Performance index provides engineers with tool to select PDM, track performance

G. Robello Samuel
EnerTech-Landmark Graphics
Houston

Stefan Miska
University of Tulsa
Tulsa

Jeff Li
BJ-Nowsco
Calgary

A newly developed performance index allows engineers to select the most efficient positive displacement motor (PDM) and to track performance during drilling.

The fixed design parameters for a positive-displacement motor (PDM)-such as eccentricity, shaft diameter, housing/shaft pitch, and overall length-depend on a motor's geometry and construction.

The variable that can be changed is the flow rate. Flow rate controls the PDM's operating parameters such as motor speed, torque, and power output, each of which is derived from the motor.

Because of the various parameters involved, it is necessary to combine different motor variables and operating parameters with the aid of the Buckingham Pi theorem. To compare the design parameters, which affect the operating parameters, common parametric curves have been generated to show the effect of the various design parameters.

This paves the way for coupling the vertical cross-sectional parameters with the horizontal cross-sectional parameters of the PDM.

Incompressible fluid system

For PDMs that operate on incompressible fluid system, a dimensionless index called the performance index (PI) for the motor has been defined and is given as the product of dimensionless speed and dimensionless torque. ^1-4

The speed and torque are the most important variables sought to describe the motor characteristics.

Therefore, the different variables are put in the form:

f(D_s, V_v, p_h, e, N, Dp, T) = PI

With the dimensionless parameters consisting of:

Dimensionless speed, N_d = ND / Vv

Dimensionless torque, T_d = T / (Dpep_h D_s)

Performance Index, PI = N_d T_d

There is a positive relation between the various parameters involved in the vertical cross section, but no relation exits between the parameters of vertical and horizontal cross section.

To bring forth all the variables of the multilobe motor and express the dimensionless parameters as a function of the winding ratio of the motor, the optimum relationship between the diameter of the shaft and the pitch of the housing has been established.

This also establishes an optimal relationship between the vertical and horizontal cross-sectional parameters of the motor.

The optimum relationship between the pitch of the housing and diameter of the housing are as follows:

P_h = 2^1/2 / 2 (pD_h)

Using the above relationship in addition to equations provided in the Equation box [359,405 bytes], it is possible to calculate the parameters for dimensionless speed, dimensionless torquem, and the PI as function of the PDM's configuration.

Fig. 1 [57,386 bytes], Fig. 2 [58,306 bytes], Fig. 3 [79,092 bytes], and Fig. 4 [79,871 bytes] show the plots of dimensionless speed, dimensionless torque, PI, and the combined plot of dimensionless parameters for various winding numbers.

Because the motor can be manufactured with various dimensions, the dimensionless groups can be used as a standard to compare and characterize the performance of the actual motor under consideration with that of the optimized ideal motor.

The figures can also be used as a guideline to observe PDM performance. The calculated PI of the considered motor may either be greater than, equal to, or less than the optimal PI. Thus, the PI of the motor can be used to establish a benchmark or an operational criterion for purposes of comparison.

Compressible fluid system

The compressible fluid system encounters problems amenable to analytical attack. Therefore, it is necessary to resort to experimental tests or make use of the results of experimental investigations conducted by other researchers. Thus, different dimensionless quantities have been developed for the PDM that operate in a compressible fluid system. The parameters and definitions of greatest interest are provided in the Equation box including the p theorem. Until now, it has been assumed that all the important variables are known. Further, to compare the performance of different designs operating under different conditions, it is important to develop p terms, which may be of special value in presenting the performance parameters.

The Mach number (M) is the ratio of the fluid velocity in the power section to that of the velocity of sound in the fluid. Using the perfect gas law, M can also be expressed as shown in Equation 9.

Besides the above dimensionless parameter, certain alternate variables grouped together in the form of dimensionless numbers may be used to predict operating conditions. One such component is the adiabatic expansion energy (Equation 10).

The dimensionless parameter derived from the adiabatic power relationship is provided in Equation 11. Other dimensionless terms that can be grouped together and still satisfy the p theorem are shown in Equations 12 and 13.

Canadian Fracmaster Co. conducted extensive bench tests using compressible fluid under varying conditions. Because the required parameters could not be measured during all the tests, additional plots could not be generated using the various above-mentioned dimensionless parameters.

Nevertheless, a few data were made available from which an attempt was made to plot other parameters used to observe the behavior and performance of the PDM.

Fig. 5 [68,340 bytes] and Fig. 6 [64,373 bytes] show dimensionless plots for two back pressures representing bottom-hole pressures. With the known configuration of the motor and mass flow rate, the outlet and inlet pressure ratios can be estimated.

With the known pressure ratio, Fig. 6 helps to estimate the operating output horsepower of the motor. This change observed in both the curves represents the stall condition of the motor. This change can be used to estimate the required parameters so that the motor is operated below the stall condition.

In a similar manner, several other useful plots can also be generated to analyze the performance of any motor configuration operating under different conditions. In addition, such an analysis identifies the influence of essential parameters, helping to define the operating window.

Practical application

The following examples show the practical usefulness for calculating the dimensionless speed, dimensionless torque, PI, and the corresponding dimensional graphs.

