Single-Phase Pipe Flow

Introduction

Single-phase pipe flow describes the movement of a single fluid (liquid or gas) through pipelines. This is the foundation for:

Surface flowlines — horizontal or near-horizontal transport
Injection tubing — water or gas injection wells
Single-phase risers — gas-only or water-only risers
Process piping — facilities and plant pipelines

The pressure drop in single-phase flow has two components:

Frictional losses — energy dissipated due to fluid viscosity and pipe wall roughness
Elevation changes — potential energy changes due to vertical pipe segments

Fundamental Equations

Total Pressure Drop

$\Delta P_{total} = \Delta P_{friction} + \Delta P_{elevation}$

For producers (flow upward), both terms are positive (pressure decreases from bottom to top).

For injectors (flow downward), friction is positive but elevation is negative (hydrostatic head assists flow).

Reynolds Number

The Reynolds number ( $N_{Re}$ ) is a dimensionless ratio of inertial to viscous forces:

$N_{Re} = \frac{\rho \cdot v \cdot d}{\mu}$

It determines the flow regime:

Reynolds Number	Flow Regime	Characteristics
$N_{Re} < 2100$	Laminar	Smooth, orderly flow
$2100 < N_{Re} < 4000$	Transition	Unstable, intermittent turbulence
$N_{Re} > 4000$	Turbulent	Chaotic, fully mixed flow

Liquid Flow (Incompressible)

For liquid (incompressible) flow:

$N_{Re} = \frac{1.48 \cdot Q_L \cdot \rho_L}{d \cdot \mu_L}$

Parameter	Symbol	Units
Liquid flow rate	$Q_L$	bbl/d
Liquid density	$\rho_L$	lb/ft³
Pipe inner diameter	$d$	in
Liquid viscosity	$\mu_L$	cP

Excel Function: ReynoldsNumberLiquid

=ReynoldsNumberLiquid(Ql, Rho_l, pipeID, Ul)

Gas Flow (Compressible)

For gas (compressible) flow:

$N_{Re} = \frac{20.09 \cdot Q_g \cdot \gamma_g}{d \cdot \mu_g}$

Parameter	Symbol	Units
Gas flow rate	$Q_g$	mscf/d
Gas specific gravity	$\gamma_g$	dimensionless (air = 1.0)
Pipe inner diameter	$d$	in
Gas viscosity	$\mu_g$	cP

Excel Function: ReynoldsNumberGas

=ReynoldsNumberGas(Qg, SGgas, pipeID, Ug)

Friction Factor

The Fanning friction factor ( $f$ ) relates wall shear stress to the fluid's kinetic energy.

Laminar Flow ($N_

For laminar flow, the friction factor is independent of pipe roughness:

$f = \frac{16}{N_{Re}}$

Turbulent Flow ($N_

For turbulent flow, the Chen equation (1979) provides an explicit approximation to the implicit Colebrook-White equation:

$\frac{1}{\sqrt{f}} = -4 \log_{10}\left[\frac{\varepsilon/d}{3.7065} - \frac{5.0452}{N_{Re}} \log_{10}\left(\frac{(\varepsilon/d)^{1.1098}}{2.8257} + \left(\frac{7.149}{N_{Re}}\right)^{0.8981}\right)\right]$

Where:

$\varepsilon/d$ = relative pipe roughness (dimensionless)

Pipe Roughness Values

Pipe Material	Absolute Roughness $\varepsilon$ (ft)	Typical $\varepsilon/d$
Drawn tubing	0.000005	0.00001 - 0.0001
Commercial steel	0.00015	0.0001 - 0.001
Galvanized iron	0.0005	0.0005 - 0.002
Cast iron	0.00085	0.001 - 0.005
Concrete	0.001 - 0.01	0.002 - 0.02
Riveted steel	0.003 - 0.03	0.005 - 0.05

Note: Relative roughness $\varepsilon/d$ = (absolute roughness) / (pipe inner diameter)

Frictional Pressure Drop

Fanning Equation

The Fanning equation calculates frictional pressure drop for incompressible flow:

$\Delta P_f = \frac{2 \cdot f \cdot \rho \cdot v^2 \cdot L}{g_c \cdot d}$

Where:

$f$ = Fanning friction factor
$\rho$ = fluid density, lb/ft³
$v$ = flow velocity, ft/s
$L$ = pipe length, ft
$d$ = pipe inner diameter, ft
$g_c$ = gravitational constant = 32.174 (lbm·ft)/(lbf·s²)

The result is in psi when proper unit conversions are applied.

