ESP Pump Performance - Head, Staging, Efficiency, and Horsepower

Overview

ESP pump performance is characterized by the relationship between flow rate, generated head, power consumption, and efficiency. These relationships are described by pump performance curves provided by the manufacturer, which are measured with water and must be adjusted for actual well fluids.

The fundamental design task is to match the pump's hydraulic capability to the Total Dynamic Head (TDH) required by the well.

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Total Dynamic Head (TDH)

Definition

Total Dynamic Head is the total pressure (expressed as feet of fluid) that the pump must generate to lift fluid from the pump intake to the surface at the desired rate and wellhead pressure.

$$TDH = H_{lift} + H_{friction} + H_{whp}$$

Where:

Net Lift Component

The net lift accounts for the hydrostatic column the pump must overcome:

$$H_{lift} = D_{pump} - \frac{P_{intake} \times 144}{\rho_f}$$

Where:

• $D_{pump}$ = pump setting depth (ft, measured from surface)
• $P_{intake}$ = pump intake pressure (psi)
• $\rho_f$ = fluid density (lb/ft^3)

In terms of fluid gradient:

$$H_{lift} = D_{pump} - \frac{P_{intake}}{0.433 \times SG_f}$$

Where $SG_f$ is the specific gravity of the produced fluid mixture.

Friction Head Component

Friction losses in the production tubing depend on flow rate, tubing diameter, and fluid properties:

$$H_{friction} = \frac{f \cdot L \cdot v^2}{2 \cdot g \cdot d}$$

Where:

• $f$ = Moody friction factor (dimensionless)
• $L$ = tubing length (ft)
• $v$ = fluid velocity (ft/s)
• $g$ = gravitational acceleration (32.174 ft/s^2)
• $d$ = tubing inner diameter (ft)

For practical ESP design, friction losses are often computed from published friction loss charts or using the Hazen-Williams equation for liquid flow.

Wellhead Pressure Head

$$H_{whp} = \frac{P_{whp} \times 144}{\rho_f} = \frac{P_{whp}}{0.433 \times SG_f}$$

Complete TDH Equation

Combining all components:

$$TDH = \left(D_{pump} - \frac{P_{intake}}{0.433 \times SG_f}\right) + H_{friction} + \frac{P_{whp}}{0.433 \times SG_f}$$

Typical TDH Values

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Head-Capacity Curves

Pump Performance Characterization

Each ESP pump model has a characteristic set of curves plotted against flow rate:

Head (ft/stage)
    │
    │  ●
    │    ●
    │      ●                    Head-Capacity Curve
    │        ●                  (decreasing head with
    │          ●                 increasing rate)
    │            ●
    │              ●
    │                ●
    │                  ●
    └──────────────────────────▶ Flow Rate (BPD)

Efficiency (%)
    │
    │          ┌──●──┐
    │        ●│  BEP │●
    │      ●  └──────┘  ●       Efficiency Curve
    │    ●                ●     (peaked at BEP)
    │  ●                    ●
    │●                        ●
    └──────────────────────────▶ Flow Rate (BPD)

BHP (HP/stage)
    │
    │                      ●
    │                  ●        Brake Horsepower Curve
    │              ●            (increasing with rate)
    │          ●
    │      ●
    │  ●
    └──────────────────────────▶ Flow Rate (BPD)

Head per Stage

The head generated by a single pump stage varies with flow rate. At shutoff (zero flow), head is maximum. As flow rate increases, head decreases:

$$H_{stage}(Q) = A_0 + A_1 Q + A_2 Q^2$$

Where $A_0$, $A_1$, $A_2$ are polynomial coefficients from the manufacturer's curve fit. Most pump curves are well-represented by a second or third-order polynomial.

Total Head from Multiple Stages

The total head generated by $N$ stages is:

$$H_{total} = N \times H_{stage}(Q)$$

Stages operate in series, so their heads add while flow rate remains constant through each stage.

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Pump Staging

Number of Stages Required

Given the TDH requirement and the head produced per stage at the design rate:

$$N_{stages} = \frac{TDH}{H_{stage}(Q_{design})}$$

Round up to the next whole number. In practice, add a safety margin of 5-10%:

$$N_{stages} = \left\lceil \frac{TDH}{H_{stage}(Q_{design})} \times 1.05 \right\rceil$$

Staging Considerations

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Best Efficiency Point (BEP)

Definition

The Best Efficiency Point (BEP) is the flow rate at which the pump operates at maximum hydraulic efficiency. At BEP:

• Energy transfer from impeller to fluid is optimal
• Radial and axial hydraulic forces are minimized
• Vibration and wear are lowest
• Run life is maximized

Recommended Operating Range

ESP manufacturers typically recommend operating within a range around the BEP:

BEP and Pump Selection

When selecting a pump, the design flow rate should fall near the BEP:

$$0.8 \times Q_{BEP} \leq Q_{design} \leq 1.2 \times Q_{BEP}$$

If the design rate falls outside this range, a different pump model should be selected.

