Vertical Multiphase Flow Correlations

Overview

Accurate prediction of pressure gradients in vertical and near-vertical wells is essential for:

• Well performance analysis — bottom-hole pressure from wellhead pressure
• Gas lift design — injection pressure and depth optimization
• Artificial lift sizing — pump setting depth and required power
• Flowing gradient curves — production capacity at various wellhead pressures

This document covers two major correlations developed specifically for vertical two-phase flow:

1. Hagedorn-Brown (1965) — General vertical flow, wide range of conditions
2. Gray (1974) — Optimized for vertical gas wells with mist/annular flow

Both correlations use dimensionless groups that account for fluid properties, flow rates, and pipe geometry, making them applicable across different field conditions.

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Hagedorn-Brown Correlation (1965)

Development Background

Hagedorn and Brown conducted an extensive experimental study using a 1,500-ft vertical test well equipped with:

• Three tubing sizes: 1.0 in., 1.25 in., 1.5 in.
• Electronic pressure transmitters at multiple depths
• Air-water systems with varying liquid viscosities
• 475 tests covering wide flow conditions

Key innovation: Separate correlations for liquid holdup and friction factor using dimensionless groups based on similarity analysis.

Pressure Gradient Equation

The total vertical pressure gradient consists of three components:

$$\frac{dp}{dh} = \left(\frac{dp}{dh}\right)_{\text{elevation}} + \left(\frac{dp}{dh}\right)_{\text{friction}} + \left(\frac{dp}{dh}\right)_{\text{acceleration}}$$

Expanding each term:

$$144 \frac{dp}{dh} = \bar{\rho}_m + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar{\rho}_m} + \frac{\Delta(v_m^2/2g_c)}{\Delta h}$$

Where the mixture density accounting for slippage is:

$$\bar{\rho}_m = \rho_L H_L + \rho_g (1 - H_L)$$

Physical interpretation:

• Elevation term — Hydrostatic head, increased by liquid holdup
• Friction term — Wall friction and interfacial shear
• Acceleration term — Usually negligible except at low pressures near surface

Dimensionless Groups

Hagedorn-Brown identified four primary dimensionless numbers governing vertical two-phase flow:

1. Liquid Velocity Number ($N_

Represents the ratio of liquid kinetic forces to surface tension forces:

$$N_{LV} = 1.938 v_{SL} \sqrt[4]{\frac{\rho_L}{\sigma}}$$

Where:

• $v_{SL}$ = superficial liquid velocity, ft/s
• $\rho_L$ = liquid density, lb/ft³
• $\sigma$ = surface tension, dynes/cm

Physical meaning: High $N_{LV}$ → turbulent liquid flow, increased droplet atomization

2. Gas Velocity Number ($N_

Represents gas velocity effects:

$$N_{GV} = 1.938 v_{SG} \sqrt[4]{\frac{\rho_L}{\sigma}}$$

Where $v_{SG}$ is superficial gas velocity, ft/s.

Physical meaning: High $N_{GV}$ → high gas flux, mist flow regime likely

3. Pipe Diameter Number ($N_D$)

Normalizes pipe size with respect to fluid properties:

$$N_D = 120.872 D \sqrt{\frac{\rho_L}{\sigma}}$$

Where $D$ is pipe inside diameter, ft.

Physical meaning: Relates pipe diameter to capillary length scale

4. Liquid Viscosity Number ($N_L$)

Characterizes viscous forces relative to surface tension:

$$N_L = 0.15726 \mu_L \left(\frac{1}{\rho_L \sigma^3}\right)^{1/4}$$

Where $\mu_L$ is liquid viscosity, cP.

Physical meaning: High $N_L$ → viscous forces dominate (heavy oils), affects flow pattern

Liquid Holdup Correlation

The liquid holdup $H_L$ (fraction of pipe volume occupied by liquid) is determined from:

$$\frac{H_L}{\psi} = f\left(\frac{N_{LV}}{N_{GV}^{0.575}} \left(\frac{p}{p_a}\right)^{0.1} \frac{CN_L}{N_D}\right)$$

Where:

• $\psi$ = secondary correction factor
• $p$ = pressure, psia
• $p_a$ = 14.7 psia (atmospheric pressure)
• $C$ = viscosity correction coefficient (from chart)

Step 1: Calculate primary holdup ($H_L/\psi$)

The primary holdup correlation is given graphically (Fig. 7 in original paper) or by:

$$\frac{H_L}{\psi} = \frac{A}{1 + B \cdot X^C}$$

Where $X$ is the correlating parameter:

$$X = \frac{N_{LV}}{N_{GV}^{0.575}} \left(\frac{p}{p_a}\right)^{0.1} \frac{CN_L}{N_D}$$

And empirical constants are (curve-fit to published chart):

• $A \approx 1.0$
• $B$ and $C$ determined from chart correlation

Step 2: Determine viscosity correction (C)

For low viscosity liquids ($N_L \to 0$): $CN_L \to 0.061$

For higher viscosity, $CN_L$ is read from Fig. 12 (original paper) as function of $N_L$.

