Two-Phase Gas-Liquid Flow Regimes & Lockhart-Martinelli Pressure Drop

Executive Summary & Engineering Relevance

Simultaneous gas-liquid two-phase flow is ubiquitous across the process and energy industries: offshore oil & gas production flowlines, reboiler return lines, wet steam distribution systems, refrigeration evaporators, condensate return lines, and chemical reactor effluent piping.

Unlike single-phase flow, where fluid velocity is uniform across the cross section, two-phase systems exhibit phase slip—the less dense gas phase generally travels at a substantially higher velocity than the liquid phase ($v_g > v_l$). This velocity difference generates complex phase distributions termed Flow Regimes (stratified, wavy, slug, annular, dispersed bubble, and mist flow).

Among these regimes, Slug Flow (Intermittent Flow) poses severe mechanical and operational hazards. Slugs of dense liquid traveling at gas velocities generate high momentum impact forces on pipe elbows, creating violent cyclic piping vibration, structural fatigue failure, and surging into downstream three-phase separators and slug catchers.

Predicting the hydrodynamic flow regime, in-situ liquid volume fraction (Liquid Holdup, $H_L$), and frictional pressure gradient requires empirical multiphase correlations. The Lockhart-Martinelli (1949) correlation, augmented by Chisholm (1967), remains the most robust and universally applied method in chemical process engineering.

1. Two-Phase Flow Regimes in Horizontal & Vertical Pipes

The spatial distribution of the gas and liquid phases depends on the superficial velocities ($v_{sg}$ and $v_{sl}$), fluid densities, viscosities, surface tension, pipe diameter, and line inclination.

Horizontal Two-Phase Flow Regimes (Increasing Gas Velocity ->):
1. Stratified Smooth:        2. Stratified Wavy:
   +-------------------+        +-------------------+
   |   Gas (vsg)       |        |   Gas Phase       |
   |~~~~~~~~~~~~~~~~~~~|        | ~ ~ ~ ~ ~ ~ ~ ~ ~ | (Interfacial Waves)
   |   Liquid Film     |        |   Liquid (vsl)    |
   +-------------------+        +-------------------+

3. Slug / Intermittent Flow:  4. Annular / Annular-Mist:
   +-------------------+        +-------------------+
   |  Gas  | LIQUID | G |        |==== Liquid Film ===|
   | Pocket|  SLUG  | a |        | ---> High-Speed <---| (Gas Core + Mist)
   |       | (Dense)| s |        |==== Liquid Film ===|
   +-------------------+        +-------------------+

1.1 Summary of Horizontal Flow Regimes

2. Superficial Velocities & Liquid Holdup ($H_L$)

2.1 Superficial Velocities ($v_{sl}, v_{sg}$)

In two-phase engineering, velocities are defined on a superficial basis—the velocity each phase would have if it flowed alone across the entire pipe cross-section ($A = \frac{\pi}{4} D^2$):

$$v_{sl} = \frac{Q_l}{A} = \frac{W_l}{\ho_l \cdot A}$$

$$v_{sg} = \frac{Q_g}{A} = \frac{W_g}{\ho_g \cdot A}$$

The total mixture velocity is simply: $$v_m = v_{sl} + v_{sg}$$

2.2 In-Situ Liquid Holdup ($H_L$) vs. No-Slip Holdup ($\lambda_l$)

The No-Slip Holdup ($\lambda_l$) is the input volume fraction:

$$\lambda_l = \frac{Q_l}{Q_l + Q_g} = \frac{v_{sl}}{v_{sl} + v_{sg}}$$

Because gas moves faster than liquid ($v_g > v_l$), liquid accumulates in the pipe. The actual in-situ liquid volume fraction ($H_L$) is always larger than the input fraction ($\mathbf{H_L > \lambda_l}$):

$$H_L = \frac{A_l}{A} \iff \text{Void Fraction } \alpha = 1 - H_L = \frac{A_g}{A}$$

The phase slip ratio ($S$) is defined as: $$S = \frac{v_g}{v_l} = \left(\frac{v_{sg}}{1 - H_L} \right) \cdot \left(\frac{H_L}{v_{sl}} \right) = \left(\frac{\lambda_g}{\lambda_l} \right) \cdot \left(\frac{H_L}{1 - H_L} \right)$$

3. The Lockhart-Martinelli Method & Chisholm Formulation

In 1949, R. W. Lockhart and R. C. Martinelli correlated two-phase frictional pressure drops by relating the two-phase gradient $(\Delta P / L)_{TP}$ to the single-phase pressure gradients calculated if each phase flowed alone in the pipe.

