Pipeline Flow Equations for Gas, Liquid, and Two-Phase Systems

Overview

Pipeline calculations determine fluid transport capacity and pressure losses in surface lines. These calculations are critical for sizing gathering lines, designing gas transmission pipelines, and evaluating system capacity under changing production conditions.

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Single-Phase Gas Pipelines

General Energy Equation

All gas pipeline equations derive from the steady-state energy balance for compressible flow:

$$Q_{sc} = C \cdot \left[\frac{(P_1^2 - P_2^2) \cdot D^5}{f \cdot \gamma_g \cdot \bar{T} \cdot \bar{Z} \cdot L}\right]^{1/2}$$

Where $Q_{sc}$ = gas rate (scf/d), $P_1$/$P_2$ = inlet/outlet pressure (psia), $D$ = pipe diameter (in.), $f$ = Moody friction factor, $\gamma_g$ = gas specific gravity, $\bar{T}$ = average temperature (R), $\bar{Z}$ = average compressibility factor, and $L$ = length (miles).

The three classical equations differ in how they treat the friction factor.

Weymouth Equation (1912)

Assumes friction factor depends only on diameter: $f = 0.032/D^{1/3}$

$$Q_{sc} = 433.5 \cdot \left[\frac{(P_1^2 - P_2^2) \cdot D^{16/3}}{\gamma_g \cdot \bar{T} \cdot \bar{Z} \cdot L}\right]^{1/2}$$

Conservative for large-diameter pipes. Best for short, small-diameter lines (< 12 in.).

Panhandle A Equation (1956)

Incorporates Reynolds number effects implicitly for partially turbulent flow:

$$Q_{sc} = 435.87 \cdot E \cdot \left[\frac{(P_1^2 - P_2^2)}{\gamma_g^{0.8539} \cdot \bar{T} \cdot \bar{Z} \cdot L}\right]^{0.5394} \cdot D^{2.6182}$$

Where $E$ is the pipeline efficiency (0.90--0.95 new, 0.80--0.90 older pipe). Best for medium-diameter lines (4--12 in.).

Panhandle B Equation (1956)

Optimized for fully turbulent flow in large-diameter pipelines:

$$Q_{sc} = 737.02 \cdot E \cdot \left[\frac{(P_1^2 - P_2^2)}{\gamma_g^{0.9608} \cdot \bar{T} \cdot \bar{Z} \cdot L}\right]^{0.5100} \cdot D^{2.5300}$$

Best for large-diameter transmission lines (16--48 in.) over long distances.

Gas Equation Comparison

Average Pressure

The correct average pressure along a gas pipeline is:

$$\bar{P} = \frac{2}{3}\left(P_1 + P_2 - \frac{P_1 \cdot P_2}{P_1 + P_2}\right)$$

The average Z-factor should be evaluated at $\bar{P}$ and $\bar{T}$.

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Single-Phase Liquid Pipelines

Darcy-Weisbach Equation

For incompressible liquid flow:

$$\Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho_L v^2}{2 g_c}$$

In oilfield units:

$$\Delta P = \frac{f \cdot L \cdot \rho_L \cdot v^2}{25.8 \cdot D}$$

Where $\Delta P$ = pressure drop (psi), $f$ = Moody friction factor, $L$ = length (ft), $D$ = diameter (in.), $\rho_L$ = liquid density (lb/ft^3), $v$ = velocity (ft/s).

Friction Factor

Reynolds number: $Re = \rho_L v D / \mu_L$

Laminar ($Re < 2100$): $f = 64/Re$

Turbulent ($Re > 4000$) --- Colebrook-White (implicit):

$$\frac{1}{\sqrt{f}} = -2 \log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$$

Swamee-Jain approximation (explicit, valid $5000 < Re < 10^8$):

$$f = \frac{0.25}{\left[\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{5.74}{Re^{0.9}}\right)\right]^2}$$

Typical Pipe Roughness

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Two-Phase Pipeline Flow

When gas and liquid flow together, pressure drop depends on liquid holdup, flow regime, and phase slippage. Empirical correlations are required.

