Multiphase Flow

Hagedorn-Brown Correlation

The industry-standard empirical method for calculating pressure gradients in vertical gas wells. Predicts liquid holdup and bottom-hole pressure using dimensionless groups derived from extensive field data.

Launch Calculator Liquid Holdup

Data Source

1,000+ wells

Field measurements from 1.25" to 3.5" tubing at University of Tulsa (1965).

Best Application

Vertical gas wells

High GLR wells with condensate or water production (<15° deviation).

Accuracy

± 10-15%

For vertical wells within original data range; field calibration recommended.

Contents

Use this method when:

  • Calculating BHP from wellhead conditions
  • Predicting liquid loading in gas wells
  • Sizing velocity strings or artificial lift
  • Performing nodal analysis

1. Method Overview

The Hagedorn-Brown correlation, published in JPT April 1965 (SPE-940-PA), remains one of the most widely used methods for vertical multiphase flow calculations. It provides a unified approach that handles all flow regimes without discontinuities at regime transitions.

Development Background

A.R. Hagedorn and K.E. Brown developed this correlation at the University of Tulsa using:

  • 1,500-ft test well with 1.0", 1.25", and 1.5" tubing
  • 475 pressure traverse measurements across various flow conditions
  • Back-calculated holdup values from measured pressure drops
  • Dimensionless groups adapted from Duns and Ros (1963)
Key insight: Unlike flow-regime-specific methods (Duns-Ros, Orkiszewski), Hagedorn-Brown uses a single continuous correlation for all flow patterns, avoiding discontinuities that can cause numerical instabilities in iterative calculations.
Vertical gas well cross-section schematic showing wellhead with pressure gauge at surface, tubing string running down center, casing with cement annulus, packer seal near bottom, and perforations with flow arrows indicating multiphase production from formation to surface with depth markers
Vertical well schematic showing tubing, casing, and pressure measurement points for traverse analysis.

Application Range

Parameter Original Data Range Recommended Limits
Tubing ID 1.0" - 1.5" 1.0" - 4.0"
Liquid Rate 0 - 2,000 bbl/d 0 - 10,000 bbl/d
GLR 50 - 3,000 scf/bbl 50 - 50,000 scf/bbl
Deviation Vertical only < 15° from vertical
Oil Viscosity 0.5 - 110 cP 0.1 - 200 cP

2. Dimensionless Numbers

The correlation uses four dimensionless groups that characterize the flow conditions. These numbers relate fluid properties, flow velocities, and pipe geometry in a way that allows correlation of experimental data.

Liquid Velocity Number (NLV)

Definition: NLV = 1.938 × vSL × (ρL / σ)0.25 Where: vSL = Superficial liquid velocity (ft/s) ρL = Liquid density (lb/ft³) σ = Liquid-gas surface tension (dyne/cm) 1.938 = Unit conversion factor for field units

Gas Velocity Number (NGV)

Definition: NGV = 1.938 × vSG × (ρL / σ)0.25 Where: vSG = Superficial gas velocity at flowing conditions (ft/s) Note: Gas velocity must be calculated at actual P and T: vSG = Qg,std × (Pb/P) × (T/Tb) × Z / (86400 × A) Where Pb = 14.7 psia, Tb = 520°R (standard conditions)

Pipe Diameter Number (ND)

Definition: ND = 120.872 × D × (ρL / σ)0.5 Where: D = Tubing inside diameter (ft) 120.872 = Unit conversion factor for field units

Liquid Viscosity Number (NL)

Definition: NL = 0.15726 × μL × (1 / (ρL × σ³))0.25 Where: μL = Liquid viscosity (cP) 0.15726 = Unit conversion factor for field units
Hagedorn-Brown dimensionless numbers diagram showing four boxes explaining N-LV liquid velocity number representing liquid inertia versus surface tension, N-GV gas velocity number, N-D diameter number for pipe scale effect, and N-L viscosity number for viscous versus surface forces with formulas
Hagedorn-Brown dimensionless groups showing physical interpretation of each parameter.