Step 1: Calculate Single-Phase Frictional Pressure Drops

Using the superficial velocities, compute superficial Reynolds numbers:

$$Re_l = \frac{\ho_l \cdot v_{sl} \cdot D}{\mu_l}, \quad Re_g = \frac{\ho_g \cdot v_{sg} \cdot D}{\mu_g}$$

Calculate single-phase friction factors ($f_l, f_g$) and pressure drops:

$$\left(\frac{\Delta P}{L} \right)_l = \frac{2 f_l \ho_l v_{sl}^2}{D}$$

$$\left(\frac{\Delta P}{L} \right)_g = \frac{2 f_g \ho_g v_{sg}^2}{D}$$

Step 2: Calculate the Lockhart-Martinelli Parameter ($X$)

The Martinelli parameter $X$ represents the square root of the ratio of liquid-to-gas single-phase frictional pressure gradients:

$$X = \sqrt{\frac{(\Delta P / L)_l}{(\Delta P / L)_g}}$$

For turbulent liquid and turbulent gas ($tt$ flow), which occurs in over 90% of industrial process pipelines:

$$X_{tt} = \left(\frac{W_l}{W_g} \right)^{0.9} \left(\frac{\ho_g}{\ho_l} \right)^{0.5} \left(\frac{\mu_l}{\mu_g} \right)^{0.1}$$

Step 3: Two-Phase Multiplier ($\Phi_l^2$) via Chisholm's Equation

Duncan Chisholm (1967) established the analytical relationship for the two-phase friction multiplier $\Phi_l^2$:

$$\left(\frac{\Delta P}{L} \right)_{TP} = \Phi_l^2 \cdot \left(\frac{\Delta P}{L} \right)_l$$

$$\Phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}$$

Alternatively, relative to the gas phase: $$\left(\frac{\Delta P}{L} \right)_{TP} = \Phi_g^2 \cdot \left(\frac{\Delta P}{L} \right)_g, \quad \Phi_g^2 = 1 + C \cdot X + X^2$$

Chisholm Factor ($C$) Matrix

Step 4: Liquid Holdup ($H_L$) Evaluation

Lockhart and Martinelli provided empirical data for liquid holdup, accurately represented by Butterworth's generalized model:

$$H_L = \frac{1}{\sqrt{1 + \frac{C}{X} + \frac{1}{X^2}}} = \frac{1}{\Phi_l}$$

4. Flow Regime Transitions: Mandhane Map Formulations

Mandhane, Gregory, and Aziz (1974) compiled over 5,900 flow regime observations into the industry-standard horizontal regime boundary map, plotted in terms of superficial velocities $(v_{sg}, v_{sl})$ with property correction factors:

Mandhane Horizontal Regime Boundaries:
vsl [m/s]
  ^
10|           DISPERSED BUBBLE
  |-------------------------------------
 1|         |        SLUG FLOW          |
  |         |      (Intermittent)       |
.1| STRAT.  |---------------------------|  ANNULAR-MIST
  |  SMOOTH |     STRATIFIED WAVY       |
  +---------+---------------------------+------------> vsg [m/s]
  0.1       1.0                         10           100

Key Transition Equations:

• Stratified to Wavy Boundary: $$v_{sg} \ge 1.52\ m/s$$
• Wavy to Slug Boundary (Taitel-Dukler Kelvin-Helmholtz instability): $$v_{sg} \ge \left(1 - \frac{h_l}{D} \right) \sqrt{\frac{(\ho_l - \ho_g) g \cos \theta A_g}{\ho_g \cdot \frac{dA_l}{dh_l}}}$$
• Transition to Annular Flow: $$v_{sg} \ge 3.1 \left[ \frac{\sigma g (\ho_l - \ho_g)}{\ho_g^2} \right]^{0.25}$$

