Multiphase Flow in Wells

Gray Vertical Two-Phase Flow Correlation

Calculate bottomhole pressure, predict liquid holdup, and analyze gas well performance using Gray's dimensionless number approach. Industry-standard method for high-GLR gas wells per API 14B.

Launch Calculator Dimensionless Numbers

Original Reference

Gray (1974)

API User's Manual for API 14B Subsurface Safety Valve Sizing

Best Application

GLR > 5,000 scf/bbl

High gas-liquid ratio wells in mist or annular flow regimes

Typical Accuracy

±10-15%

For vertical gas wells; less accurate for slug flow or high liquid loading

Contents

Use this guide when you need to:

  • Calculate bottomhole pressure from wellhead conditions
  • Predict liquid loading onset in gas wells
  • Size tubing for gas well completions
  • Evaluate gas lift requirements

1. Overview & Applications

The Gray correlation (1974) is an empirical method for calculating pressure drop in vertical two-phase gas-liquid flow. Developed by H.E. Gray for the API 14B standard on subsurface safety valve sizing, it uses dimensionless groups to correlate liquid holdup with flow conditions.

Primary Use

BHP from Wellhead

Calculate bottomhole flowing pressure given wellhead pressure, flow rates, and fluid properties.

Liquid Loading

Critical Velocity

Determine if gas velocity is sufficient to lift liquids from wellbore (Turner criteria).

Tubing Design

Size Optimization

Balance pressure drop vs. liquid lifting capacity when selecting tubing size.

Gas Lift

Injection Rate

Calculate required gas injection to achieve target bottomhole pressure.

Key Concepts

  • Two-phase flow: Simultaneous flow of gas and liquid in a conduit; behavior differs from single-phase due to phase interactions
  • Liquid holdup (HL): Fraction of pipe cross-section occupied by liquid; determines mixture density
  • Slippage: Gas travels faster than liquid due to buoyancy; causes liquid to accumulate
  • No-slip holdup (λ): Input liquid fraction if phases traveled at same velocity; λ = Ql/(Ql+Qg)
Vertical two-phase flow regime progression showing five pipe sections for bubble, slug, churn, annular, and mist flow patterns with increasing gas velocity from left to right, including velocity ranges, typical holdup values, and Gray correlation accuracy ratings color-coded from poor to excellent
Vertical flow regime progression with Gray correlation applicability ratings for each pattern.

Flow Regime Applicability

Flow Regime Vsg Range Characteristics Gray Accuracy
Bubble < 3 ft/s Discrete gas bubbles in liquid Poor (not designed for this)
Slug 3-10 ft/s Alternating liquid slugs/gas pockets Moderate (±20-30%)
Churn 10-20 ft/s Chaotic oscillating flow Moderate (±15-25%)
Annular 20-50 ft/s Liquid film on wall, gas core Good (±10-15%)
Mist > 50 ft/s Liquid droplets in gas Excellent (±5-10%)
When to use Gray: Vertical or near-vertical wells (<15° deviation) with high GLR (>5,000 scf/bbl) operating in annular or mist flow. For oil wells or high liquid loading, use Hagedorn-Brown instead.

2. Gray Dimensionless Numbers

Gray's correlation uses four dimensionless groups that incorporate fluid properties, pipe geometry, and flow velocities. These numbers normalize the complex physics into universal correlating parameters.

Liquid Velocity Number (Nvl): Nvl = Vsl × (ρL / (g × σ))0.25 Where: Vsl = Superficial liquid velocity (ft/s) = Ql / A ρL = Liquid density (lbm/ft³) g = Gravitational acceleration (32.17 ft/s²) σ = Surface tension (lbf/ft) [multiply dyne/cm × 6.85×10⁻⁵] Gas Velocity Number (Nvg): Nvg = Vsg × (ρL / (g × σ))0.25 Where: Vsg = Superficial gas velocity (ft/s) = Qg / A Qg = Gas volumetric rate at flowing P,T Pipe Diameter Number (Nd): Nd = D × (ρL × g / σ)0.5 Where: D = Pipe inside diameter (ft) Liquid Viscosity Number (Nl): Nl = μL × (g / (ρL × σ³))0.25 Where: μL = Liquid viscosity (lbm/ft·s) [multiply cP × 6.72×10⁻⁴]

Physical Interpretation

Number Physical Meaning Typical Range
Nvl Ratio of liquid inertia to surface tension forces 0.001 - 1.0
Nvg Ratio of gas inertia to surface tension forces 1 - 500
Nd Ratio of gravitational to surface tension forces (pipe scale) 50 - 500
Nl Ratio of viscous to surface tension forces 0.001 - 0.1
Gray dimensionless number schematic showing vertical pipe section with two-phase flow in center, surrounded by four boxes connected by arrows showing how physical variables combine: N-vl for liquid velocity effect, N-vg for gas velocity effect, N-d for pipe diameter scale, and N-l for liquid viscosity effect with formulas
Gray dimensionless numbers showing how physical variables combine into correlating parameters.