In Example 1, a positive displacement motor with a 1:2 winding ratio was considered for analysis as follows:

• Diameter of the shaft = 3.8 in.
• Configuration of the motor = 1:2
• Rotor eccentricity = 0.96 in.
• Diameter of the Housing = 5.75 in.
• Pitch of the housing = 12 in.

Pressure drop across the motor was determined to be 465 psi at a circulation flowrate of 600 gpm.

The dimensionless parameters for the actual motor were:

D_s = 3.8 in., e = 0.96 in., p_h = 12 in., D_h = 5.75 in., Dp = 465 psi, Q = 600 gpm.

Dimensional speed: N^*_d = D_s / p_h = 3.8 / 12 = 0.32

The torque developed by the motor was calculated as follows:

T = (0.01 x 500 x 0.5 x 1.5 x 5.75² x 12 / 1.5²) = 661 ft-lb

Substituting the respective values in the Dimensional Torque of the motor under consideration, T_d^* can be calculated as follows:

T^*_d = T x 12 / (1.75 x 2.5 x 12 x 465) = 661 x 12 / (21,888) = 0.36

Consequently, the PI of the motor is:

PI^* = 0.36 3 0.32 5 0.115

The dimensionless speed for the optimized motor can be calculated as follows:

N_d = 2^1/2(1-i) / pi(2-i) = 2^1/2 (1-0.5) / (p x 0.5 (2 - 0.5)) = 0.3

In a similar manner, the dimensionless torque can also be calculated:

T_d = 0.25 x 0.5(1 + 0.5) / (1 - 0.5) = 0.38

Consequently, the PI is:

PI = 0.3 x 0.38 = 0.11

The estimated dimensionless speed, dimensionless torque, and the PI for the motor with the above dimensions coincide with that of the respective dimensionless quantities of the optimal motor.

After comparing the motor's PI with that of the optimal design, the PI, which is the representation of the horsepower of the optimal motor, showed a good comparison with that of the actual motor. This also explains that the dimensions of the above motor in the vertical cross-section are ideally related to the horizontal cross sectional dimensions.

Example 2

This example compares the operating PI of the motor with the optimum PI of the PDM using the following dimensions and parameters:

• Diameter of the shaft = 1.5 in.
• Configuration of the motor = 4:5
• Shaft eccentricity = 0.35 in.
• Housing diameter = 2.75 in.
• Pitch of the housing = 6 in.

The pressure drop across the motor was determined to be 250 psi at a circulation flowrate of 200 gpm with an overall efficiency of 70%.

The dimensionless parameters for the actual motor are as follows:

D_s = 1.5 in., e = 0.35 in., p_h = 6 in., D_h = 2.75 in., Dp = 250 psi, Q = 200 gpm

The dimensional speed: N_d = D_s / (n x p_h) = 1.5/( 4 x 6) = 0.06

Torque: T = (0.01 x 250 x 0.8 x 1.8 x 2.75² x 6 x 0.7) / 1.2² = 80 ft-lb

Substituting the respective values in the dimensional torque of the motor under consideration, T_d^* can be calculated as follows:

T^*_d = T x 12 /(0.35 x 1.5 x 6 x 250) = 80 x 12 / 787.5 = 1.22

Consequently the PI of the motor under consideration in this illustration is given by:

PI^* = 0.06 x 1.22 = 0.073

The dimensionless speed for the optimized motor can be calculated as follows:

N_d = 2^1/2(1-i) / pi(2-i) = 2^1/2(1-0.8) / p x 0.8 x (2-0.8) = 0.09

In a similar manner, the dimensionless torque can be calculated as follows:

T_d = 0.25 x 0.8(1 + 0.8) / (1 - 0.8) = 1.8

Thus, PI = 0.09 x 1.8 = 0.16

This example shows the deviation in the dimensionless parameters of the motor from that of the optimal dimensionless quantities. After analyzing the various dimensionless quantities, it was found that the relative difference between the performance indices of the two motors is 0.55, of which the relative difference in the dimensionless speed is 0.33 and the dimensionless torque is 0.33.

Therefore, it is shown that dimensionless speed and dimensionless torque contributes equally to the reduced PI. The dimensionless speed is a function of the winding number of the shaft/housing, diameter of the shaft, and the pitch of the shaft/housing.

To achieve an optimum design with the same pressure drop, eccentricity, pitch, and motor diameter, the low configuration ratio becomes an optimal choice.

The efficiency of the motor also has a significant effect on the dimensionless torque. When the efficiency is reduced from 100% to 70%, it causes a change in the relative difference of dimensionless torque from 0.33 to 0.03.

References

1. Miska, S., Qui, W., and Samuel, G.R., "Advanced Horizontal Coiled Tubing Drilling System," Annual Report, U.S. Department of Energy, BDM, 1997.
2. Samuel, G.R., TUDRP Advisory Board Meeting Reports, May and November 1996, May and November 1997.
3. Stefan, M., "Analytical Study of the Performance of Positive Displacement Motor (PDM): Modeling for Incompressible Fluid," SPE paper 39026, presented at the Latin American Petroleum Conference, Rio De Janeiro, Sept. 30-Aug. 3, 1997.
4. Samuel, G.R., and Saveth, K., "Progressing Cavity Pump (PCP): New Performance Equations for Optimal Design," SPE paper 39756, presented at the Permian Basin Conference, Midland, Mar. 17-25, 1998.

The Authors