5. Comprehensive Step-by-Step Industrial Worked Example

Design Problem Statement:

A crude oil and natural gas production line from an offshore wellhead platform to a central separator facility: * Pipe Inside Diameter: DN200 (8-inch Sch 40), $D = 202.7\ mm$ ($0.2027\ m$) * Line Length: $L = 1,200\ m$ * Liquid Flow Rate (Light Crude): $W_l = 45,000\ kg/h$ ($12.5\ kg/s$) * Gas Flow Rate (Associated Gas): $W_g = 3,200\ kg/h$ ($0.8889\ kg/s$) * Liquid Properties: $\ho_l = 820\ kg/m^3$, $\mu_l = 2.50\ cP$ ($0.0025\ Pa\cdot s$) * Gas Properties: $\ho_g = 18.5\ kg/m^3$, $\mu_g = 0.013\ cP$ ($1.3 \times 10^{-5}\ Pa\cdot s$)

Calculate: 1. Superficial velocities ($v_{sl}, v_{sg}$) and total mixture velocity ($v_m$). 2. Single-phase pressure gradients for liquid and gas alone. 3. Lockhart-Martinelli parameter ($X$) and Chisholm multiplier ($\Phi_l^2$). 4. Liquid holdup ($H_L$) and in-situ liquid volume in the pipeline. 5. Total two-phase pressure drop ($\Delta P_{TP}$) across the 1,200 m pipeline. 6. Predicted flow regime (Mandhane).

Step 1: Superficial Velocities

Cross-sectional area: $$A = \frac{\pi}{4} (0.2027)^2 = 0.03227\ m^2$$

Superficial liquid velocity: $$v_{sl} = \frac{W_l}{\ho_l \cdot A} = \frac{12.5}{820 \times 0.03227} = \frac{12.5}{26.46} = \mathbf{0.472\ m/s}$$

Superficial gas velocity: $$v_{sg} = \frac{W_g}{\ho_g \cdot A} = \frac{0.8889}{18.5 \times 0.03227} = \frac{0.8889}{0.597} = \mathbf{1.489\ m/s}$$

Mixture velocity: $$v_m = 0.472 + 1.489 = \mathbf{1.961\ m/s}$$

No-slip liquid fraction: $$\lambda_l = \frac{0.472}{1.961} = \mathbf{0.2407} \quad (24.1\%)$$

Step 2: Single-Phase Pressure Drops

Liquid superficial Reynolds number: $$Re_l = \frac{\ho_l \cdot v_{sl} \cdot D}{\mu_l} = \frac{820 \times 0.472 \times 0.2027}{0.0025} = \frac{78.45}{0.0025} = \mathbf{31,380} \quad \text{(Turbulent)}$$

Gas superficial Reynolds number: $$Re_g = \frac{\ho_g \cdot v_{sg} \cdot D}{\mu_g} = \frac{18.5 \times 1.489 \times 0.2027}{0.000013} = \frac{5.584}{0.000013} = \mathbf{429,540} \quad \text{(Turbulent)}$$

Both phases are turbulent $\implies$ $tt$ Flow Regime $\implies C = 20$.

Fanning friction factors (Blasius $f = 0.079 Re^{-0.25}$): $$f_l = 0.079 \times (31,380)^{-0.25} = 0.079 \times 0.07519 = 0.00594$$ $$f_g = 0.079 \times (429,540)^{-0.25} = 0.079 \times 0.03906 = 0.00309$$

Single-phase pressure gradients: $$\left(\frac{\Delta P}{L}\right)_l = \frac{2 \times 0.00594 \times 820 \times (0.472)^2}{0.2027} = \frac{2.171}{0.2027} = \mathbf{10.71\ Pa/m}$$

$$\left(\frac{\Delta P}{L}\right)_g = \frac{2 \times 0.00309 \times 18.5 \times (1.489)^2}{0.2027} = \frac{0.2533}{0.2027} = \mathbf{1.25\ Pa/m}$$