Excel Function: FrictionPressureDropLiquid

=FrictionPressureDropLiquid(Ql, Ul, Rho_l, pipeID, pipeRoughness, pipeLength)

Parameter	Description	Units
`Ql`	Liquid rate	bbl/d
`Ul`	Liquid viscosity	cP
`Rho_l`	Liquid density	lb/ft³
`pipeID`	Pipe inner diameter	in
`pipeRoughness`	Relative roughness	dimensionless
`pipeLength`	Pipe length	ft

Elevation Pressure Drop

Potential Energy Change

For inclined or vertical pipes, the elevation pressure drop accounts for gravitational potential energy:

$\Delta P_{elev} = \frac{\rho \cdot L \cdot \sin(\theta)}{144}$

Parameter	Symbol	Units
Fluid density	$\rho$	lb/ft³
Pipe length	$L$	ft
Pipe angle from horizontal	$\theta$	degrees

Sign Convention:

$\theta = 0°$ — Horizontal flow (no elevation change)
$\theta = +90°$ — Vertical upward (producer)
$\theta = -90°$ — Vertical downward (injector)

Excel Function: PotentialEnergyPressureDropLiquid

=PotentialEnergyPressureDropLiquid(Rho_l, pipeLength, pipeAngle)

Complete Pressure Calculations

Liquid Flow

Inlet Pressure from Outlet:

$P_{in} = P_{out} + \Delta P_{friction} + \Delta P_{elevation}$

Excel Function: InletPipePressureLiquid

=InletPipePressureLiquid(P_out, Ql, Ul, Rho_l, pipeID, pipeRoughness, pipeLength, pipeAngle)

Outlet Pressure from Inlet:

$P_{out} = P_{in} - \Delta P_{friction} - \Delta P_{elevation}$

Excel Function: OutletPipePressureLiquid

=OutletPipePressureLiquid(P_in, Ql, Ul, Rho_l, pipeID, pipeRoughness, pipeLength, pipeAngle)

Gas Flow

Gas flow requires special treatment because gas is compressible — density varies with pressure along the pipe.

Horizontal Gas Flow ( $\theta = 0$ ):

$P_1^2 = P_2^2 + \frac{1.007 \times 10^{-4} \cdot \gamma_g \cdot f \cdot \bar{Z} \cdot \bar{T} \cdot Q_g^2 \cdot L}{d^5}$

Inclined Gas Flow ( $\theta \ne 0$ ):

$P_1^2 = e^{-s} \cdot P_2^2 - 2.685 \times 10^{-3} \cdot (1 - e^{-s}) \cdot \frac{f \cdot (\bar{Z} \cdot \bar{T} \cdot Q_g)^2}{\sin\theta \cdot d^5}$

Where the elevation parameter:

$s = -\frac{0.0375 \cdot \gamma_g \cdot \sin\theta \cdot L}{\bar{Z} \cdot \bar{T}}$

Parameter	Symbol	Units
Inlet pressure	$P_1$	psia
Outlet pressure	$P_2$	psia
Gas specific gravity	$\gamma_g$	dimensionless
Fanning friction factor	$f$	dimensionless
Mean Z-factor	$\bar{Z}$	dimensionless
Mean temperature	$\bar{T}$	°R
Gas flow rate	$Q_g$	mscf/d
Pipe length	$L$	ft
Pipe diameter	$d$	in

Excel Function (Inlet from Outlet): InletPipePressureGas

=InletPipePressureGas(Qg, P_out, pipeLength, pipeID, pipeAngle, pipeRoughness, zFactor, T, SGgas, Ug)

Excel Function (Outlet from Inlet): OutletPipePressureGas

=OutletPipePressureGas(Qg, P_in, pipeLength, pipeID, pipeAngle, pipeRoughness, zFactor, T, SGgas, Ug)

Input Validation

Parameter	Valid Range	Typical Values
Flow rate (liquid)	≥ 0 bbl/d	100 - 50,000
Flow rate (gas)	≥ 0 mscf/d	100 - 100,000
Pipe ID	> 0 in	2 - 24
Pipe length	≥ 0 ft	100 - 50,000
Pipe angle	-90° to +90°	0° (horizontal)
Roughness	> 0	0.0001 - 0.01
Density	> 0 lb/ft³	40 - 70 (oil), 62 (water)
Viscosity	> 0 cP	0.5 - 100 (liquid), 0.01 - 0.03 (gas)
Z-factor	> 0	0.7 - 1.0
Gas SG	> 0	0.55 - 1.2

Limitations

Single-Phase Assumptions

No phase change — liquid stays liquid, gas stays gas
Newtonian fluid — viscosity independent of shear rate
Steady-state flow — constant flow rate and conditions
Isothermal — temperature constant along pipe (or use average)

When to Use Multiphase Correlations

Oil and gas flow together
Condensate drops out of gas
Water cuts present
Two-phase flow expected

See PipeFlow Overview for multiphase correlation selection.

PipeFlow Overview — Correlation selection guide
PVT Gas Properties — Z-factor, gas viscosity
PVT Overview — Fluid property selection

References

Chen, N.H. (1979). "An Explicit Equation for Friction Factor in Pipe." Industrial & Engineering Chemistry Fundamentals, Vol. 18, No. 3, pp. 296-297.
Economides, M.J., Hill, A.D., Ehlig-Economides, C., and Zhu, D. (2013). Petroleum Production Systems, 2nd Edition. Prentice Hall.
Brill, J.P. and Mukherjee, H. (1999). Multiphase Flow in Wells. SPE Monograph Vol. 17.
Brown, K.E. (1984). The Technology of Artificial Lift Methods, Vol. 1. PennWell Books.
Moody, L.F. (1944). "Friction Factors for Pipe Flow." Transactions of the ASME, Vol. 66, pp. 671-684.