Rig hydraulics · field units

Triplex pump output formula and volumetric efficiency

The output of a triplex mud pump in field units is: output (bbl/stk) = 0.000243 × liner ID² (in) × stroke length (in) × volumetric efficiency. A 6 in liner on a 12 in stroke displaces 0.000243 × 6² × 12 = 0.1050 bbl per stroke at 100% efficiency, or 0.1018 bbl/stk at 97%. Multiply by 42 for gallons per stroke and by pump speed (SPM) for flow rate: at 120 SPM that pump delivers 513 gpm. This page sets out the formula, where the constant comes from, a table by liner size, how to measure volumetric efficiency, and how pump output turns volumes into strokes and minutes.

Reference material only. Stroke counts used in well control must come from the pump output figure on your rig's own kill sheet, measured and agreed on the rig, and operations must follow your company's approved procedures. DrillX calculators are advisory.

1. What pump output means

A triplex mud pump has three pistons running in replaceable liners. It is single-acting: each piston displaces mud on its forward stroke only. One "stroke" on the rig's stroke counter is one full turn of the crankshaft, in which all three pistons deliver once. Pump output, also called pump displacement or pump factor, is the volume delivered per stroke, in barrels per stroke (bbl/stk) or gallons per stroke.

Pump output is the number that links volume to strokes. Every stroke figure on a kill sheet, every bottoms-up time and every displacement schedule depends on it.

2. The formulas

D is the liner inside diameter in inches, L is the stroke length in inches, and Eff is volumetric efficiency as a decimal (0.97 for 97%). SPM is pump speed in strokes per minute.

Where 0.000243 comes from. Each piston sweeps a cylinder of π ÷ 4 × D² × L cubic inches. Three pistons give 3 × 0.7854 × D² × L. One oilfield barrel is 42 US gallons of 231 cubic inches each, or 9,702 cubic inches. So the constant is 3 × 0.7854 ÷ 9,702 = 0.0002429, rounded to 0.000243. Multiplying by 42 gives the gallons-per-stroke constant, 0.0102.

3. Worked example

Hydraulic horsepower is pump pressure (psi) × flow rate (gpm) ÷ 1,714. It is the power the pump puts into the mud, and it must stay within the rating for the liner size fitted.

4. Output by liner size

Output rises with the square of liner diameter, so a liner change moves the number a long way. The table is for a triplex pump with a 12 in stroke; the flow rate column is at 100 SPM and 97% efficiency.

For a different stroke length, scale in proportion: a 10 in stroke delivers 10 ÷ 12 of these figures. Larger liners move more mud per stroke but lower the maximum pressure the pump can hold, because the same rod load is spread over a bigger piston area. The pump manufacturer's data plate gives the pressure limit for each liner size.

5. Volumetric efficiency

The formula gives theoretical displacement. A real pump delivers slightly less, and the ratio of the two is volumetric efficiency:

The losses come from valves that close late or leak, worn pistons and liners, and air or gas in the mud that compresses instead of being displaced. A pump with a healthy charge pump on the suction side fills its cylinders more completely than one drawing straight from the pit. Triplex pumps in good condition are commonly assumed to run in the mid to high 90s per cent; the only figure that should go on a kill sheet is one measured on that pump.

Measuring it. Line the pump up to take suction from a calibrated tank, such as the trip tank, pump a known volume at the speed you intend to use, and count strokes. If 20.0 bbl leaves the tank in 200 strokes, measured output is 20.0 ÷ 200 = 0.1000 bbl/stk. Against a theoretical 0.1050 bbl/stk, efficiency is 0.1000 ÷ 0.1050 = 95.3%.

6. Duplex, quintuplex and metric units

A duplex pump has two double-acting pistons that displace mud on both the forward and the return stroke. On the return side the piston rod, of diameter d, takes up part of the cylinder, which is why the rod term is subtracted. A quintuplex pump has five single-acting pistons, so its constant is five-thirds of the triplex constant. In metric units the worked example above comes to 0.1018 × 158.99 = 16.19 litres per stroke.

7. Where pump output is used

• Kill sheets. Surface-to-bit and bit-to-surface strokes set the pressure step-down schedule. See kill sheet calculations.
• Bottoms-up and lag time. Annular volume divided by pump output gives the strokes for cuttings or gas to reach surface. See annular velocity and annular capacity.
• Hole cleaning. Flow rate in gpm sets annular velocity.
• ECD. Flow rate drives annular pressure loss and so equivalent circulating density.
• Displacements, pills and cement. Spotting a pill or displacing cement to the right depth is done by counting strokes.

8. Mistakes to avoid

• Using the triplex constant for a duplex pump, or forgetting the rod term on a duplex.
• Leaving the old pump output on the kill sheet after a liner change. Every stroke figure on the sheet changes with it.
• Assuming 100% efficiency. The mud arrives later than calculated, and kill mud reaches the bit after the schedule says it has.
• Confusing piston rod diameter with liner diameter, or using the liner's outside diameter.
• Mixing barrels per stroke with gallons per stroke. The two differ by a factor of 42.
• Adding the SPM of two pumps without checking they have the same liner size. Add the flow rates, not the stroke rates.

How DrillX helps

Flow rate from the pumps also sets how much pressure is spent at the bit: see bit hydraulics: nozzle TFA, jet velocity and HSI. Pump output per stroke is also what turns a cement job's displacement volume into strokes to bump the plug: see cement volume calculation.

DrillX includes a triplex pump output and volumetric efficiency calculator alongside hydrostatic pressure, ECD, annular pressure loss, MAASP from leak-off test, annular capacity and velocity, buoyancy and string weight, and a Driller's and Wait & Weight kill sheet. Values update as you type, in field units, and in English or Arabic. DrillX is in final testing; email info@drillx.app for a demo or early access, or see what DrillX does.

Sources and further reading

• International Association of Drilling Contractors, WellSharp well-control training: iadc.org/wellsharp
• International Well Control Forum, well-control certification and standard kill sheets: iwcf.org
• SLB Energy Glossary, triplex pump: glossary.slb.com