Step 3: Lockhart-Martinelli Parameter ($X$) & Multiplier ($\Phi_l^2$)

$$X = \sqrt{\frac{10.71}{1.25}} = \sqrt{8.568} = \mathbf{2.927}$$

Chisholm two-phase liquid multiplier ($C = 20$): $$\Phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2} = 1 + \frac{20}{2.927} + \frac{1}{(2.927)^2}$$ $$\Phi_l^2 = 1 + 6.833 + 0.1168 = \mathbf{7.950}$$

Step 4: Two-Phase Pressure Drop Across 1,200 m Line

Two-phase pressure gradient: $$\left(\frac{\Delta P}{L}\right)_{TP} = \Phi_l^2 \cdot \left(\frac{\Delta P}{L}\right)_l = 7.950 \times 10.71\ Pa/m = \mathbf{85.14\ Pa/m}$$

Total line pressure loss: $$\Delta P_{TP} = 85.14\ Pa/m \times 1200\ m = 102,174\ Pa = \mathbf{1.022\ \text{bar}} \quad (14.82\ psi)$$

Step 5: Liquid Holdup & Total Liquid Inventory

Liquid holdup: $$H_L = \frac{1}{\Phi_l} = \frac{1}{\sqrt{7.950}} = \frac{1}{2.8196} = \mathbf{0.3547} \quad (35.5\%)$$

Note that $H_L = 35.5\%$ is significantly higher than the input fraction $\lambda_l = 24.1\%$ due to gas slip!

Total liquid inventory stored inside the pipeline during steady operation: $$V_{ \text{liquid}} = A \cdot L \cdot H_L = 0.03227\ m^2 \times 1200\ m \times 0.3547 = \mathbf{13.73\ m^3} \quad (11,260\ kg)$$

Separator Design Consequence: If the pipeline is pigged or gas flow suddenly surges, up to $13.7\ m^3$ of liquid can be swept forward into the production separator. The downstream slug catcher must be sized for at least $15 - 20\ m^3$ of surge volume!

Step 6: Flow Regime Identification

For $v_{sg} = 1.489\ m/s$ and $v_{sl} = 0.472\ m/s$: * $v_{sg} \approx 1.5\ m/s$ lies right at the boundary between Stratified Wavy and Slug / Intermittent Flow. * In practice, any minor undulations in seabed or ground topography will cause waves to bridge the pipe bore, resulting in terrain-induced slugging. * Pipe supports and slug catcher sizing must be rated for intermittent slug impact forces.

6. Best Practice Engineering Rules of Thumb

1. Avoid Sizing Lines in the Slug Flow Regime: If a design falls into the intermittent/slug envelope, evaluate whether increasing the pipe diameter (to drop into stratified flow) or decreasing it (to sweep into annular flow) is feasible.
2. Account for Topographical Slugging: Even if a pipeline is predicted to be in stratified flow along a horizontal grade, any uphill slope ($> 0.5^\circ$) dramatically increases liquid holdup, triggering severe hydrodynamic slugging at low points.
3. Verify API RP 14E Erosional Velocity Limits: Ensure total mixture velocity does not exceed the erosional limit: $$v_e = \frac{C_{ \text{API}}}{\sqrt{\ho_m}}$$ Where $C_{ \text{API}} = 100 - 125$ for continuous service.

References & Academic Literature

1. Lockhart, R. W., & Martinelli, R. C. (1949). Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engineering Progress, 45(1), 39–48.
2. Chisholm, D. (1967). A theoretical basis for the Lockhart-Martinelli correlation for two-phase flow. International Journal of Heat and Mass Transfer, 10(12), 1767–1778.
3. Mandhane, J. M., Gregory, G. A., & Aziz, K. (1974). A flow pattern map for gas—liquid flow in horizontal pipes. International Journal of Multiphase Flow, 1(4), 537–553.
4. Taitel, Y., & Dukler, A. E. (1976). A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AIChE Journal, 22(1), 47–55.
5. American Petroleum Institute. (2007). API RP 14E: Recommended Practice for Design and Installation of Offshore Production Platform Piping Systems. API Publishing Services.