow of the gas and the specific
weight and volumetric flow rate of gas at any position in the annulus is given by
_ w g ¼ g Q g ¼ g Q: (6-19)
g
Substituting Equations (6-6) and (6-14) into the two terms on the right side of
Equation (6-19) gives a relationship between the specific weight and volumetric
flow rate at the surface and the specific weight and volumetric flow rate at any
position in the annulus. This is
P g S g PS g
Q g ¼ Q: (6-20)
R e T g R e T av
Solving Equation (6-20) for Q yields

P g T av
Q ¼ Q g : (6-21)
P T g
The three-phase flow of gas, incompressible fluid, and rock cuttings up the annu-
lus can be described by a mixed specific weight term, which is a function of its
position in the annulus. This mixed specific weight g mix is

_ w t
g mix ¼ : (6-22)
P g T av
Q g þ Q m
P T g
In the derivation of Equation (6-22), the volume contribution of the solids (the
rock cuttings) is assumed to be small and negligible relative to the volumes of
the gas and the incompressible fluid in the mixture (i.e., contributes only to
the _ w t term).
The velocity of this mixture changes as a function of its position in the annu-
lus. The velocity V of the three-phase flow in the annulus is