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Horsepower Calculations

Hydraulic Horsepower

The theoretical power required to lift fluid:

$$HHP = \frac{Q \times TDH \times SG_f}{3960}$$

Where:

• $HHP$ = hydraulic horsepower (HP)
• $Q$ = flow rate (BPD)
• $TDH$ = total dynamic head (ft)
• $SG_f$ = fluid specific gravity
• 3960 = unit conversion constant (for BPD and ft)

Brake Horsepower

The actual shaft power required, accounting for pump inefficiency:

$$BHP = \frac{HHP}{\eta_{pump}} = \frac{Q \times TDH \times SG_f}{3960 \times \eta_{pump}}$$

Where $\eta_{pump}$ is the pump efficiency at the operating rate (decimal fraction, typically 0.40 to 0.75).

Alternatively, from the manufacturer's BHP per stage curve:

$$BHP = N_{stages} \times BHP_{stage}(Q)$$

Power Components

The relationship is:

$$MHP \geq \frac{BHP}{\eta_{motor}}$$

Where $\eta_{motor}$ is the motor efficiency (typically 0.80 to 0.92).

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Pump Efficiency

Definition

Pump efficiency is the ratio of useful hydraulic power output to shaft power input:

$$\eta_{pump} = \frac{HHP}{BHP} = \frac{Q \times TDH \times SG_f}{3960 \times BHP}$$

Factors Affecting Efficiency

Typical Efficiency Ranges

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Performance Adjustment for Actual Conditions

Speed Correction (Affinity Laws)

When a Variable Speed Drive (VSD) operates the pump at a frequency other than 60 Hz, the affinity laws apply:

$$\frac{Q_2}{Q_1} = \frac{N_2}{N_1}$$$$\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^2$$$$\frac{BHP_2}{BHP_1} = \left(\frac{N_2}{N_1}\right)^3$$

Where $N_1$ and $N_2$ are the rotational speeds (or equivalently, drive frequencies).

Fluid Property Corrections

Pump curves are published for water (SG = 1.0, viscosity approximately 1 cP). For actual well fluids:

• Specific gravity: BHP scales linearly with SG. Head (in feet) is unaffected.
• Viscosity: Head, efficiency, and capacity are all derated. See ESP Viscosity Corrections.
• Gas: Free gas reduces effective head and can cause gas locking. See ESP Gas Handling.

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Design Example

Given Data

Calculations

Net Lift:

$$H_{lift} = 7{,}500 - \frac{800}{0.433 \times 0.95} = 7{,}500 - 1{,}945 = 5{,}555 \text{ ft}$$

Wellhead Pressure Head:

$$H_{whp} = \frac{150}{0.433 \times 0.95} = 365 \text{ ft}$$

Total Dynamic Head:

$$TDH = 5{,}555 + 250 + 365 = 6{,}170 \text{ ft}$$

Number of Stages:

$$N = \lceil 6{,}170 / 28 \times 1.05 \rceil = \lceil 231 \rceil = 231 \text{ stages}$$

Hydraulic Horsepower:

$$HHP = \frac{3{,}000 \times 6{,}170 \times 0.95}{3{,}960} = 4{,}441 \text{ HP}$$

Brake Horsepower:

$$BHP = \frac{4{,}441}{0.62} = 7{,}163 \text{ HP} \ldots \text{(this would be unusually large)}$$

Note: This example illustrates a deep, high-TDH well. In practice, the designer would evaluate whether the required HP is within available motor sizes and consider staged pumping or alternative lift methods.

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Related Topics

• ESP System Design Overview -- Complete design workflow and component selection
• ESP Gas Handling -- Gas interference and separator sizing
• ESP Viscosity Corrections -- Performance derating for viscous fluids
• ESP Motor and Cable Sizing -- Motor HP and cable selection
• Well Flow Overview -- Determining pump intake pressure from IPR

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References

1. Takacs, G. (2009). Electrical Submersible Pumps Manual: Design, Operations, and Maintenance. Gulf Professional Publishing.

2. Stepanoff, A.J. (1957). Centrifugal and Axial Flow Pumps: Theory, Design, and Application, 2nd Edition. John Wiley & Sons.

3. Hydraulic Institute. (2012). Rotodynamic Pumps -- Guideline for Effects of Liquid Viscosity on Performance. ANSI/HI 9.6.7.

4. Brown, K.E. (1984). The Technology of Artificial Lift Methods, Vol. 2b. PennWell Books.

5. Centrilift (Baker Hughes). (2008). Submersible Pump Handbook, 9th Edition. Baker Hughes.