Typical Values

Number Low GLR Oil Well Gas-Condensate Well High-Rate Gas Well
NLV 5 - 50 0.5 - 5 0.01 - 0.5
NGV 10 - 100 50 - 500 200 - 2000
ND 20 - 200 (geometry dependent)
NL 0.001 - 0.1 0.0001 - 0.01 0.0001 - 0.001

3. Liquid Holdup Correlation

Liquid holdup (HL) is the fraction of the pipe cross-section occupied by liquid at any instant. It differs from the input liquid fraction (λL) because gas flows faster than liquid—this velocity difference is called "slip."

Holdup vs. No-Slip Holdup

No-slip (input) liquid fraction: λL = vSL / (vSL + vSG) = vSL / vm Actual liquid holdup: HL ≥ λL (always, due to slip) Physical meaning: • λL = 0.05 means 5% of inlet flow is liquid • HL = 0.20 means 20% of pipe volume is liquid • Difference indicates liquid accumulation due to slower liquid velocity

Hagedorn-Brown Holdup Procedure

The correlation uses three steps with polynomial curve fits:

Step 1: Calculate CNL from NL

Viscosity correction factor: CNL = 0.061 × NL³ - 0.0929 × NL² + 0.0505 × NL + 0.0019 This polynomial fits the original H-B Figure 3 correlation chart.

Step 2: Calculate correlation groups H and B

Primary correlating group (H): H = (NLV / NGV0.575) × (P / 14.7)0.1 × CNL / ND Secondary correlating group (B): B = (NGV × NLV0.38) / ND2.14

Step 3: Calculate ψ factor and HL

ψ factor (piecewise polynomial): If B ≤ 0.025: ψ = 27170×B³ - 317.52×B² + 0.5472×B + 0.9999 If 0.025 < B ≤ 0.055: ψ = -533.33×B² + 58.524×B + 0.1171 If B > 0.055: ψ = 2.5714×B + 1.5962 Holdup ratio from H group: HL/ψ = √[(0.0047 + 1123.32×H + 729489.64×H²) / (1 + 1097.1566×H + 722153.97×H²)] Final liquid holdup: HL = (HL/ψ) × ψ Constraint: HL ≥ λL (physical requirement)
Hagedorn-Brown holdup correlation chart showing semi-log plot of H-L over psi ratio on y-axis from 0 to 1.0 versus correlating parameter H on logarithmic x-axis, with S-shaped correlation curve rising from near zero at low H values to asymptotically approaching 1.0 at high H values
Hagedorn-Brown holdup correlation relating HL/ψ to the correlating parameter H.

Griffith Bubble Flow Modification

For bubble flow regime, the original H-B method can underpredict holdup. A modification using the Griffith correlation is applied:

Bubble flow criterion: LB = max(1.071 - 0.2218 × vm² / D, 0.13) If λg < LB, use Griffith holdup: Griffith holdup: HL = 1 - 0.5 × [1 + vm/vs - √((1 + vm/vs)² - 4×vSG/vs)] Where vs = 0.8 ft/s (slip velocity for large bubbles)

Holdup and Flow Patterns

Flow Pattern Typical HL GLR Range Characteristics
Bubble 0.70 - 0.95 < 500 Discrete gas bubbles in liquid
Slug 0.40 - 0.70 500 - 2,000 Alternating liquid slugs and gas pockets
Churn/Transition 0.25 - 0.40 2,000 - 5,000 Chaotic, oscillatory flow
Annular/Mist 0.05 - 0.25 > 5,000 Liquid film on wall, gas core with droplets
Four vertical two-phase flow patterns side by side showing bubble flow with dispersed gas bubbles in liquid, slug flow with alternating Taylor bubbles and liquid slugs, churn flow with chaotic oscillating mixture, and annular-mist flow with thin liquid film on wall and gas core with entrained droplets, with typical holdup values labeled
Vertical multiphase flow patterns showing progression from bubble to annular/mist with increasing gas velocity.

4. Pressure Gradient Calculation

The total pressure gradient consists of three components: elevation (hydrostatic), friction, and acceleration. For most gas well applications, acceleration is negligible.

