Rod Pump Displacement and Efficiency

Overview

Pump displacement determines the maximum theoretical production rate of a rod pump system. The actual rate is always lower due to gas interference, leakage, rod stretch, and incomplete barrel filling. Understanding these efficiency losses is essential for proper pump sizing.

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Theoretical Pump Displacement

Equation

$$PD = 0.1166 \times A_p \times S_p \times N$$

The constant 0.1166 converts in³/stroke × SPM to STB/d.

Standard Pump Sizes

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Effective Plunger Stroke

Stroke Losses

The polished rod stroke $S$ at surface is not fully transmitted to the plunger. The effective plunger stroke accounts for rod stretch:

$$S_p = S - \delta_{fluid} - \delta_{rod}$$

Stretch due to fluid load:

$$\delta_{fluid} = \frac{W_f \cdot L}{A_r \cdot E}$$

Stretch due to rod weight:

$$\delta_{rod} = \frac{w_r \cdot L^2}{2 \cdot E}$$

Where:

• $W_f$ = fluid load (lbs)
• $L$ = rod string length (inches)
• $A_r$ = rod cross-sectional area (in²)
• $E$ = Young's modulus for steel = 30 × 10⁶ psi
• $w_r$ = rod weight per unit length (lbs/in)

Typical Stroke Losses

Note: At greater depths, stroke losses can consume 30-50% of the surface stroke. Longer stroke lengths help compensate.

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Fillage

Definition

Fillage is the fraction of the pump barrel filled with liquid on the downstroke:

$$\text{Fillage} = \frac{V_{liquid}}{V_{barrel}}$$

Causes of Incomplete Fillage

Fillage and Dynamometer Cards

Fillage is diagnosed from dynamometer cards (surface or downhole). A full pump shows a rectangular card; incomplete fillage shows a characteristic "gas pound" or "fluid pound" pattern.

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Volumetric Efficiency

Definition

$$\eta_v = \frac{q_{actual}}{PD}$$

Volumetric efficiency accounts for all losses:

$$\eta_v = \eta_{fillage} \times \eta_{leakage} \times \eta_{shrinkage}$$

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Pump Intake Pressure

Submergence

The pump intake pressure (PIP) determines whether the pump can fill:

$$PIP = 0.433 \times \gamma_f \times h_{sub}$$

Where:

• $PIP$ = pump intake pressure (psi)
• $\gamma_f$ = fluid specific gravity
• $h_{sub}$ = fluid submergence above pump (feet)

Minimum PIP

The PIP must be sufficient to:

1. Overcome bubble point pressure effects (prevent gas breakout)
2. Maintain adequate fillage
3. Provide enough NPSH for pump operation

Typical minimum PIP: 100-300 psi above bubble point for oil wells.

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Design Considerations

Rate vs. Depth Trade-Off

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Related Topics

• Rod Pump Overview — System components and design workflow
• Rod String Analysis — Loads, stress, and fatigue
• IPR Overview — Matching pump rate to inflow

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References

1. Takacs, G. (2015). Sucker-Rod Pumping Handbook. Gulf Professional Publishing.

2. API Recommended Practice 11L (2008). "Design Calculations for Sucker Rod Pumping Systems." American Petroleum Institute.

3. Brown, K.E. (1980). The Technology of Artificial Lift Methods, Vol. 2a. PennWell Books.

4. Gipson, F.W. and Swaim, H.W. (1988). "The Beam Pumping Design Chain." In Petroleum Engineering Handbook. SPE.

5. Lea, J.F., Nickens, H.V., and Wells, M.R. (2008). Gas Well Deliquification, 2nd Edition. Gulf Professional Publishing.