STANDARD HANDBOOK OF PETROLEUM & NATURAL GAS ...

Downhole Motors 887
operating speed of 408 rpm. The torque and the horsepower of the positive
displacement motor are both linear with the pressure drop across the motor.
Therefore, as more weight is placed on the drill bit (via the motor), the greater
is the resisting torque of the rock. The mud pumps can compensate for this
increased torque by increasing the pressure on the constant flowrate through
the motor. In this example the limit in pressure drop across the motor is about
580 psi. Beyond this limit there will be either extensive leakage or damage to
the motor, or both.
If the positive displacement motor is lifted off the bottom of the borehole
and circulation continues, the motor will simply continue to rotate at 408 rpm. The
differential pressure, however, will drop to the value necessary to overcome
internal friction and rotate, about 100 psi. In this situation the motor produces
no drilling torque or horsepower.
As the positive displacement motor is lowered and weight is placed on the
motor and thus the bit, the motor speed continues but the differential pressure
increases, resulting in an increase in torque and horsepower. As more weight is
added to the positive displacement motor and bit, the torque and horsepower
will continue to increase with increasing differentiated pressure (Le., standpipe
pressure). The amount of torque and power can be determined by the pressure
change at the standpipe at the surface between the unloaded condition and the
loaded condition. If too much weight is placed on the motor, the differential
pressure limit for the motor will be reached and there will be leakage or a
mechanical failure in the motor.
The rotor of the Moineau-type positive displacement motor has a helical
design. The axial wave number of the rotor is one less than the axial wave
number for the stator for a given chamber. This allows the formation of a series
of fluid cavities as the rotor rotates. The number of stator wave lengths n, and
the number of rotor wave lengths nr per chamber are related by [79,86]
n, = nr + 1 (4-1 54)
The rotor is designed much like a screw thread. The rotor pitch is equivalent
to the wavelength of the rotor. The rotor lead is the axial distance that a wave
advances during one full revolution of the rotor. The rotor pitch and the stator
pitch are equal. The rotor lead and stator lead are proportional to their
respective number of waves. Thus, the relationship between rotor pitch tr (in.)
and stator pitch, ts (in.) is [86]
tr = t$ (4-155)
The rotor lead Lr (in.) is
Lr = nrtr
(4- 156)
The stator lead Ls (in.) is
Ls = nsts (4- 157)
The specific displacement per revolution of the rotor is equal to the crosssectional
area of the fluid multiplied by the distance the fluid advances. The
specific displacement s (in.3) is
s = nrnstrA (4- 158)