Page 180 - Air and gas Drilling Field Guide 3rd Edition

7.2 General Derivation 171

The absolute average temperature T av over the first depth interval below the
surface is
T r þ T h1
T av1 ¼ : (7-17)
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 :::; (7-18)
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 interval
temperatures are determined in sequence in a similar method as shown 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 inside the drill string is
given by
_ w g ¼ g Q g ¼ gQ: (7-19)
g
Substituting Equations (7-6) and (7-14) into the two terms on the right side of
Equation (7-19) gives a relationship between the specific weight and volumetric
flow rate at the surface and the specific weight of volumetric flow rate at any
position inside the drill string. This is
P g S g PS g
Q g ¼ Q: (7-20)
R e T g R e T av
Solving Equation (7-20) for Q yields

P g T av
Q ¼ Q g : (7-21)
P T g
The three-phase flow of gas, incompressible fluid, and rock cuttings up the
inside of the drill string can be described by a mixed specific weight term, which
is a function of its position in the drill string. This mixed specific weight, g mix ,is
_ w t
g mix ¼ : (7-22)
P g T av
Q g þ Q m
P T g
In the derivation of Equation (7-22), the volume 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 inside the
drill string. The velocity, V, of the three-phase flow inside the drill string is

Q þ Q m
V ¼ ; (7-23)
p 2
D
4 i