Page 267 - Air and Gas Drilling Manual

6-12 Air and Gas Drilling Manual
2
where P bh is bottomhole pressure (lb/ft , abs),
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] 2
 2 
w ˙ g (  + w ˙ m)   1 1 
P ai = P bh +  2   −  (6-34)
 gA n    γ mixbh γ mixai  
 
2
where A is the total cross-sectional area of the drill bit orifices (or nozzles) (ft ),
n
3
γ mixai is the mixture specific weight above the drill bit (lb/ft ).
Equation 6-34 must be solved by trial and error techniques since γ mixai depends on
the pressure P ai. Note that this equation does not account for friction flow loss
through the orifices and nozzles.
For fluid mixtures that are nearly all gas (with little incompressible fluid), the
pressure above the drill bit inside the drill string will depend upon whether the
critical flow conditions exist in the orifices or nozzle throats. The critical pressure
through the bit orifices or nozzles is [1]
k
 P bh   2 k  −1
  =   (6-35)
 P  c  k + 1 
ai
where k is the ratio of specific heats for the gas.
The right-hand side of Equation 6-35 is determined only by the value of the
specific heat ratio constant of a gas (e.g., for air k = 1.4 and for natural gas k =
1.28). Thus, for air the critical pressure ratio is 0.528 and for natural gas the critical
ratio is 0.549. Therefore, if P ai is determined to be
P bh
for air P ≥ (6-36)
ai
.
0 528
P bh
for natural gas P ai ≥ (6-37)
.
0 549
the flow through the orifice or nozzle throat is sonic. Under these sonic flow
conditions, the upstream pressure, P ai, does not depend on downstream pressure, P bh.
For these sonic flow conditions the upstream pressure P ai is