Beggs and Brill Multiphase Flow Correlation

Overview

The Beggs and Brill (1973) correlation is one of the most widely used methods for predicting pressure gradients and liquid holdup during two-phase gas-liquid flow in pipes at any angle of inclination. Unlike earlier correlations limited to horizontal or vertical flow, Beggs and Brill developed a unified approach applicable to:

• Horizontal pipes (θ = 0°)
• Vertical pipes (θ = ±90°)
• Inclined pipes (-90° ≤ θ ≤ +90°)

This versatility makes it particularly valuable for:

• Gathering lines in hilly terrain
• Directional wells with deviations from vertical
• Offshore pipelines laid along sloping sea floors
• Gas lift systems with varying well trajectories

Physical Basis

The Two-Phase Flow Problem

Multiphase flow differs fundamentally from single-phase flow due to:

1. Phase slippage — Gas and liquid travel at different velocities
2. Flow pattern changes — Interface geometry varies with angle and flow rates
3. Liquid holdup — Actual liquid fraction in pipe differs from input fraction
4. Angle-dependent behavior — Pressure recovery in downhill flow is not 100%

The Beggs-Brill correlation addresses all these phenomena through empirical correlations developed from experimental data.

Pressure Gradient Equation

The total pressure gradient consists of three components:

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

Elevation gradient:

$$\left(\frac{dp}{dL}\right)_{\text{elevation}} = \frac{g}{g_c} \sin\theta \left[\rho_L H_L + \rho_g(1 - H_L)\right]$$

Friction gradient:

$$\left(\frac{dp}{dL}\right)_{\text{friction}} = \frac{f_{tp} G_m v_m}{2g_c d}$$

Acceleration gradient:

$$\left(\frac{dp}{dL}\right)_{\text{acceleration}} = \frac{\rho_L H_L + \rho_g(1 - H_L)}{g_c} \frac{v_m dv_m}{dL}$$

Where:

• $H_L$ = liquid holdup (fraction of pipe volume occupied by liquid)
• $f_{tp}$ = two-phase friction factor
• $\theta$ = pipe inclination angle from horizontal (+ upward, - downward)
• $G_m$ = total mass flow rate
• $v_m$ = mixture velocity

Flow Pattern Identification

Beggs and Brill identified three primary flow patterns based on horizontal flow behavior:

1. Segregated Flow

Characteristics:

• Gas and liquid are clearly separated
• Occurs at low liquid and gas velocities
• Includes stratified and wavy flow regimes

Criterion:

$$\lambda_L < 0.01 \text{ and } F_r < L_1$$

or

$$\lambda_L \geq 0.01 \text{ and } F_r < L_2$$

2. Intermittent Flow

Characteristics:

• Alternating gas and liquid slugs
• Includes plug and slug flow
• Common in small-diameter pipes

Criterion:

$$0.01 \leq \lambda_L < 0.4 \text{ and } L_2 < F_r \leq L_3$$

or

$$\lambda_L \geq 0.4 \text{ and } L_3 < F_r \leq L_1$$

3. Distributed Flow

Characteristics:

• Gas is dispersed as bubbles in liquid (bubble flow)
• Or liquid dispersed as droplets in gas (mist flow)
• Occurs at high mixture velocities

Criterion:

$$\lambda_L < 0.4 \text{ and } F_r \geq L_3$$

or

$$\lambda_L \geq 0.4 \text{ and } F_r > L_1$$

Flow Pattern Boundaries

The transition boundaries are defined by:

$$L_1 = 316 \lambda_L^{0.302}$$$$L_2 = 0.0009252 \lambda_L^{-2.4684}$$$$L_3 = 0.10 \lambda_L^{-1.4516}$$

Where $\lambda_L$ is the no-slip liquid holdup (input liquid fraction):

$$\lambda_L = \frac{v_{SL}}{v_{SL} + v_{SG}} = \frac{q_L}{q_L + q_g}$$

And $F_r$ is the Froude number:

$$F_r = \frac{v_m^2}{g d}$$

Liquid Holdup Correlation

The core of the Beggs-Brill method is the liquid holdup correlation, which accounts for phase slippage.

Horizontal Flow Holdup

For horizontal pipes (θ = 0°), the liquid holdup is correlated as:

$$\frac{H_{L(0)}}{{\lambda_L}^a} = c$$

Where the coefficients depend on flow pattern:

The exponent is:

$$a = \frac{b}{F_r^c}$$

Solving for horizontal holdup:

$$H_{L(0)} = c \lambda_L^a$$

Inclined Flow Correction

For inclined pipes (θ ≠ 0°), the holdup is corrected:

$$H_L = H_{L(0)} \psi$$

Where the inclination correction factor $\psi$ depends on flow pattern and inclination:

$$\psi = 1 + C \left[\sin(1.8\theta) - 0.333 \sin^3(1.8\theta)\right]$$

The coefficient C is:

For uphill flow (θ > 0°):

$$C = (1 - \lambda_L) \ln\left[d \lambda_L^e f^g N_{LV}^h\right]$$

For downhill flow (θ < 0°):

$$C = (1 - \lambda_L) \ln\left[d \lambda_L^e f^g N_{LV}^h\right]$$

With pattern-dependent coefficients:

The liquid velocity number is:

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

Two-Phase Friction Factor

The friction factor accounts for energy losses due to wall friction and interfacial shear.

