Drilling hydraulics · field and metric units

Annular velocity and annular capacity formulas

Annular velocity is the speed at which drilling fluid travels up the space between the drill string and the hole or casing. In oilfield units, annular velocity (ft/min) = 24.5 × flow rate (gpm) ÷ (Dh² − Dp²), and annular capacity (bbl/ft) = (Dh² − Dp²) ÷ 1029.4, with the hole diameter Dh and the pipe outside diameter Dp in inches. At 600 gpm in a 12¼ in. hole around 5 in. drill pipe, annular velocity is 117.5 ft/min. This page sets out both formulas in field and metric units, shows where the constants come from, and works through annular volume, bottoms-up strokes and the mistakes that most often spoil the answer.

Reference material only. Hydraulics and hole-cleaning decisions must follow your company's approved procedures and the drilling programme for the well. DrillX calculators are advisory.

1. What the annulus is and why its size matters

The annulus is the ring-shaped space between the outside of the drill string and the wall of the hole or the inside of the casing. Two numbers describe it. Annular capacity is the volume it holds per foot of length, and it is what you need for annular volume, bottoms-up strokes and kick height. Annular velocity is how fast the mud moves up it at a given pump rate, and it is what carries cuttings out of the hole and what drives the friction behind equivalent circulating density (ECD). Both depend on the same term: the hole diameter squared minus the pipe outside diameter squared.

2. The formulas

Dh is the hole diameter in open hole, or the casing inside diameter in a cased section. Dp is the outside diameter of the drill pipe, drill collars or tubing. Both are in inches. The two velocity formulas are the same calculation: 1 bbl is 42 US gallons, and 1029.4 ÷ 42 = 24.5. The same relationships in other unit sets:

3. Where 1029.4 and 24.5 come from

Both constants are cylinder geometry and nothing else. One foot of a cylinder with a diameter of D inches has a volume of π ÷ 4 × D² × 12 = 9.4248 × D² cubic inches. A US oilfield barrel is 42 gallons of 231 cubic inches each, which is 9,702 cubic inches. So the capacity in barrels per foot is D² × 9.4248 ÷ 9,702 = D² ÷ 1029.4, and in gallons per foot it is D² × 9.4248 ÷ 231 = D² ÷ 24.51, usually rounded to 24.5. An annulus is a large cylinder with a smaller one taken out, which is why the formulas use Dh² − Dp². In metric units the same reasoning gives 4 ÷ π × 1,000,000 = 1,273,240 for diameters in millimetres and capacity in m³/m.

4. Worked example: annular velocity around pipe and collars

The pump rate is the same everywhere, but the mud moves about 45% faster past the collars than past the drill pipe because the gap is smaller. The lowest velocity in a well is always in the widest annulus, which is usually the top of the hole or the riser, and that is where hole cleaning has to be checked. To turn the formula round and find the flow rate for a target velocity, use Q = AV × (Dh² − Dp²) ÷ 24.5: holding 120 ft/min around the 5 in. pipe here needs 120 × 125.06 ÷ 24.5 = 613 gpm.

5. Worked example: annular volume and bottoms-up

Take an 8½ in. hole at 10,000 ft with 9,400 ft of 5 in. drill pipe above 600 ft of 6½ in. drill collars, and a pump delivering 0.100 bbl/stroke at 100 strokes per minute. To keep the arithmetic short the whole annulus is treated as 8.5 in.; on a real well the cased section uses the casing inside diameter.

The volumes above use the unrounded capacities (0.04590 and 0.02914 bbl/ft). Bottoms-up strokes are the figure a driller watches when circulating out gas, a sweep or a sample, and the same annular volumes appear on a kill sheet as strokes from bit to shoe and bit to surface.

6. Metric example

For a 215.9 mm hole (8½ in.) around 127 mm drill pipe (5 in.) at 1,700 L/min: Dh² − Dp² = 46,612.81 − 16,129 = 30,483.81. Annular capacity is 30,483.81 ÷ 1,273,240 = 0.02394 m³/m, or 23.94 litres per metre. Annular velocity is 1,273.24 × 1,700 ÷ 30,483.81 = 71.0 m/min, which is about 233 ft/min. The same well in field units is 449 gpm, and 24.5 × 449 ÷ 47.25 gives the same answer to within rounding.

7. Annular velocity and hole cleaning

Annular velocity has to beat the rate at which cuttings settle through the mud. If it is too low, cuttings build up in the annulus; if it is too high opposite open hole, it can erode the wellbore wall and it raises annular friction and ECD. There is no single correct number. SPE PetroWiki's summary of the research notes that about 200 ft/min was once the usual minimum in practice, that later work suggested 50 ft/min should give satisfactory transport in a vertical well with a typical drilling mud, and that deviated wells are a different problem: cuttings beds were seen to form at inclinations above about 35° from vertical, and raising the annular velocity by increasing the flow rate reduced the bed height significantly. Velocity is one input alongside mud rheology, hole angle, pipe rotation and rate of penetration, so treat a calculated figure as a check on the hydraulics programme, not a replacement for it.

8. Mistakes to avoid

• Subtracting before squaring. The term is Dh² − Dp², not (Dh − Dp)². For 8.5 in. and 5 in. the correct value is 47.25; the wrong one is 12.25, which overstates velocity almost four times.
• Using the pipe inside diameter. The annulus is bounded by the pipe's outside diameter. The inside diameter belongs in the drill string capacity, ID² ÷ 1029.4.
• Using bit size inside casing. In a cased section Dh is the casing inside diameter, which depends on the casing weight as well as its nominal size.
• Mixing gpm and bbl/min. Use 24.5 with gallons per minute and 1029.4 with barrels per minute.
• One velocity for the whole well. Every change of hole size or pipe size is a separate section with its own capacity and velocity.
• Assuming the hole is in gauge. A washed-out hole is larger than the bit, so the real annular volume is higher and the real velocity lower than calculated.
• Assuming the pump delivers its theoretical output. Flow rate is strokes per minute × output per stroke × volumetric efficiency.

How DrillX helps

DrillX includes an annular capacity and velocity calculator alongside hydrostatic pressure, ECD, annular pressure loss, MAASP from leak-off test, triplex pump output, 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. For the mud inside the string and the volume its steel displaces on trips, see drill pipe capacity and displacement.

Sources and further reading

• Society of Petroleum Engineers, PetroWiki, Cuttings transport: petrowiki.spe.org
• SLB Energy Glossary, annular velocity: glossary.slb.com
• Drilling Manual, annular capacity formulas in field and metric units: drillingmanual.com
• Wyo-Ben, Annular Velocity technical sheet (PDF): wyoben.com