Total Pressure Gradient

General form: (dP/dL)total = (dP/dL)elevation + (dP/dL)friction + (dP/dL)acceleration For steady-state gas wells (acceleration ≈ 0): (dP/dL)total ≈ (dP/dL)h + (dP/dL)f Units: psi/ft

Elevation (Hydrostatic) Component

Elevation gradient: (dP/dL)h = ρm / 144 Where mixture density: ρm = ρL × HL + ρg × (1 - HL) Units: ρ in lb/ft³, result in psi/ft For deviated wells: (dP/dL)h,deviated = (dP/dL)h × cos(θ) Where θ = deviation angle from vertical

Friction Component

Friction gradient: (dP/dL)f = (f × ρm × vm²) / (2 × gc × D × 144) Where: f = Darcy friction factor (from Colebrook-White) vm = vSL + vSG (mixture velocity, ft/s) gc = 32.174 lbm·ft/(lbf·s²) D = Tubing ID (ft) Two-phase Reynolds number: Re = ρm × vm × D / μTP Two-phase viscosity (geometric mean): μTP = μLHL × μg(1-HL)

Colebrook-White Friction Factor

Implicit equation (requires iteration): 1/√f = -2 × log₁₀(ε/(3.7×D) + 2.51/(Re×√f)) Where: ε = Pipe roughness (0.0006 in for commercial steel tubing) D = Pipe diameter (in, same units as ε) Typical friction factors: Smooth pipe: f = 0.015 - 0.020 Tubing: f = 0.018 - 0.025 Corroded: f = 0.025 - 0.040

Component Comparison

Well Type Elevation % Friction % Dominant Factor
Low-rate oil well 95 - 99% 1 - 5% Hydrostatic (high HL)
Gas-condensate well 85 - 95% 5 - 15% Hydrostatic
High-rate gas well 60 - 85% 15 - 40% Both significant
Very high rate 40 - 60% 40 - 60% Friction increases

5. Worked Example

Calculate the bottom-hole pressure for a gas well producing through 2.875" (2.441" ID) tubing.

Given Data

Well geometry: Measured depth: 8,000 ft Tubing ID: 2.441 in = 0.2034 ft Deviation: 0° (vertical) Operating conditions: Wellhead pressure: 500 psia Surface temperature: 100°F Gas rate: 2 MMscfd Liquid rate: 50 bbl/d (water) Fluid properties: ρL = 62.4 lb/ft³ (water) ρg = 2.5 lb/ft³ (at avg conditions) μL = 0.8 cP μg = 0.012 cP σ = 50 dyne/cm Z = 0.92

Step 1: Calculate Velocities

Pipe area: A = π × D² / 4 = π × (0.2034)² / 4 = 0.0325 ft² Superficial liquid velocity: vSL = (50 × 5.615) / (86400 × 0.0325) = 0.100 ft/s Superficial gas velocity (at avg P = 800 psia, T = 160°F): Qg,actual = 2,000,000 × (14.7/800) × (620/520) × 0.92 = 40,200 scf/d vSG = 40,200 / (86400 × 0.0325) = 14.3 ft/s Mixture velocity: vm = 0.100 + 14.3 = 14.4 ft/s No-slip holdup: λL = 0.100 / 14.4 = 0.0069 (0.69%)

Step 2: Calculate Dimensionless Numbers

Convert surface tension: σ = 50 dyne/cm × 6.852×10⁻⁵ = 0.00343 lb/ft Dimensionless numbers: NLV = 1.938 × 0.100 × (62.4 / 0.00343)0.25 = 1.938 × 0.100 × 36.7 = 7.11 NGV = 1.938 × 14.3 × 36.7 = 1017 ND = 120.872 × 0.2034 × (62.4 / 0.00343)0.5 = 120.872 × 0.2034 × 134.9 = 3317 NL = 0.15726 × 0.8 × (1 / (62.4 × 0.00343³))0.25 = 0.0023