No-Slip Friction Factor

First, calculate the single-phase (no-slip) friction factor using mixture properties:

Reynolds number:

$$N_{Re} = \frac{1488 \rho_m v_m d}{\mu_m}$$

Where the no-slip mixture density and viscosity are:

$$\rho_m = \rho_L \lambda_L + \rho_g (1 - \lambda_L)$$$$\mu_m = \mu_L \lambda_L + \mu_g (1 - \lambda_L)$$

Friction factor:

• Laminar flow (Re < 2000): $f_n = \frac{16}{N_{Re}}$
• Turbulent flow (Re ≥ 2000): $f_n = \frac{0.0056}{1 + (Re)^{0.32}}$ (smooth pipe approximation)

Two-Phase Friction Factor

The two-phase friction factor is related to the no-slip factor by:

$$\frac{f_{tp}}{f_n} = e^S$$

Where:

$$S = \frac{\ln(y)}{\left(-0.0523 + 3.182 \ln(y) - 0.8725 [\ln(y)]^2 + 0.01853 [\ln(y)]^4\right)}$$

And:

$$y = \frac{\lambda_L}{H_L^2}$$

Physical interpretation: When $H_L < \lambda_L$ (slippage occurs), the friction factor increases because the liquid velocity is higher than the no-slip assumption.

Calculation Procedure

To calculate pressure gradient at a given location:

1. Calculate flow properties:

   • Mixture velocity: $v_m = v_{SL} + v_{SG}$
   • No-slip liquid holdup: $\lambda_L = v_{SL} / v_m$
   • Froude number: $F_r = v_m^2 / (gd)$

2. Identify flow pattern:

   • Calculate $L_1$, $L_2$, $L_3$ from $\lambda_L$
   • Compare $F_r$ with boundaries

3. Calculate horizontal holdup:

   • Use flow pattern coefficients (a, b, c)
   • Compute $H_{L(0)}$

4. Apply inclination correction:

   • Calculate $C$ from flow pattern and angle
   • Compute $\psi$ and $H_L$

5. Calculate friction factor:

   • Compute $N_{Re}$ and $f_n$
   • Calculate $y = \lambda_L / H_L^2$ and $S$
   • Compute $f_{tp} = f_n e^S$

6. Evaluate pressure gradient:

   • Elevation gradient from $H_L$ and $\theta$
   • Friction gradient from $f_{tp}$
   • Acceleration gradient (often negligible)

Applicability and Limitations

Validated Range

The Beggs-Brill correlation was developed from:

• Pipe diameters: 1.0 in., 1.5 in.
• Inclination angles: -90° to +90° (all angles)
• Fluids: Air and water
• Pressures: 35 to 95 psia
• Liquid flow rates: 0 to 30 gal/min
• Gas flow rates: 0 to 300 Mscf/D
• Total data points: 584 tests

Extrapolation to Field Conditions

The correlation has been successfully applied to:

• Larger pipe diameters: 2 in. to 12 in. (with some accuracy loss)
• Oil and gas: Instead of air and water
• Higher pressures: Typical reservoir conditions

Known Limitations

1. Flow pattern transitions — Accuracy decreases near transition boundaries
2. Small liquid holdups — May overpredict holdup at very high gas rates
3. Downhill flow — Assumes less than 100% pressure recovery (conservative)
4. Pipe roughness — Uses smooth pipe friction factor (may underestimate in rough pipe)
5. High viscosity liquids — Developed for water (μ ≈ 1 cP), less accurate for heavy oils

When to Use Beggs-Brill

Advantages:

• Only multiphase correlation valid for all inclination angles
• Well-established and widely accepted in industry
• Incorporated in most commercial software packages
• Conservative (tends to overpredict pressure drop slightly)

Best applications:

• Directional wells with varying angles
• Gathering systems in hilly terrain
• Offshore pipelines with elevation changes
• When inclination varies along pipe length

Alternatives to consider:

• Hagedorn-Brown — More accurate for vertical wells (θ ≈ 90°)
• Gray — Better for vertical gas wells with high GLR
• OLGA/PIPESIM — Mechanistic models for critical design

Related Documentation

• Pipeflow Overview — Correlation selection guide
• Vertical Correlations — Hagedorn-Brown, Gray methods
• Single-Phase Flow — Reduces to this at 100% liquid or gas
• PVT Properties — Required fluid property correlations

References

1. Beggs, H.D. and Brill, J.P. (1973). "A Study of Two-Phase Flow in Inclined Pipes." Journal of Petroleum Technology, 25(5), pp. 607-617. SPE-4007-PA. DOI: 10.2118/4007-PA.

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

3. 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 3: Multiphase Flow in Pipes.

4. 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 Correlations.

5. Standing, M.B. (1981). "A Set of Equations for Computing Equilibrium Ratios of a Crude Oil/Natural Gas System at Pressures Below 1,000 psia." Journal of Petroleum Technology, 33(9), pp. 1193-1195.