Pump Output Formula — Triplex & Duplex Displacement

Overview

A positive-displacement piston pump delivers a fixed volume per stroke determined by piston (liner) diameter, stroke length, and the number of pistons. Triplex pumps have three single-acting cylinders; each piston displaces fluid only on the forward stroke. Duplex pumps have two double-acting cylinders; each piston displaces fluid on both forward and back strokes, but the rod takes up volume on the back stroke. Volumetric efficiency captures real-world losses from valve leakage, fluid compressibility (especially in gas-cut mud), and suction-side losses.

Theory

A piston of diameter D sweeping a stroke S sweeps a volume of (π/4)*D²*S per stroke. The constant 0.000243 used in oilfield formulas is the conversion from cubic inches to barrels for a single-acting cylinder times the geometry constant, derived as π/(4 * 9702) where 9702 in³ = 1 bbl. Multiplying by the number of cylinders gives the per-rotation displacement; multiplying by strokes per minute gives flow rate.

Formulas

Triplex (3 single-acting cylinders)

Output (bbl/stroke) = 0.000243 * D^2 * S * 3 * Eff/100

D = liner ID (in), S = stroke length (in), Eff = volumetric efficiency (%). Per-cylinder displacement times 3 cylinders gives the per-revolution output.

Duplex (2 double-acting cylinders)

Output (bbl/stroke) = 0.000243 * (2*D^2 - d^2) * S * Eff/100

D = liner ID (in), d = piston rod OD (in), S = stroke length (in), Eff = volumetric efficiency (%). The (2D² - d²) term reflects the back-stroke giving up the rod volume.

Unit Conversions

bbl/min = bbl/stroke * SPM
GPM     = bbl/min * 42
L/min   = GPM * 3.7854
m^3/min = bbl/min * 0.15899

Pump Displacement vs Discharge Volume

Q_discharge = Q_theoretical * Eff_volumetric

Eff_volumetric = 0.90 - 0.98 typical
Eff_volumetric = 0.80 - 0.88 for worn pumps or gas-cut mud

Hydraulic Horsepower

HHP = (P * Q) / 1714

P = discharge pressure (psi)
Q = pump output (GPM)
HHP = hydraulic horsepower (delivered by the pump)

Key Symbols

Worked Example

Given: Triplex pump, D = 6.0 in liner, S = 12 in stroke, SPM = 100, Eff = 95%.

Step 1 — Output per stroke:

bbl/stk = 0.000243 * 6^2 * 12 * 3 * 0.95
        = 0.000243 * 36 * 12 * 3 * 0.95
        = 0.2992 bbl/stroke

Step 2 — Flow rate:

bbl/min = 0.2992 * 100 = 29.92 bbl/min
GPM     = 29.92 * 42  = 1257 GPM
L/min   = 1257 * 3.785 = 4756 L/min

Step 3 — Bottoms-up time at 9,500 ft TD with 95 bbl total hole volume:

Bottoms-up = 95 bbl / 29.92 bbl/min = 3.18 min

Step 4 — Hydraulic horsepower at 3,000 psi standpipe:

HHP = (3000 * 1257) / 1714 = 2200 HP

Valid Ranges & Typical Numbers

When the formula is not enough

• Gas-cut mud or surface-only fluid loss can mask a real efficiency degradation; monitor delta between calculated and measured discharge over time.
• High-pressure cementing pumps and pressure-test pumps may use the same equation but with very different efficiency factors; consult the vendor data sheet.
• Centrifugal charge pumps on the suction side feed positive-displacement triplex/duplex pumps; they do not displace fixed volume per stroke and require head-flow curves, not these formulas.

References

1. Bourgoyne, A.T., Millheim, K.K., Chenevert, M.E., Young, F.S. (1986). Applied Drilling Engineering. SPE Textbook Series, Vol. 2. Chapter 4 (Drilling hydraulics).
2. IADC Drilling Manual, 12th Edition, Chapter Pumps and Hydraulics.
3. API Spec 7K — Drilling and Well Servicing Equipment (mud pump specifications).
4. PetroWiki — Drilling fluid hydraulics: https://petrowiki.spe.org/Drilling_fluid_hydraulics