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Mud Hydraulics Optimization 83

maximum possible horsepower. The conditions for maximum bit horse-
power were derived by Kendall and Goins (1960).
The pressure drop at the bit is expressed as

m
Δp b = p p − cq (4.1)
Substituting this expression into Eq. (2.88) gives
m
P Hb = qðp p − cq Þ (4.2)
1,714

Using calculus to determine the flow rate at which the bit horsepower is
a maximum gives
m
p p − ðm + 1Þcq
dp Hb
= = 0 (4.3)
dq 1,714
Solving for the root of this equation yields
m
p p = ðm + 1Þcq = ðm + 1ÞΔp d (4.4)
or
Δp d = p p (4.5)
m + 1
2
It can be shown that d p Hb
2 < 0 at this root, so the root corresponds to
dq
a maximum. Thus, bit hydraulic horsepower is a maximum when the
parasitic pressure loss is 1 times the pump pressure. Since
m + 1
Δp b = p p − Δp d = p p − p p = m p p (4.6)
m + 1 m + 1
bit hydraulic horsepower is a maximum when the pressure drop at that
bit is m times the pump pressure.
m + 1
4.2.2 The Maximum Jet Impact Force Criterion

The maximum jet impact force 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 exert the maximum possible jet
impact force to clean the bottomhole.
The conditions for maximum jet impact force were also derived by
Kendall and Goins (1960). Substituting Eq. (4.1) into Eq. (2.89) gives
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
F j = 0:01823C d q ρðp p − cq Þ (4.7)