Worked Example: Calculate Dimensionless Numbers

Given: Tubing ID = 2.441 in = 0.2034 ft Liquid rate = 50 bbl/day condensate Gas rate = 5,000 Mscfd Wellhead P = 500 psia, T = 100°F Condensate density = 45 lbm/ft³ Gas SG = 0.65 Surface tension = 20 dyne/cm Liquid viscosity = 3 cP Step 1: Calculate pipe area A = π × D²/4 = π × (0.2034)²/4 = 0.0325 ft² Step 2: Calculate liquid velocity Ql = 50 bbl/day × 5.615 ft³/bbl × (1 day/86,400 s) = 0.00325 ft³/s Vsl = Ql/A = 0.00325/0.0325 = 0.10 ft/s Step 3: Calculate gas velocity at flowing conditions ρg at 500 psia ≈ 2.0 lbm/ft³ (from real gas law with Z ≈ 0.92) Qg,std = 5,000,000 scfd = 57.87 scf/s Qg,actual = Qg,std × (14.7/500) × (560/520) × (0.92/1.0) = 1.83 ft³/s Vsg = 1.83/0.0325 = 56.3 ft/s Step 4: Convert units σ = 20 × 6.85×10⁻⁵ = 0.00137 lbf/ft μL = 3 × 6.72×10⁻⁴ = 0.00202 lbm/(ft·s) Step 5: Calculate dimensionless numbers Nvl = 0.10 × (45/(32.17 × 0.00137))^0.25 = 0.10 × (1019)^0.25 = 0.10 × 5.65 = 0.565 Nvg = 56.3 × 5.65 = 318 Nd = 0.2034 × (45 × 32.17/0.00137)^0.5 = 0.2034 × 1023 = 208 Nl = 0.00202 × (32.17/(45 × 0.00137³))^0.25 = 0.00202 × 8150 = 0.0165

3. Liquid Holdup (R-Factor Method)

Gray's liquid holdup correlation uses an empirical R-factor to account for slippage between gas and liquid phases. The actual (in-situ) holdup is higher than the no-slip holdup because gas flows faster than liquid.

No-Slip Liquid Holdup: λ = Vsl / (Vsl + Vsg) = Ql / (Ql + Qg) This is the holdup if both phases traveled at the same velocity. Gray R-Factor: R = (Nvl/Nvg)0.982 × (Nl/Nd)0.124 The R-factor correlates slip behavior with dimensionless numbers. Psi (ψ) Correlation Factor: ψ = 1 + R × (R + 0.0814 × ln(R + 0.000925)) This factor amplifies the no-slip holdup to account for slippage. In-Situ Liquid Holdup: HL = ψ × λ Physical limits: 0 < HL ≤ 1.0 Slip Ratio: S = HL / λ (typically 2-10× for gas wells)

Effect of Holdup on Pressure Drop

Liquid holdup directly affects mixture density, which dominates pressure gradient in vertical wells:

Two-Phase Mixture Density: ρm = ρL × HL + ρg × (1 - HL) Example: If ρL = 45 lbm/ft³, ρg = 2 lbm/ft³, HL = 0.05: ρm = 45 × 0.05 + 2 × 0.95 = 2.25 + 1.9 = 4.15 lbm/ft³ Compare to no-slip (λ = 0.002): ρm,no-slip = 45 × 0.002 + 2 × 0.998 = 2.09 lbm/ft³ Error = (4.15 - 2.09)/4.15 = 50% underprediction if ignoring slip!

Holdup Sensitivity Table

Vsg (ft/s) Vsl (ft/s) λ (%) HL (%) Slip Ratio Flow Regime
100.10.998.28.3×Churn
250.10.404.110.3×Annular
500.10.202.311.5×Mist
250.51.9612.46.3×Annular
500.50.996.86.9×Mist
Gray correlation liquid holdup versus gas velocity chart showing H-L decreasing with increasing superficial gas velocity V-sg, with three curves for different liquid velocities, vertical dashed lines marking flow regime transitions from churn to annular to mist flow
Gray correlation liquid holdup as a function of gas velocity for different liquid loading conditions.

4. Pressure Gradient Calculation

Total pressure gradient in vertical two-phase flow has three components: elevation (hydrostatic head), friction, and acceleration. For most gas well conditions, elevation dominates (80-95% of total).

