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== Aspects in Sucker Rod Pump Design Based on API TR 11L ==


''Downhole Pump Displacement''
Due to the elasticity of the rod, the rod string might strength or contract through the pumping cycle. This results in a downhole stroke length at the plunger "''S''<sub>p</sub>" that slightly differs from the design stroke length ''S''. This difference results in an actual flow "''q<sub>a</sub>''" that is different from the design flow rate ''"q"''. Based on API TR 11L, the rod stretch is predicted. "''q<sub>a</sub>''" is then calculated and is compared to the desired ''"q"''. The optimum ''"q"'' can then be reached with an iterative procedure. In the following, the procedure for this calculation is clarified:
1 - Determine the theoretical flow rate ''"q"'' from the pump speed "''N''", surface stroke length "''S''" , and plunger size "''d''<sub>p</sub>" as follows,
''q'' = (π/4''d''<sub>p</sub><sup>2</sup>''S'')''NF'',   (1)
where the term between brackets represents the volume displaced by the pump during a single cycle, while ''F'' is downhole pump efficiency.
2 - A primary selection of rod string design is required. Firstly, the total length of rod string approximately equals the pump setting depth ''L'' in non-literal wells. Moreover, the configuration of rod string diameters is determined from a standard set of configurations provided in API RP 11L. The standard provides a table of the characteristics of the tapered rod string. Figure 1 shows a portion of table 4.1 provided for rod string configuration and properties. The figure is a snapshot of the digitized table provided by the Petroleum Extension (PETEX®) of The University of Texas at Austin, which can be found [https://petex.utexas.edu/publications/600-beamlift-toolbox here] under the title “Beam Lift System Design Calculators.” In Figure 1, the last six columns are API sizes of rod diameters. The table numbers represent the percentage of each size in the making of the rod string. the first column is a list of configuration identifiers. Based on ''d''<sub>p</sub> determined in the previous step, a rod configuration is selected. The other information provided by the table for each rod configuration is
a. Rod weight ''W''<sub>r</sub>  (lb/ft).
b. Elastic Constant ''E''<sub>r</sub>  (in/lb.ft).
c. Frequency Factor ''F''<sub>c</sub>  (-).
<gallery>
Figure1.png|Figure 1. A snapshot of the digitized table 4.1 of API RP 11L provided by the Petroleum Extension (PETEX®) of The University of Texas at Austin.
</gallery>
The dimensionless number ''S''<sub>p</sub>/''S'' is defined in API TR 11L as a function of two other dimensionless numbers, namely ''N''/''N''<sub>o</sub>' and ''F''<sub>0</sub>/''Sk''<sub>r</sub>. ''N''/''N''<sub>o</sub>' condenses the effect of pumping speed and natural frequency in the tapered rod strings. The natural frequency of non-tapered rod string ''N''<sub>o</sub> is defined by Griffin (1968) as the number of strokes that propagates through the rod string at four times the velocity of sound during the unit time. Therefore, it takes the frequency unit, namely, strokes per unit time. It is mathematically written as,
''N''<sub>o</sub> = ''v''<sub>s</sub>/4''L'',         (2)
where ''v''<sub>s</sub> is the velocity of sound and ''L'' is the rod string length. API RP 11L suggests the following formula based on a typical value for ''v''<sub>s</sub> in steel, the formula results require ''L'' in ft and results in ''N''<sub>o</sub> in strokes per minute Takacs (2015).
''N''<sub>o</sub> = 245,000/''L''.         (3)
Although rod string diameter is not involved in Eq. (3), the variation of diameter in tapered string affects the natural frequency. For a tapered rod string, the natural frequency ''N''<sub>o</sub>' is defined as,
''N''<sub>o</sub>' =  ''F''<sub>c</sub>''N''<sub>o</sub>.         (4)
Recall that ''F''<sub>c</sub> is the frequency factor found in Rod table 4.1 of the standard (Figure 1).
''F''<sub>0</sub>/''Sk''<sub>r</sub> condenses the effects of elastic rod stretch due to fluid load. ''F''<sub>0</sub> is the fluid load on the plunger defined as (lbs),
''F''<sub>0</sub> =  0.052''ρ''<sub>L</sub>''L''(π/4''d''<sub>p</sub><sup>2</sup>) (5)
where ''ρ''<sub>L</sub> is liquid density (lb/ft3), ''L'' is in ft and ''d''<sub>p</sub> is in inch. ''k''<sub>r</sub> is the Spring Constant of the total rod string and represents the load required to stretch the total rod string for unit length. ''k''<sub>r</sub> is defined as,
''k''<sub>r</sub> = 1/''E''<sub>r</sub>''L'',         (6)
where ''E''<sub>r</sub> is the elastic constant of the tubing. It takes the dimension  1/''F''<sub>u</sub> , where ''F''<sub>u</sub> is unit force.
