Drilling hydraulics · field units

Annular pressure loss formula, with a worked example

Annular pressure loss (APL) is the friction pressure the mud loses as it flows up the annulus from the bit to surface. For a Bingham plastic mud in laminar flow, the standard field-unit formula is APL (psi) = [PV × V ÷ (1000 × (Dh − Dp)²) + YP ÷ (200 × (Dh − Dp))] × L, with PV in cp, YP in lb/100 ft², annular velocity V in ft/s, hole and pipe diameters in inches and section length L in feet. For 10,000 ft of 5 in drill pipe in 8½ in hole at 400 gpm with PV 20 and YP 15, APL is about 271 psi, which adds about 0.52 ppg to a 10.0 ppg mud as equivalent circulating density. This page sets out the laminar and turbulent formulas, how to tell which applies, a two-section worked example, and the common shortcuts and their limits.

Reference material only. Hydraulics for a real well should come from your hydraulics program, the mud engineer's measured rheology and your company's approved procedures. DrillX calculators are advisory.

1. What annular pressure loss is and why it matters

When the pumps are off, bottom-hole pressure is just the hydrostatic pressure of the mud column. When the pumps are on, the mud has to be pushed up the annulus against friction, and that friction pressure acts on the open hole as well. The formation therefore sees hydrostatic pressure plus annular pressure loss. Expressed as a density, that total is the equivalent circulating density (ECD).

APL is small compared with the total standpipe pressure, most of which is lost inside the drill string and through the bit nozzles, but it is the part that matters for the well. Too much APL can push ECD past the fracture gradient at the weakest exposed formation and cause losses; a well that is only overbalanced while circulating can flow when the pumps stop.

2. The formulas

Q is flow rate in US gallons per minute, Dh is the hole or casing inside diameter and Dp is the pipe or collar outside diameter, both in inches. PV is plastic viscosity in centipoise, YP is yield point in lb/100 ft², MW is mud weight in ppg and L is the measured length of the annular section in feet. The result is pressure loss in psi per foot, multiplied by the section length to get psi. Multiplying the velocity by 60 gives ft/min, the unit used on the annular velocity page.

These are the Bingham plastic equations for flow in a concentric annulus, treating it as a slot, as set out in Bourgoyne and co-authors' Applied Drilling Engineering. The laminar formula has two parts: the first term grows with velocity and plastic viscosity, the second is the pressure needed just to overcome the mud's yield point and does not depend on flow rate. Some field handbooks write the laminar equation with V in ft/min, which turns the 1000 into 60,000, and a few use 225 instead of 200 in the yield-point term; check which version a spreadsheet uses before comparing results.

3. Laminar or turbulent?

Calculate the annular velocity and the critical velocity for each section. If V is below Vc the flow is laminar and the laminar formula applies; if V is above Vc it is turbulent. Around drill pipe in a normal-size hole the flow is usually laminar. Around drill collars and other large-OD tools, where the gap is narrow and velocity is high, it is often turbulent. When in doubt, some engineers calculate both and use the larger loss, which errs on the side of a higher ECD.

4. Worked example

Total annular pressure loss = 254.5 + 41.7 = 296 psi. Converted to a density with the ECD formula: ECD = 10.0 + 296 ÷ (0.052 × 10,000) = 10.57 ppg. If the whole 10,000 ft were 5 in pipe, the loss would be 0.0271 × 10,000 ≈ 271 psi and ECD about 10.52 ppg, which shows how much a short collar section in a tight annulus adds.

In the drill pipe section, the yield-point term (0.0214 psi/ft) is almost four times the viscous term (0.0056 psi/ft). That is typical in laminar flow, and it is why yield point, not just pump rate, drives ECD in narrow-margin wells.

5. What raises annular pressure loss

• Pump rate. Higher flow means higher velocity; in turbulent flow the loss rises with velocity to the power 1.75.
• A narrower annulus. The clearance appears squared or to the 1.25 power in the denominators, so large-OD collars, stabilisers, liners and small hole sizes raise APL sharply.
• Mud rheology. Higher PV and YP raise laminar losses. Gelled mud also needs extra pressure to break circulation after a connection or trip.
• Hole length. Friction builds over measured length, so long horizontal and extended-reach sections add APL without adding vertical depth.
• Cuttings load. Cuttings in the annulus raise the effective density and the friction, which a clean-fluid calculation does not capture.
• Pipe rotation, eccentricity and temperature. These change the real loss, in either direction, and are why measured downhole pressure is preferred where a pressure-while-drilling tool is run.

6. Quick estimates and their limits

The ECD page gives a simplified field equation from Lapeyrouse, APL = 1.4327 × 10⁻⁷ × MW × L × V² ÷ (Dh − Dp), with V in ft/min. It ignores PV and YP, so it is only a rough sense-check. For the 10,000 ft drill pipe case above it gives about 176 psi against 271 psi from the Bingham laminar formula, because most of the laminar loss here comes from the yield point. Bingham plastic often over-predicts loss at low shear rates; power-law and Herschel-Bulkley models, as used in API RP 13D, usually match field data better. Use quick formulas to check a hydraulics model, not to replace one.

7. Mistakes to avoid

• Using standpipe pressure as annular pressure loss. Most standpipe pressure is lost inside the string and at the bit.
• Using measured depth instead of TVD when converting APL to ECD, or TVD instead of measured length when calculating APL itself.
• Using nominal bit size when the hole is washed out. A larger hole lowers velocity and APL; an under-gauge hole raises them.
• Mixing velocity units: the laminar formula above takes ft/s; the 60,000 version takes ft/min.
• Ignoring the collar and BHA section, which is short but often has the highest loss per foot.
• Using lab rheology at surface temperature for a hot, deep well without correction.

How DrillX helps

Annular loss is one part of the parasitic pressure loss; the share of standpipe pressure left for the bit nozzles is covered in bit hydraulics: nozzle TFA, jet velocity and HSI.

The same laminar equation, driven by pipe movement instead of the pumps, gives a quick estimate of the pressure change while tripping: see swab and surge pressure, and the trip margin formula.

DrillX includes an annular pressure loss calculator alongside ECD, hydrostatic pressure, annular capacity and velocity, 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.

Sources and further reading

• A. T. Bourgoyne Jr., K. K. Millheim, M. E. Chenevert and F. S. Young Jr., Applied Drilling Engineering, SPE Textbook Series Vol. 2 (Society of Petroleum Engineers, 1986), chapter 4, drilling hydraulics: source of the Bingham plastic annular equations and critical velocity.
• API RP 13D, Rheology and Hydraulics of Oil-well Drilling Fluids (American Petroleum Institute).
• Norton J. Lapeyrouse, Formulas and Calculations for Drilling, Production and Workover (Gulf Professional Publishing): simplified annular pressure loss equation.
• SLB Energy Glossary, equivalent circulating density and related terms: glossary.slb.com