Step 3: Calculate Holdup

CNL: CNL = 0.061×(0.0023)³ - 0.0929×(0.0023)² + 0.0505×(0.0023) + 0.0019 = 0.00202 H group: H = (7.11 / 10170.575) × (800/14.7)0.1 × 0.00202 / 3317 H = (7.11 / 64.8) × 1.48 × 0.00202 / 3317 = 9.85×10⁻⁸ B group: B = (1017 × 7.110.38) / 33172.14 = (1017 × 2.24) / 3.23×10⁷ = 7.05×10⁻⁵ ψ factor (B < 0.025): ψ = 27170×(7.05×10⁻⁵)³ - 317.52×(7.05×10⁻⁵)² + 0.5472×(7.05×10⁻⁵) + 0.9999 = 1.000 HL/ψ: HL/ψ = √[(0.0047 + 1123.32×9.85×10⁻⁸ + 729489.64×(9.85×10⁻⁸)²) / (1 + 1097.1566×9.85×10⁻⁸ + 722153.97×(9.85×10⁻⁸)²)] HL/ψ ≈ 0.069 Final holdup: HL = 0.069 × 1.000 = 0.069 (constrained: max(0.069, 0.0069) = 0.069)

Step 4: Calculate Pressure Gradient

Mixture density: ρm = 62.4 × 0.069 + 2.5 × (1 - 0.069) = 4.31 + 2.33 = 6.64 lb/ft³ Elevation gradient: (dP/dL)h = 6.64 / 144 = 0.0461 psi/ft Two-phase viscosity: μTP = 0.80.069 × 0.0120.931 = 0.987 × 0.0136 = 0.0134 cP Reynolds number: Re = (6.64 × 14.4 × 0.2034) / (0.0134 × 6.72×10⁻⁴) = 2.16×10⁶ Friction factor (iterating Colebrook-White): ε/D = 0.0006 / 2.441 = 0.000246 f ≈ 0.0175 (Darcy) Friction gradient: (dP/dL)f = (0.0175 × 6.64 × 14.4²) / (2 × 32.174 × 0.2034 × 144) = 0.0012 psi/ft Total gradient: (dP/dL)total = 0.0461 + 0.0012 = 0.0473 psi/ft

Step 5: Final Result

Total pressure drop: ΔP = 0.0473 × 8000 = 378 psi Bottom-hole pressure: BHP = 500 + 378 = 878 psia Summary: • Flow pattern: Annular/Mist (HL = 6.9%) • Elevation ΔP: 369 psi (97.6%) • Friction ΔP: 9 psi (2.4%) • Well is efficiently lifting liquids
Interpretation: Low liquid holdup (6.9%) indicates the well is in annular/mist flow with efficient liquid removal. The pressure gradient is dominated by the elevation component. No artificial lift is needed at these conditions.

6. Correlation Selection Guide

Multiple multiphase flow correlations exist for different applications. Selection depends on well geometry, flow regime, and required accuracy.

Correlation Comparison

Correlation Year Best For Limitations
Hagedorn-Brown 1965 Vertical gas wells, all flow regimes Vertical only; empirical
Beggs-Brill 1973 Deviated/horizontal wells Less accurate for vertical
Duns-Ros 1963 Vertical, regime-specific Discontinuities at transitions
Orkiszewski 1967 Combines best methods Complex implementation
Gray 1974 Quick gas well estimates Lower accuracy
Mechanistic (OLGA) Modern Complex systems, transients Requires software

Selection Flowchart

Decision process: 1. Is well vertical or near-vertical (<15°)? YES → Use Hagedorn-Brown NO → Use Beggs-Brill 2. Is it a high-rate gas well? YES → Hagedorn-Brown is excellent NO → Continue evaluation 3. Need flow regime identification? YES → Consider Duns-Ros or Orkiszewski NO → Hagedorn-Brown (unified approach) 4. Deviated well (15-60°)? → Use Beggs-Brill with inclination correction 5. Horizontal well? → Use Beggs-Brill or Eaton-Brown Recommendation: • Preliminary: Gray correlation (quick estimates) • Design: Hagedorn-Brown (industry standard) • Critical: Mechanistic model + field calibration

Accuracy Summary

Application H-B Accuracy When to Use Alternative
High-rate vertical gas well ± 10% Never—this is H-B's strength
Gas-condensate well ± 15% Only if field data shows bias
Loaded gas well ± 15-20% Consider Turner criterion also
Oil well (low GOR) ± 20% Consider Duns-Ros or Orkiszewski
Deviated well (>15°) Poor Use Beggs-Brill
Best practice: For critical applications, compare 2-3 correlations and validate against field pressure surveys. Hagedorn-Brown typically provides reliable middle-range predictions for vertical gas wells. When possible, tune the correlation to field data by adjusting holdup predictions.

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