Step 3: Calculate secondary correction ($\psi$)

The correction factor $\psi$ accounts for additional holdup effects:

$$\psi = f\left(\frac{N_{GV}}{N_{LV}^{0.380}} N_L^{0.10}\right)$$

Given graphically in Fig. 9 (original paper). For most practical cases: $\psi \approx 1.0$ to 1.2.

Step 4: Final holdup

$$H_L = \frac{H_L}{\psi} \times \psi$$

Friction Factor Correlation

The two-phase friction factor is based on mixture Reynolds number:

$$N_{Re} = \frac{C_1 D (v_{SL} + v_{SG}) [\rho_L H_L + \rho_g(1-H_L)]}{\mu_L^{H_L} \mu_g^{1-H_L}}$$

Where the mixture viscosity uses the Arrhenius power-law mixing rule:

$$\mu_m = \mu_L^{H_L} \mu_g^{1-H_L}$$

Physical basis: Gas-liquid mixtures show "concave" viscosity-concentration curves (viscosity drops rapidly with gas fraction), better represented by logarithmic mixing than linear.

The friction factor $f$ is then obtained from standard Moody chart correlations:

• Laminar flow ($N_{Re} < 2000$): $$f = \frac{64}{N_{Re}}$$

• Turbulent flow ($N_{Re} \geq 2000$):

  • Smooth pipe: Colebrook-White or explicit approximations
  • Typical range: $f = 0.015$ to 0.035

Griffith Modification

For bubble flow regime (low gas rates, high liquid rates), the Griffith correlation provides better holdup prediction:

Criterion for bubble flow:

$$v_m < L_B \text{ and } \lambda_L > 0.13$$

Where $L_B$ is bubble-slug transition velocity (approximately 10 ft/s for typical conditions).

Bubble flow holdup:

$$H_L = \lambda_L + \frac{v_m - v_b}{C_0 v_m}$$

Where:

• $\lambda_L$ = no-slip liquid holdup = $v_{SL}/(v_{SL} + v_{SG})$
• $v_b$ = bubble rise velocity (approximately 0.8 to 1.2 ft/s)
• $C_0$ = distribution parameter (approximately 1.2)

Application: Use Griffith correlation when bubble flow criteria are met; otherwise use Hagedorn-Brown holdup correlation.

Calculation Procedure

To calculate pressure at depth $h + \Delta h$ given pressure at depth $h$:

1. Calculate superficial velocities: $$v_{SL} = \frac{q_L}{A_t}, \quad v_{SG} = \frac{q_g}{A_t}$$

2. Evaluate dimensionless groups:

   • Calculate $N_{LV}$, $N_{GV}$, $N_D$, $N_L$

3. Determine liquid holdup:

   • Check bubble flow criterion (Griffith)
   • If bubble flow: Use Griffith $H_L$
   • Otherwise: Calculate correlating parameter $X$
   • Read $H_L/\psi$ from correlation chart
   • Calculate $\psi$ from secondary correlation
   • Compute $H_L = (H_L/\psi) \times \psi$

4. Calculate friction factor:

   • Compute mixture Reynolds number $N_{Re}$
   • Determine $f$ from Moody correlation

5. Evaluate pressure gradient:

   • Elevation: $(\Delta p/\Delta h)_{elev} = \bar{\rho}_m / 144$
   • Friction: $(\Delta p/\Delta h)_{fric}$ from equation
   • Total: $dp/dh$ (usually acceleration term negligible)

6. Calculate outlet pressure: $$p_{out} = p_{in} + \frac{dp}{dh} \times \Delta h$$

7. Iterate if necessary: Update fluid properties at new pressure and repeat for accuracy.

Applicability and Limitations

Validated Range:

