# Beggs and Brill correlation

## Contents

## Brief

**Beggs and Brill** is an empirical two-phase flow correlation published in **1972** ^{[1]}.

It distinguish between 4 flow regimes.

**Beggs and Brill** is the default VLP correlation in sPipe.

## Math & Physics

### Fluid flow energy balance

^{[1]}

where

^{[1]}

### Friction factor

No slip Reynolds two phase number:

^{[2]}

Colebrook–White ^{[3]} equation for the Darcy's friction factor:

^{[4]}

Corrected two phase friction factor:

^{[2]}

where

^{[2]}

and

^{[2]}

with constraint:

^{[2]}

## Discussion

Why ** Beggs and Brill**?

The best correlation for the horizontal flow.— pengtools.com

## Flow Diagram

## Workflow H_{L}

^{[5]}

^{[5]}

^{[5]}

^{[5]}

^{[5]}

^{[5]}

^{[5]}

^{[2]}^{[2]}^{[2]}^{[2]}

^{[1]}

**Determine the flow pattern:**

- SEGREGATED:
^{[2]} - TRANSITION:
^{[2]} - INTERMITTENT:
^{[2]} - DISTRIBUTED:
^{[2]}

**Calculate **

- SEGREGATED:
^{[2]} - INTERMITTENT:
^{[2]} - DISTRIBUTED:
^{[2]}

- with the constraint
^{[2]}

^{[2]}

^{[2]}

**C Uphill:**

- SEGREGATED:
^{[2]} - INTERMITTENT:
^{[2]} - DISTRIBUTED:
^{[2]}

**C Downhill:**

- ALL:
^{[2]}

- with the restriction
^{[2]}

**Finally:**

- SEGREGATED, INTERMITTENT, DISTRIBUTED:

^{[2]}

- TRANSITION:

^{[2]}

where:

^{[2]}

## Modifications

1. Force approach gas at low C_{L}. If C_{L}<0.001 Then f'=f.

2. Force approach to single phase fluid. If H_{L}>1 Then H_{L}=1.

3. Use calculated water density instead of the constant value of 62.4 lbm/ft3.

## Nomenclature

- = correlation variable, dimensionless
- = flow area, ft2
- = formation factor, bbl/stb
- = correlation variable, dimensionless
- = non-slip liquid holdup factor, dimensionless
- = pipe diameter, ft
- = total flux weight, lb
_{m}/ft^{2}/sec - = depth, ft
- = liquid holdup factor, dimensionless
- = liquid holdup factor when flow is horizontal, dimensionless
- = friction factor, dimensionless
- = corrected friction factor, dimensionless
- = gas-liquid ratio, scf/bbl
- = correlation variables, dimensionless
- = Froude number, dimensionless
- = liquid velocity number, dimensionless
- = pressure, psia
- = conversion constant equal to 32.174049, lb
_{m}ft / lb_{f}sec^{2} - = flow rate, bbl/d - liquid, scf/d - gas
- = Reynolds number, dimensionless
- = solution gas-oil ratio, scf/stb
- = correlation variable, dimensionless
- = specific gravity, dimensionless
- = temperature, °R or °K, follow the subscript
- = velocity, ft/sec
- = watercut, fraction
- = correlation variable, dimensionless
- = gas compressibility factor, dimensionless

### Greek symbols

- = absolute roughness, ft
- = viscosity, cp
- = density, lb
_{m}/ft^{3} - = integrated average density at flowing conditions, lb
_{m}/ft^{3} - = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
- = inclination correction factor, dimensionless
- = inclination angle, ° from horizontal

### Subscripts

g = gas

K = °K

L = liquid

m = gas/liquid mixture

ns = non-slip

o = oil

R = °R

SL = superficial liquid

SG = superficial gas

w = water

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Beggs, H. D.; Brill, J. P. (May 1973). "A Study of Two-Phase Flow in Inclined Pipes".*Journal of Petroleum Technology*. AIME.**255**(SPE-4007-PA). - ↑
^{2.00}^{2.01}^{2.02}^{2.03}^{2.04}^{2.05}^{2.06}^{2.07}^{2.08}^{2.09}^{2.10}^{2.11}^{2.12}^{2.13}^{2.14}^{2.15}^{2.16}^{2.17}^{2.18}^{2.19}^{2.20}^{2.21}^{2.22}^{2.23}^{2.24}^{2.25}^{2.26}Brill, J. P.; Beggs, H. D. (1991).*Two-Phase Flow In Pipes*(6 ed.). Oklahoma: U. of Tulsa Tulsa. - ↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws".
*Journal of the Institution of Civil Engineers*. London, England.**11**: 133–156. - ↑ Moody, L. F. (1944). "Friction factors for pipe flow".
*Transactions of the ASME*.**66**(8): 671–684. - ↑
^{5.0}^{5.1}^{5.2}^{5.3}^{5.4}^{5.5}^{5.6}Lyons, W.C. (1996).*Standard handbook of petroleum and natural gas engineering*.**2**. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.