Multiphase Flow in Wells

Gray Correlation for Vertical Two-Phase Flow: Engineering Fundamentals

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)

H.E. Gray original work (1974); published in the API 14B (1978) Subsurface Controlled Safety Valve Sizing manual

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%)
Gray is a single continuous correlation: unlike Duns-Ros, Gray does not switch equations by flow regime. The regimes above are shown only as a qualitative guide to where Gray is most reliable. The flow-pattern label in the calculator is an informational Duns-Ros-style add-on, not part of the Gray pressure-traverse math.
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 two dimensionless groups built from the no-slip mixture density, the mixture superficial velocity, the density difference and surface tension. The leading constant 453.592 absorbs the unit conversion so that surface tension is supplied directly in dyne/cm. Note that Nv uses the mixture velocity raised to the fourth power.

No-Slip Mixture Density (ρns): ρns = ρL × λL + ρg × (1 − λL) where λL = Vsl / (Vsl + Vsg) (no-slip liquid holdup) and Vm = Vsl + Vsg (mixture superficial velocity) Gray Velocity Number (Nv): Nv = 453.592 × ρns² × Vm⁴ / (gc × σ × (ρL − ρg)) Where: ρns = No-slip mixture density (lbm/ft³) Vm = Mixture superficial velocity (ft/s) — note the 4th power gc = 32.17 lbm·ft/(lbf·s²) σ = Surface tension (dyne/cm, used directly) Gray Pipe Diameter Number (Nd): Nd = 453.592 × gc × (ρL − ρg) × D² / σ Where: D = Pipe inside diameter (ft) σ = Surface tension (dyne/cm)

Physical Interpretation

Number Physical Meaning Typical Range (gas wells)
Nv Mixture-inertia to surface-tension / buoyancy forces (Vm⁴ dependence) 10² - 10⁶
Nd Buoyancy (gravitational) to surface-tension forces at pipe scale 500 - 5,000
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 representative (mid-well) conditions At the mid-well pressure ≈ 755 psia, T ≈ 132°F, Z ≈ 0.92: ρg ≈ 2.42 lbm/ft³ (real gas law) Qg,std = 5,000,000 scfd = 57.87 scf/s Qg,actual = Qg,std × (14.7/755) × (592/520) × (0.92/1.0) = 1.19 ft³/s Vsg = 1.19/0.0325 = 36.5 ft/s Vm = Vsl + Vsg = 0.10 + 36.5 = 36.6 ft/s Step 4: No-slip holdup and no-slip mixture density λL = Vsl/Vm = 0.10/36.6 = 0.00273 ρL = 45×(1−0.10) + 62.4×0.10 = 46.74 lbm/ft³ (10% water cut) ρns = 46.74 × 0.00273 + 2.42 × (1 − 0.00273) = 2.54 lbm/ft³ Step 5: Calculate Gray dimensionless numbers (σ = 20 dyne/cm direct) Nv = 453.592 × 2.54² × 36.6⁴ / (32.17 × 20 × (46.74 − 2.42)) Nv = 453.592 × 6.45 × 1.79×10⁶ / 28,510 ≈ 1.84×10⁵ Nd = 453.592 × 32.17 × (46.74 − 2.42) × 0.2034² / 20 ≈ 1,338

3. Liquid Holdup (Gray Closed-Form Method)

Gray's liquid holdup is a single closed-form expression in the dimensionless numbers Nv and Nd and the superficial velocity ratio R. There is no flow-regime switching and no iterative chart lookup — the gas void fraction fg falls out directly, and the in-situ liquid holdup is HL = 1 − fg. The result is always at least the no-slip holdup, because gas slips past the slower liquid.

No-Slip Liquid Holdup: λL = Vsl / (Vsl + Vsg) = Ql / (Ql + Qg) This is the holdup if both phases traveled at the same velocity. Superficial Velocity Ratio: R = Vsl / Vsg Gray B coefficient: B = 0.0814 × (1 − 0.0554 × ln(1 + 730·R/(R + 1))) Gray A exponent: A = −2.314 × ( Nv × (1 + 205/Nd) )B Gas Void Fraction and Liquid Holdup: fg = (1 − eA) / (R + 1) HL = 1 − fg (clamped to λL ≤ HL ≤ 1) Slip Ratio: S = HL / λL (typically ~1.2-3× in high-GLR mist/annular wells; larger at lower gas velocity) Worked Example (mid-well, from Section 2): Nv = 1.84×10⁵, Nd = 1,338, R = 0.10/36.5 = 0.00274 B = 0.0814 × (1 − 0.0554 × ln(1 + 730×0.00274/1.00274)) = 0.0765 A = −2.314 × (1.84×10⁵ × (1 + 205/1338))0.0765 = −5.91 fg = (1 − e−5.91) / 1.00274 = 0.995 HL = 1 − 0.995 = 0.0054 (≈ 0.54% — vs no-slip λL = 0.27%, an ~2× slip ratio)

