Beggs-Brill Multiphase Flow Formula

The Beggs-Brill (1973) correlation calculates pressure drop for two-phase (gas-liquid) flow in pipes at any inclination angle. It is one of the most widely used multiphase flow correlations in production engineering, applicable to tubing, flowlines, and pipelines. The method determines flow pattern,...

Overview

Theory

In multiphase flow, gas and liquid travel at different velocities (slip), creating holdups that affect both pressure gradient and flow behavior. Beggs-Brill identifies four flow patterns — segregated, intermittent, distributed, and transition — and applies pattern-specific correlations for liquid holdup and friction factor.

Formulas

Input Parameters

λL = vsl / vm        (input liquid fraction, no-slip holdup)
NFR = vm^2 / (g * D)  (Froude number)
vm = vsl + vsg        (mixture velocity)
vsl = QL / A           (superficial liquid velocity)
vsg = QG / A           (superficial gas velocity)

Flow Pattern Boundaries

L1 = 316 * λL^0.302
L2 = 0.0009252 * λL^(-2.4684)
L3 = 0.10 * λL^(-1.4516)
L4 = 0.5 * λL^(-6.738)

Segregated: λL < 0.01 and NFR < L1, OR λL ≥ 0.01 and NFR < L2
Intermittent: 0.01 ≤ λL < 0.4 and L3 ≤ NFR ≤ L1, OR λL ≥ 0.4 and L3 ≤ NFR ≤ L4
Distributed: λL < 0.4 and NFR ≥ L1, OR λL ≥ 0.4 and NFR > L4
Transition: L2 ≤ NFR < L3

Horizontal Liquid Holdup (HL0)

Segregated:

HL0 = 0.98 * λL^0.4846 / NFR^0.0868

Intermittent:

HL0 = 0.845 * λL^0.5351 / NFR^0.0173

Distributed:

HL0 = 1.065 * λL^0.5824 / NFR^0.0609

Constrained: HL0 ≥ λL

Inclination Correction

ψ = 1 + C * (sin(1.8θ) - 0.333 * sin³(1.8θ))
HL = HL0 * ψ

where C depends on flow pattern and NLV (liquid velocity number).

Pressure Gradient

Elevation:

(dP/dz)_el = ρm * sin(θ) / 144  (psi/ft)
ρm = ρL * HL + ρG * (1 - HL)

Friction:

(dP/dz)_f = ftp * ρns * vm^2 / (2 * g * D * 144)

where ftp = two-phase friction factor, ρns = no-slip density.

Total:

dP/dz = ((dP/dz)_el + (dP/dz)_f) / (1 - Ek)

Ek = acceleration term (usually small, <5%).

Two-Phase Friction Factor

fn = 0.0056 + 0.5 * Re^(-0.32)  (no-slip)
ftp = fn * exp(S)

where S depends on y = λL / HL².

Worked Example

Given: 3.5" tubing (ID = 2.992"), vertical well. QL = 500 bbl/d oil (ρL = 50 lb/ft³), QG = 200 Mscf/d (ρG = 4 lb/ft³), σ = 25 dynes/cm.

Step 1: Areas and velocities:

A = π/4 * (2.992/12)^2 = 0.0488 ft²
vsl = (500 * 5.615) / (86,400 * 0.0488) = 0.666 ft/s
vsg = (200,000 * 1/86,400) / 0.0488 * (14.7/(P*Z)) ... (depends on P, T, Z)

(Simplified for illustration: vsg ≈ 3.0 ft/s)

vm = 0.666 + 3.0 = 3.666 ft/s
λL = 0.666 / 3.666 = 0.182
NFR = 3.666^2 / (32.2 * 2.992/12) = 13.44 / 8.03 = 1.67

Step 2: Flow pattern:

L1 = 316 * 0.182^0.302 = 316 * 0.598 = 189
L3 = 0.10 * 0.182^(-1.4516) = 0.10 * 19.8 = 1.98
Since λL = 0.182, L3 = 1.98, NFR = 1.67 < L3 → Check L2
L2 = 0.0009252 * 0.182^(-2.4684) = 0.0009252 * 155 = 0.143
NFR = 1.67 > L2 → Transition zone

Valid Ranges

Parameter	Beggs-Brill Data Range
Pipe diameter	1 – 1.5 in (experimental), applied up to 12"
Inclination	-90° to +90° (any angle)
Liquid holdup	0 – 0.87
Gas velocity	0 – 100 ft/s
Liquid velocity	0 – 10 ft/s

Limitations

Developed from small-diameter (1–1.5") air-water data
May not be accurate for high-viscosity oils or foaming systems
Flow pattern transitions can be discontinuous
Does not account for pipe roughness explicitly (uses smooth pipe)

References

Beggs, H.D. & Brill, J.P. (1973). "A Study of Two-Phase Flow in Inclined Pipes." JPT, 25(5), 607–617.
Brill, J.P. & Mukherjee, H. (1999). Multiphase Flow in Wells. SPE Monograph, Vol. 17.
PetroWiki — Multiphase flow: https://petrowiki.spe.org/Multiphase_flow_in_pipes

Beggs-Brill Multiphase Flow