Page 302 - Air and Gas Drilling Manual

Chapter 7: Reverse Circulation Models 7-11
Equation 7-26 together with Equations 7-27 through 7-30 can be used in
sequential integration steps starting at the top of the inside of the drill string (with
the known exit pressure) and continuing for each subsequent change in cross-
sectional area inside the drill string until the pressure above the drill bit inside the
drill string is determined. These sequential calculation steps require trial and error
solutions. The trial and error process requires the selection of the upper limit of the
pressure in each integral on the right side of Equation 7-26. This upper limit
pressure selection must give a right side integral solution that is equal to the left
side integral solution.
7.2.3 Three-Phase Flow Through the Bit
There are three basic calculation techniques for determining the pressure change
through the constriction of the single drill bit (or water course) orifice.
The first technique assumes that the mixture of incompressible fluid, gas, and
rock cuttings passing through the single orifice has a high incompressible fluid
volume fraction. Under these conditions the mixture is assumed to act as an
incompressible fluid. Thus, borrowing from mud drilling technology, the pressure
change through the drill bit, ∆P b, can be approximated by [6, 7]
w ˙ 2
∆P = t π 2 (7-31)
b
2 g γ mixai C 2   D bi 4
 
4
2
where ∆P b is pressure change (lb/ft ),
γ mixai is the mixture specific weight above the drill bit inside the drill
3
string (lb/ft ),
C is the fluid flow loss coefficient for drill bit orifices or nozzles (the value of
this constant is dependent on the type of gas or aerated flow),
D bi is the drill bit single orifice inside diameter (ft).
The pressure change obtained from Equation 7-31 is subtracted from the pressure
above the drill bit inside the drill string P ai obtained from Equation 7-26. The
annulus bottomhole pressure, P bh, is

P = P − ∆ P (7-32)
bh ai b
2
where P bh is bottomhole pressure (lb/ft , abs),
2
P ai is pressure above the drill bit inside the drill string (lb/ft , abs).
For fluid mixtures that have a high gas volume fraction, the pressure above the
drill bit inside the drill string can be approximated by [8]
 w ˙ 2   
P = P −  t   1 − 1  (7-33)
bh ai gA   γ γ
2
  bi   mixbh mixai  
2
where A is the cross-sectional area of the single drill bit orifice (ft ),
bi