# Hagedorn and Brown correlation

## Contents

## Brief

**Hagedorn and Brown** is an empirical two-phase flow correlation published in **1965** ^{[1]}.

It doesn't distinguish between the flow regimes.

The heart of the **Hagedorn and Brown** method is a correlation for the liquid holdup H_{L} ^{[2]}.

**Hagedorn and Brown** is the default VLP correlation for the **oil wells** in the PQplot.

## Math & Physics

Following the law of conservation of energy the basic steady state flow equation is:

^{[1]}

where

^{[1]}

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

^{[4]}

Reynolds two phase number:

^{[1]}

## Discussion

Why ** Hagedorn and Brown**?

One of the consistently best correlations ...— Michael Economides et al^{[2]}

## Demo

Hagedorn and Brown correlation overview video:

In this video it's shown:

- What the Hagedorn and Brown correlation is
- History and practical application
- Math & Physics
- Flow diagram to get the VLP curve
- Workflow to find HL

## Flow Diagram

## Workflow H_{L}

^{[1]}

^{[5]}

^{[5]}

^{[5]}

^{[5]}

^{[1]}

^{[2]}

^{[5]}

^{[5]}

^{[1]}

^{[1]}

^{[1]}

^{[2]}

^{[6]}

^{[2]}

^{[6]}

^{[1]}

## Modifications

1. Use the no-slip holdup when the original empirical correlation predicts a liquid holdup H_{L} less than the no-slip holdup ^{[2]}.

2. Use the Griffith correlation to define the bubble flow regime^{[2]} and calculate H_{L}.

3. Use watercut instead of WOR to account for the watercut = 100%.

## Nomenclature

- = flow area, ft2
- = correlation group, dimensionless
- = formation factor, bbl/stb
- = coefficient for liquid viscosity number, dimensionless
- = pipe diameter, ft
- = depth, ft
- = correlation group, dimensionless
- = liquid holdup factor, fraction
- = friction factor, dimensionless
- = gas-liquid ratio, scf/bbl
- = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lb
_{m}/bbl - = pipe diameter number, dimensionless
- = gas velocity number, dimensionless
- = liquid viscosity number, dimensionless
- = liquid velocity number, dimensionless
- = pressure, psia
- = conversion constant equal to 32.174049, lb
_{m}ft / lb_{f}sec^{2} - = total liquid production rate, bbl/d
- = Reynolds number, dimensionless
- = solution gas-oil ratio, scf/stb
- = specific gravity, dimensionless
- = temperature, °R or °K, follow the subscript
- = velocity, ft/sec
- = water-oil ratio, bbl/bbl
- = 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)
- = secondary correlation factor, dimensionless

### Subscripts

- g = gas
- K = °K
- L = liquid
- m = gas/liquid mixture
- o = oil
- R = °R
- SL = superficial liquid
- SG = superficial gas
- w = water

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}^{1.7}^{1.8}^{1.9}Hagedorn, A. R.; Brown, K. E. (1965). "Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits".*Journal of Petroleum Technology*. 17(04) (SPE-940-PA): 475–484. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}^{2.5}^{2.6}Economides, M.J.; Hill, A.D.; Economides, C.E.; Zhu, D. (2013).*Petroleum Production Systems*(2 ed.). Westford, Massachusetts: Prentice Hall. ISBN 978-0-13-703158-0. - ↑ 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}Lyons, W.C. (1996).*Standard handbook of petroleum and natural gas engineering*.**2**. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5. - ↑
^{6.0}^{6.1}Trina, S. (2010).*An integrated horizontal and vertical flow simulation with application to wax precipitation*(Master of Engineering Thesis). Canada: Memorial University of Newfoundland.