Effect of Holdup on Pressure Drop

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

Slip (in-situ) Mixture Density — used for hydrostatic head: ρs = ρL × HL + ρg × (1 − HL) Example: If ρL = 46.7 lbm/ft³, ρg = 2.42 lbm/ft³, HL = 0.0054: ρs = 46.7 × 0.0054 + 2.42 × 0.9946 = 0.25 + 2.41 = 2.66 lbm/ft³ Compare to no-slip (λL = 0.00273): ρns = 46.7 × 0.00273 + 2.42 × 0.997 = 2.54 lbm/ft³ Error = (2.66 − 2.54)/2.66 ≈ 5% in this high-velocity mist case (slip ≈ 2×). At lower gas velocity slip — and the head error from ignoring it — grow quickly (see the sensitivity table below), which is why Gray solves for HL rather than assuming no-slip.

Holdup Sensitivity Table

Vsg (ft/s) Vsl (ft/s) λ (%) HL (%) Slip Ratio Flow Regime
100.10.992.943.0×Churn
250.10.400.942.4×Annular
500.10.200.341.7×Mist
250.51.962.581.3×Annular
500.50.991.181.2×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) — uses SLIP density: (dP/dL)elev = ρs × (g/gc) × cos(θ) / 144 [psi/ft] Where: ρs = Slip (in-situ) mixture density = ρL·HL + ρg·(1 − HL) g/gc = 1.0 (gravitational constant ratio, dimensionless in lbm/lbf system) θ = Deviation from vertical (degrees) 144 = Conversion factor (in²/ft²) Friction Component — uses NO-SLIP density and Gray's pseudo roughness: (dP/dL)fric = f × ρns × Vm² / (2 × gc × D × 144) [psi/ft] Where: f = Darcy friction factor from Colebrook/Jain, evaluated with the Gray effective relative roughness ε/D = ke/D (NOT bare pipe roughness): ke = (28.5/453.592) × σ / (ρns × Vm²) for R = Vsl/Vsg ≥ 0.007 (blended toward the bare wall roughness ε for R < 0.007; floor ke ≥ 2.77×10⁻⁵ ft) ρns = No-slip mixture density (lbm/ft³) 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 dL (marched surface → bottom) Example Calculation (multi-segment march): Given: PWH = 500 psia, TVD = 8,000 ft, D = 0.2034 ft The calculator integrates the Gray gradient in 60 depth steps, updating Z, ρg, Vsg, holdup and friction at the local P,T of each step. At the representative mid-well point (P ≈ 755 psia, T ≈ 132°F): ρs = 4.15 lbm/ft³, ρns = 2.54 lbm/ft³ Vm = 36.6 ft/s, f ≈ 0.019 (with ke ≈ 1.75×10⁻⁴ ft, ε/D ≈ 8.6×10⁻⁴) Elevation gradient (slip density, g/gc = 1.0): (dP/dL)elev = 4.15 / 144 = 0.0289 psi/ft Friction gradient (no-slip density, pseudo roughness): (dP/dL)fric = 0.019 × 2.54 × 36.6² / (2 × 32.17 × 0.2034 × 144) (dP/dL)fric ≈ 0.0345 psi/ft Mid-well total gradient ≈ 0.063 psi/ft. Integrating the full traverse (gradient rises with depth as pressure and holdup increase) gives: Bottomhole pressure: PBH ≈ 1,010 psia (total ΔP ≈ 510 psi over 8,000 ft) Note: A single-segment calculation seeded at an over-high average pressure inflates both holdup and BHP (the legacy engine returned ~1,572 psia). Marching from surface to bottom with property updates at each step is self-consistent and lands BHP in the defensible ≈ 950-1,100 psia range for these conditions.

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 (used directly — the 1.912 field-unit coefficient already embeds the dyne/cm basis; do NOT convert to lbf/ft) ρL = 45 lbm/ft³, ρg = 2 lbm/ft³ Vcrit = 1.912 × [20 × (45-2) / 4]^0.25 Vcrit = 1.912 × [215]^0.25 = 1.912 × 3.83 = 7.3 ft/s If Vsg = 56 ft/s → Vsg/Vcrit = 7.7 → 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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Frequently Asked Questions

When should the Gray correlation be used instead of other multiphase flow methods?

The Gray correlation is specifically designed for high gas-liquid ratio (GLR > 5,000 scf/bbl) vertical flow in gas wells. It achieves ±10–15% accuracy for gas-condensate wells and is preferred over Hagedorn-Brown for very high GLR applications.

What are the key dimensionless numbers used in the Gray correlation?

The Gray correlation uses dimensionless numbers based on liquid velocity, gas velocity, pipe diameter, and surface tension to characterize two-phase flow behavior. These numbers are combined in Gray's closed-form expression (coefficients B and A) to determine the gas void fraction and the in-situ liquid holdup.

What is the primary application of the Gray vertical flow correlation?

The Gray correlation is primarily used for calculating bottomhole pressure from wellhead measurements in gas wells, sizing production tubing, determining critical velocity for liquid unloading, and calculating injection rates for gas lift design.