Based on an enormous number of experiments, ''S''<sub>p</sub>/''S''=''f''(''N''/''N''<sub>o</sub>',''F''<sub>0</sub>/''Sk''<sub>r</sub>) is constructed as a plot at discrete values of the independent parameters (Figure 2).
<gallery>
Figure2.png|Figure 2. Figure 4.1 of API TR 11L plotted from the digitization of Petroleum Extension (PETEX®) of The University of Texas at Austin
</gallery>
As seen from the figure, the downhole stroke ''S''<sub>p</sub> resulted from rod strain is always less than the design stroke ''S''. If tubing is not anchored, tubing strain is suspected.  The resultant ''S''<sub>p</sub> should be corrected for tubing strain as follows,
''S''<sub>p</sub>/''S'' = ''S''<sub>p</sub>/''S''|<sub>rod strain</sub> - ''S''<sub>p</sub>/''S''|<sub>tubing strain</sub> = ''S''<sub>p</sub>/''S''|<sub>rod strain</sub> - ''F''<sub>0</sub>/''Sk''<sub>t</sub> (7)
where  ''k''<sub>t</sub> is the Spring Constant of the unanchored tubing and represents the load required to stretch the unanchored portion of the tubing, between the anchor and the pump, unit length. Similar to Eq. (6), ''k''<sub>t</sub> is defined as
  ''k''<sub>t</sub> = 1/''E''<sub>t</sub>''L'',   (8)
where ''E''<sub>t</sub> is the elastic constant of the tubing. It takes the dimension  1/''F''<sub>u</sub> , where ''F''<sub>u</sub> is unit force.
3 - From ''S''<sub>p</sub>/''S'' = ''S''<sub>p</sub>/''S'' x ''S'', calculate "''q<sub>a</sub>''" using Eq. (1). If not acceptable, change "''N''", "''S''" , or "''d''<sub>p</sub>" and start from step 1 again. Increasing "''N''" to compensate for stroke length loss does not come free of expense. The more "''N''" is increased, the shorter the rod string and pump fatigue life will be. Moreover, increasing "''d''<sub>p</sub>" results in a shorter ''S''<sub>p</sub> due to inertia effects. Therefore, an optimum selection of these parameters is needed.
''Range of Polished Rod Loads''
Throughout the pump cycle, the polished rod exhibits varying loads that swing between two extremities, namely, the Maximum Polished Rod Load ''PPRL'' and the Minimum Polished Rod Load ''MPRL''. ''PPRL'' and ''MPRL'' are found as follows,
  ''PPRL'' = ''W''<sub>rf</sub> + [(''F''<sub>1</sub>/''Sk''<sub>r</sub>) x ''Sk''<sub>r</sub>], and   (9)
  ''MPRL'' = ''W''<sub>rf</sub> - [(''F''<sub>2</sub>/''Sk''<sub>r</sub>) x ''Sk''<sub>r</sub>],   (10)
where ''W''<sub>rf</sub> is the buoyant weight of the rod string. while ''F''<sub>1</sub>/''Sk''<sub>r</sub> and ''F''<sub>2</sub>/''Sk''<sub>r</sub> are functions of ''N''/''N''<sub>o</sub> and ''F''<sub>0</sub>/''Sk''<sub>r</sub> and are found from Figures 4.2 and 4.3 of API TR 11L, respectively (Figure 3 and Figure 4).
<gallery>
Figure 3.jpg|Figure 3. Figure 4.2 of API TR 11L plotted from the digitization of Petroleum Extension (PETEX®) of The University of Texas at Austin
Figure 4.jpg|Figure 4. Figure 4.3 of API TR 11L plotted from the digitization of Petroleum Extension (PETEX®) of The University of Texas at Austin
</gallery>
''W''<sub>rf</sub> for a steel rod is extimated from the following,
  ''W''<sub>rf</sub> = ''W''<sub>r</sub> (1-0.128''γ''),   (11)
''Peak Torque of The Crank''
Peak torque ''T'' is required to properly select the surface pump. The following formula is proposed in API TR 11L,
  ''T'' = (2''T''/''S''<sup>2</sup>''k''<sub>r</sub>) x ''S''<sup>2</sup>''k''<sub>r</sub> x ''S''/2 x ''T''<sub>a</sub>,   (12)
where the dimensionless property 2''T''/''S''<sup>2</sup>''k''<sub>r</sub> is obtained from the figure 4.4 of API TR 11L (Figure 5). It is a function of ''N''/''N''<sub>o</sub> and ''F''<sub>0</sub>/''Sk''<sub>r</sub>. ''T''<sub>a</sub> is a correction factor that is function of ''N''/''N''<sub>o</sub>', ''F''<sub>0</sub>/''Sk''<sub>r</sub> and ''W''<sub>rf</sub>/''Sk''<sub>r</sub>. A percentage ''p'' is found from figure 4.6 of API TR 11L based on ''N''/''N''<sub>o</sub>' and ''F''<sub>0</sub>/''Sk''<sub>r</sub>. Then, ''T''<sub>a</sub> is found from,
  ''T''<sub>a</sub> = 1 + ''p'' x (''W''<sub>rf</sub> - 0.3)/0.1.   (13)
<gallery>
Figure 5.jpg|Figure 5. Figure 4.4 of API TR 11L plotted from the digitization of Petroleum Extension (PETEX®) of The University of Texas at Austin
</gallery>
References
Griffin, F. D. 1968. Electric Analog Study of Sucker-Rod Pumping Systems. Paper presented at the Drilling and Production Practice, New York, New York.
Takacs, G. 2015. Sucker-Rod Pumping Handbook. Gulf Professional Publishing.
American Petroleum Institute. 1988. Recommended Practice For Design Calculations For Sucker Rod Pumping Systems (Conventional Units).

Latest revision as of 15:36, 24 August 2021

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