Page 303 - Air and Gas Drilling Manual

7-12 Air and Gas Drilling Manual
γ mixbh is the mixture specific weight at the bottom of the annulus (lb/ft ).
Equation 7-33 must be solved by trial and error techniques since γ mixai depends on
the pressure P bh. Note that this equation does not account for friction flow loss
through the orifices and nozzles. 3
For fluid mixtures that are nearly all gas (with little incompressible fluid and
rock cuttings) and subsonic flow conditions, the pressure above the drill bit inside
the drill string is determined from [1]
 2 k+ 1  2
 k    P  k  P  k   ˙ g
w 
ai
ai
2g   P γ bh    −    =   (7-34)
bh
 k − 1     P   P     A bi 
bh
bh
where k is the ratio of specific heats for the gas (e.g., for air k = 1.4 and for natural
gas k = 1.28).
3
γ bh is the specific weight of the gas at the bottom of the annulus (lb/ft ).
Equation 7-34 must be solved by trial and error for P bh. Note that Equation 7-34
does not account for friction flow loss through the single drill bit orifice.
The above equations will generally yield results that show that for most practical
parameters the annulus bottomhole pressure, P bh, differs very little from the pressure
above the drill bit inside the drill string, P ai. Therefore, it can usually be assumed
that
P bh ≈ P ai (7-35)
7.2.4 Two-Phase Flow in the Annulus
The downward flow condition in the annulus is two phase. The differential
pressure, dP, occurs over the incremental distance of dh. This differential pressure
can be approximated as
 2 
dP = γ mix   1 − fv p)   dh (7-36)
h (
 2 gD − D 

where D h is the inside diameter of the annulus (ft) (the borehole),
D p is the outside diameter of the drill string (pipe and collars) (ft).
The two phase flow of gas and incompressible fluid down the annulus space to
the bottom of the well can be described by a mixed specific weight term which is a
function of position in the annulus. This mixed specific weight term is

w ˙ g + w ˙ m
γ = (7-37)
mix
 P g   T av 
    Q + Q m
g
 P   T g 