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84 Part I Liquid Drilling Systems

Using calculus to determine the flow rate at which the bit impact force is
a maximum gives
0:009115C d ½2ρΔp p q − ðm + 2Þρcq m+1
= = 0 (4.8)
dF j Š
dq p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρΔp p q − ρcq
m+2
2
Solving for the root of this equation yields
Δp d = 2p p (4.9)
m + 2
2
It can be shown that d F j 2 < 0 at this root, so the root corresponds to a
dq
maximum. Thus, the jet impact force is a maximum when the parasitic
pressure loss is 2 times the pump pressure. Since
m + 2
Δp b = p p − Δp d = p p − 2p p = m p p (4.10)
m + 2 m + 2
the bit jet impact force is a maximum when the pressure drop at the bit
is m times the pump pressure.
m + 2
4.2.3 The Maximum Nozzle Velocity Criterion

The maximum nozzle velocity criterion may be stated as follows: Within
the maximum available pump pressure, the mud flow rate and the nozzle
size should be chosen so the bit will create the maximum possible jet
velocity to clean the bottomhole.
Substituting Eq. (4.1) into Eq. (2.87) gives

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
p p − cq
v n = C d −4 (4.11)
8:074 × 10 ρ
This equation implies that the nozzle velocity can be increased by redu-
cing the flow rate so the parasitic pressure loss is reduced. In field applica-
tions, the flow rate is set to the minimum flow rate determined by the
minimum annular velocity required to lift cuttings.

4.2.4 Bit Hydraulics
Regarding the question of which criterion is the best for optimizing bit
hydraulics, most people use the maximum bit hydraulic horsepower or
the maximum bit hydraulic impact force criterion at shallow to middle
depths and then shift to the maximum nozzle velocity at deeper depths.