Total Pressure Gradient: (dP/dL)total = (dP/dL)elevation + (dP/dL)friction + (dP/dL)acceleration Elevation Component (dominant in vertical flow): (dP/dL)elev = ρm × g × cos(θ) / 144 [psi/ft] Where: ρm = Two-phase mixture density (lbm/ft³) g = 32.17 ft/s² θ = Deviation from vertical (degrees) 144 = Conversion factor (in²/ft²) Friction Component: (dP/dL)fric = f × ρm × Vm² / (2 × gc × D × 144) [psi/ft] Where: f = Darcy friction factor (from Colebrook-White) Vm = Vsl + Vsg (mixture velocity, ft/s) gc = 32.17 lbm·ft/(lbf·s²) D = Pipe ID (ft) Acceleration Component: Usually negligible for steady-state vertical flow (< 1% of total)

Bottomhole Pressure Calculation

For vertical well (θ = 0°): PBH = PWH + (dP/dL)total × TVD Example Calculation: Given: PWH = 500 psia, TVD = 8,000 ft ρm = 4.15 lbm/ft³ (from previous example) f = 0.015, Vm = 56.4 ft/s, D = 0.2034 ft Elevation gradient: (dP/dL)elev = 4.15 × 32.17 × 1.0 / 144 = 0.927 psi/ft Friction gradient: (dP/dL)fric = 0.015 × 4.15 × 56.4² / (2 × 32.17 × 0.2034 × 144) (dP/dL)fric = 198 / 1883 = 0.105 psi/ft Total gradient: (dP/dL)total = 0.927 + 0.105 = 1.032 psi/ft Bottomhole pressure: PBH = 500 + 1.032 × 8,000 = 500 + 8,256 = 8,756 psia Note: This is a simplified single-segment calculation. Accurate results require iteration (properties change with depth).

Turner Critical Velocity

The minimum gas velocity to prevent liquid accumulation (loading) in vertical wells:

Turner Critical Velocity (Coleman adjustment): Vcrit = 1.912 × [σ × (ρL - ρg) / ρg²]0.25 [ft/s] Liquid Loading Assessment: • Vsg > 1.2 × Vcrit: Low risk (adequate liquid lifting) • Vsg = 1.0-1.2 × Vcrit: Moderate risk (monitor for loading) • Vsg < Vcrit: High risk (consider intervention) Example: σ = 20 dyne/cm = 0.00137 lbf/ft ρL = 45 lbm/ft³, ρg = 2 lbm/ft³ Vcrit = 1.912 × [0.00137 × (45-2) / 4]^0.25 Vcrit = 1.912 × [0.0147]^0.25 = 1.912 × 0.348 = 6.7 ft/s If Vsg = 56 ft/s → Vsg/Vcrit = 8.4 → Low risk

5. Comparison with Other Correlations

Gray is one of several empirical correlations for vertical two-phase flow. Selection depends on well type, flow regime, and accuracy requirements.

Correlation Comparison Table

Correlation Year Best Application Limitations
Gray 1974 High GLR gas wells, mist/annular flow Poor for slug flow, liquid loading
Hagedorn-Brown 1965 Oil wells, high liquid loading, slug flow Complex charts, less accurate for gas wells
Beggs-Brill 1973 Inclined/horizontal flow, all angles Developed for horizontal; less accurate for vertical
Duns-Ros 1963 Wide range, flow regime maps Discontinuities at transitions, complex
Ansari (mechanistic) 1994 All conditions, physics-based Computationally intensive

When to Use Each Method

Decision Guide: 1. Is it a gas well with GLR > 5,000 scf/bbl? → YES: Use Gray (or Duns-Ros) → NO: Go to step 2 2. Is liquid loading a concern (low gas rate)? → YES: Use Hagedorn-Brown or Ansari mechanistic → NO: Go to step 3 3. Is deviation > 15° from vertical? → YES: Use Beggs-Brill → NO: Gray or Hagedorn-Brown acceptable 4. Is high accuracy required (±5%)? → YES: Use mechanistic model (Ansari, Hasan-Kabir) → NO: Gray is acceptable for screening Industry Practice: Run multiple correlations and compare. If results differ by >20%, investigate flow regime and validate with field pressure surveys.

Accuracy Comparison (Field Studies)

Well Type Gray Error Hagedorn-Brown Error Recommended
High-rate gas (>5 MMscfd) ±8% ±15% Gray
Low-rate gas with loading ±25% ±12% Hagedorn-Brown
Gas condensate (20-50 bbl/MMscf) ±12% ±10% Either acceptable
Oil well (GOR > 1,000) ±18% ±9% Hagedorn-Brown
Summary: Gray correlation is the industry standard for vertical gas wells with high GLR operating in annular or mist flow. For oil wells, high liquid loading, or slug flow conditions, Hagedorn-Brown provides better accuracy. Always validate predictions with measured bottomhole pressure when available.

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