ESP Sizing Calculator Formula

Electrical Submersible Pumps (ESPs) are centrifugal pumps deployed downhole on production tubing, powered by an electric motor via a cable from the surface. ESPs are the preferred artificial lift for high-volume wells (100–30,000+ bbl/d). Sizing involves selecting the pump model, number of stages, m...

Overview

Theory

An ESP consists of (bottom to top): motor, seal/protector section, pump intake, pump (multiple stages of impeller + diffuser), and cable. Each stage adds a fixed amount of head (lift) at a given flow rate. The pump performance is defined by a head-capacity curve (H-Q curve) for each stage, provided by the manufacturer.

Formulas

Total Dynamic Head (TDH)

TDH = H_lift + H_friction + H_wellhead

where:

H_lift = fluid level depth (or pump setting depth if pumped off)
H_friction = friction loss in tubing (Hazen-Williams or Darcy-Weisbach)
H_wellhead = WHP / (0.433 * SG)  (convert WHP to feet of head)

Number of Stages

N_stages = TDH / head_per_stage

where head_per_stage is read from the manufacturer's pump curve at the design flow rate.

Pump Brake Horsepower

BHP_pump = N_stages * BHP_per_stage

or:

BHP_pump = (Q * SG * TDH) / (3960 * η_pump)

where Q = flow rate (gpm), η_pump = pump efficiency (typically 0.50–0.70).

Motor Selection

HP_motor ≥ BHP_pump / η_motor (with safety factor 1.1–1.2)

Cable Voltage Drop

ΔV = I * R * L / 1000

where I = motor current (amps), R = cable resistance (ohms/1000 ft), L = cable length (ft).

Surface Voltage Required

V_surface = V_motor + ΔV_cable

Tubing Friction Loss (Hazen-Williams)

hf = 0.2083 * (100/C)^1.852 * Q^1.852 / D^4.8655 (ft head per 100 ft)

where C = Hazen-Williams coefficient (typically 120 for new steel), Q in gpm, D in inches.

Pump Intake Pressure

PIP = Pr - ΔP_drawdown
PIP > Pb (to avoid gas interference, ideally PIP > 1.2 * Pb)

Worked Example

Given: Target rate = 1,500 bbl/d (44 gpm), SG = 0.90, pump depth = 6,000 ft, fluid level = 4,000 ft, WHP = 150 psi, 2-7/8" tubing.

Step 1: TDH:

H_lift = 4,000 ft
H_wellhead = 150 / (0.433 * 0.90) = 385 ft
H_friction ≈ 0.2083 * (100/120)^1.852 * 44^1.852 / 2.441^4.8655 * 60
           ≈ 120 ft (for 6,000 ft of 2-7/8" tubing at 44 gpm)
TDH = 4,000 + 120 + 385 = 4,505 ft

Step 2: Number of stages (assume 25 ft/stage at 44 gpm for selected pump):

N_stages = 4,505 / 25 = 180 stages

Step 3: Brake horsepower:

BHP = (44 * 0.90 * 4,505) / (3960 * 0.60)
    = 178,398 / 2,376
    = 75 hp

Step 4: Motor selection: 100 hp motor (with 1.3x safety factor).

Step 5: Cable (#4 AWG, R = 0.054 ohm/1000ft at downhole temp, motor current = 30A):

ΔV = 30 * 0.054 * 6,000 / 1000 = 9.7 V (per conductor; 3-phase = 3 conductors)

Valid Ranges

Parameter	Typical Range
Flow rate	100 – 30,000+ bbl/d
Pump depth	2,000 – 15,000 ft
TDH	2,000 – 12,000 ft
Motor HP	20 – 1,000 hp
Temperature limit	250 – 400°F (motor insulation dependent)
GOR	< 200 scf/STB (without gas separator)

Common Failure Modes

Gas locking — excessive free gas at intake
Scale/corrosion on pump stages
Motor burnout — overload, overheating, or cable failure
Solids wear — sand production eroding stages
Electrical failure — cable, connector, or splice issues

References

Takacs, G. (2009). Electrical Submersible Pumps Manual. Gulf Professional Publishing.
Brown, K.E. (1980). The Technology of Artificial Lift Methods, Vol. 2b. PennWell.
Centrilift (Baker Hughes) — ESP Selection Guide.
PetroWiki — ESP: https://petrowiki.spe.org/Electrical_submersible_pumps

ESP Sizing Calculator