Eaton Correlation (1967)

Empirical correlations for liquid holdup and friction factor in horizontal two-phase flow:

$$H_L = f\left(\frac{v_{SL}}{v_m}, \frac{v_m^2}{gD}, \frac{\rho_L}{\rho_g}, \frac{\mu_L}{\mu_g}\right)$$

The two-phase pressure gradient uses mixture density based on holdup:

$$\frac{dP}{dL} = \frac{f_{tp} \cdot \rho_m \cdot v_m^2}{2 g_c \cdot D} + \rho_m \cdot g \cdot \sin\theta$$

Where $\rho_m = \rho_L H_L + \rho_g(1 - H_L)$.

Dukler Correlation (1964)

A similarity analysis approach using no-slip mixture properties:

$$\rho_{ns} = \rho_L \lambda_L + \rho_g (1 - \lambda_L)$$

Where $\lambda_L = v_{SL}/(v_{SL} + v_{SG})$ is the input liquid fraction. Dukler correlates the ratio $f_{tp}/f_0$ as a function of $\lambda_L$.

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Flow Regime Identification

Baker Flow Regime Map (1954)

Uses dimensionless parameters based on mass velocities $G_L$ and $G_G$:

$$B_y = \frac{G_G}{\lambda}, \qquad B_x = \frac{G_L}{G_G} \cdot \lambda \cdot \psi$$

Where:

$$\lambda = \left[\frac{\rho_g}{\rho_{air}} \cdot \frac{\rho_L}{\rho_w}\right]^{0.5}, \qquad \psi = \frac{\sigma_w}{\sigma_L} \cdot \left[\frac{\mu_L}{\mu_w} \cdot \left(\frac{\rho_w}{\rho_L}\right)^2\right]^{1/3}$$

  B_y (Gas mass velocity / lambda)
   |
   |  DISPERSED (mist/spray)
   | ----------------------------
   |  ANNULAR
   | ------------------------
   |  SLUG
   | ------------------
   |  PLUG
   | ------------
   |  STRATIFIED (smooth and wavy)
   |
   +-------------------------------- B_x (Liquid/Gas ratio)

Mandhane Flow Regime Map (1974)

A simpler map using superficial velocities directly:

The Mandhane map is preferred for its simplicity and direct use of measurable quantities.

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Applicability and Limitations

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Related Topics

• Surface Facilities Overview --- Model selection guide
• Choke Models --- Upstream flow restriction calculations
• Pipe Flow Overview --- Wellbore multiphase flow (Beggs-Brill, Hagedorn-Brown)
• PVT Gas Properties --- Z-factor and gas viscosity for pipeline calculations

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References

1. Weymouth, T.R. (1912). "Problems in Natural Gas Engineering." Transactions of the ASME, 34, pp. 185-234.

2. Katz, D.L. and Lee, R.L. (1990). Natural Gas Engineering: Production and Storage. McGraw-Hill. Chapters 8-10.

3. Eaton, B.A., Andrews, D.E., Knowles, C.R., Silberberg, I.H., and Brown, K.E. (1967). "The Prediction of Flow Patterns, Liquid Holdup and Pressure Losses Occurring During Continuous Two-Phase Flow in Horizontal Pipelines." Journal of Petroleum Technology, 19(6), pp. 815-828. SPE-1525-PA.

4. Dukler, A.E., Wicks, M., and Cleveland, R.G. (1964). "Frictional Pressure Drop in Two-Phase Flow: A Comparison of Existing Correlations." AIChE Journal, 10(1), pp. 38-43.

5. Mandhane, J.M., Gregory, G.A., and Aziz, K. (1974). "A Flow Pattern Map for Gas-Liquid Flow in Horizontal Pipes." International Journal of Multiphase Flow, 1(4), pp. 537-553.