• Pipe diameters: 1.0 to 2.0 in.
• Vertical or near-vertical (θ > 80° from horizontal)
• Wide liquid viscosity range (0.86 to 110 cP tested)
• Liquid rates: 0 to 30 gal/min
• Gas rates: 0 to 300 Mscf/D
• Pressures: 35 to 95 psia

Strengths:

• Based on long-tube (1500 ft) experimental data
• Accounts for viscosity effects explicitly
• Reduces to single-phase flow equations at limits
• Uses only dimensionless groups (good scalability)

Limitations:

• Developed for air-water; extrapolation to oil-gas requires validation
• Charts can be difficult to read accurately (computerized curve-fits help)
• Less accurate for very small tubing (< 1 in.) or very large (> 4 in.)
• Not recommended for highly inclined or horizontal flow

When to use Hagedorn-Brown:

• Vertical or near-vertical wells (θ > 75°)
• Moderate to high liquid rates (oil wells, gas lift)
• When liquid viscosity varies significantly
• General-purpose vertical flow calculations

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Gray Correlation (1974)

Development Background

Gray developed a correlation optimized for high gas-liquid ratio vertical flow, typical of:

• Gas wells producing liquids (condensate, water)
• High-GOR oil wells
• Mist flow regime in gas lift

Source: API Manual 14B (natural gas production)

Key characteristics:

• Simplified holdup correlation for mist/annular flow
• Emphasis on gas wells (GLR > 5000 scf/STB)
• Accounts for liquid entrainment as droplets

When to Use Gray

Best applications:

• Gas wells with liquid loading
• GLR > 5000 scf/STB
• Mist or annular flow regime
• Small tubing (1.5 to 3.5 in.)

Advantages over Hagedorn-Brown for gas wells:

• Simpler calculation procedure
• Better prediction at very high gas rates
• More conservative (safer) for gas well design

Limitations:

• Less accurate for oil wells (low GLR)
• Limited validation data compared to Hagedorn-Brown
• Proprietary correlation details in API Manual 14B

Pressure Gradient Equation

Similar form to Hagedorn-Brown but with simplified holdup:

$$\frac{dp}{dh} = \frac{\rho_L H_L + \rho_g(1-H_L)}{144} + \frac{f v_m^2 \rho_m}{2 g_c D \cdot 144}$$

Gray holdup correlation:

For mist flow (high GLR):

$$H_L = \frac{v_{SL}}{v_m} \left(1 + \frac{v_b}{v_m}\right)$$

Where $v_b$ is the bubble/droplet rise velocity, correlated as function of gas velocity and pipe diameter.

Note: Full Gray correlation equations are proprietary in API 14B. Most implementations use simplified forms or vendor software (e.g., PIPESIM, PROSPER).

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Comparison: Hagedorn-Brown vs. Gray

Rule of thumb:

• Oil wells (GLR < 2000): Use Hagedorn-Brown
• Gas lift wells (GLR 2000-5000): Either method, validate
• Gas wells (GLR > 5000): Prefer Gray

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Related Documentation

• Pipeflow Overview — Correlation selection guide
• Beggs-Brill Correlation — For inclined/deviated wells
• Single-Phase Flow — Foundation equations
• PVT Properties — Required fluid property correlations
• Gas Properties — Gas density, viscosity, compressibility

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References

1. Hagedorn, A.R. and Brown, K.E. (1965). "Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits." Journal of Petroleum Technology, 17(4), pp. 475-484. SPE-940-PA.

2. Brown, K.E. and Beggs, H.D. (1977). The Technology of Artificial Lift Methods, Volume 1. Tulsa, OK: PennWell Publishing Company. Chapter 3: Multiphase Flow in Wells.

3. Gray, H.E. (1974). "Vertical Flow Correlation in Gas Wells." API User's Manual for API 14B, Subsurface Controlled Safety Valve Sizing Computer Program. Dallas, TX: American Petroleum Institute.

4. Guo, B., Lyons, W.C., and Ghalambor, A. (2007). Petroleum Production Engineering: A Computer-Assisted Approach. Burlington, MA: Gulf Professional Publishing. Chapter 3: Vertical Lift Performance.

5. Brill, J.P. and Mukherjee, H. (1999). Multiphase Flow in Wells. Monograph Series Vol. 17. Richardson, TX: Society of Petroleum Engineers. Chapter 4: Vertical Flow Correlations.

6. Economides, M.J., Hill, A.D., Ehlig-Economides, C., and Zhu, D. (2013). Petroleum Production Systems, 2nd Edition. Upper Saddle River, NJ: Prentice Hall. Chapter 2: Production from Vertical Wells.