Hagedorn and Brown correlation

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 HL [2].

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

Hagedorn and Brown in PQplot Vs Prosper & Kappa

Math & Physics

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

$144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar \rho_m} + \bar \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}$[1]

where

$\bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)$[1]

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

$\frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)$[4]

Reynolds two phase number:

$Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}$[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

Workflow HL

$M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR$[1]
$\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}$[5]
$\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R}$[5]
$\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}$[5]
$\sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}$[5]
$N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}$[1]
$CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019$[2]
$v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )$[5]
$v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_R}{520}\ \frac{z}{1}$[5]
$N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$[1]
$N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$[1]
$N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}}$[1]
$H = \frac{N_{LV}}{N_{GV}^{0.575}}\ \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D}$[2]
$\frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}}$[6]
$B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}}$[2]
$\psi = \begin{cases} 27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999, &\mbox{if } B \le 0.025 \\ -533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\ 2.5714 B +1.5962, & \mbox{if }B > 0.055 \end{cases}$[6]
$H_L = \frac{H_L}{\psi} \times \psi$[1]

Modifications

1. Use the no-slip holdup when the original empirical correlation predicts a liquid holdup HL less than the no-slip holdup [2].

2. Use the Griffith correlation to define the bubble flow regime[2] and calculate HL.

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

Nomenclature

$A_p$ = flow area, ft2
$B$ = correlation group, dimensionless
$B$ = formation factor, bbl/stb
$C$ = coefficient for liquid viscosity number, dimensionless
$D$ = pipe diameter, ft
$h$ = depth, ft
$H$ = correlation group, dimensionless
$H_L$ = liquid holdup factor, fraction
$f$ = friction factor, dimensionless
$GLR$ = gas-liquid ratio, scf/bbl
$M$ = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lbm/bbl
$N_D$ = pipe diameter number, dimensionless
$N_GV$ = gas velocity number, dimensionless
$N_L$ = liquid viscosity number, dimensionless
$N_LV$ = liquid velocity number, dimensionless
$p$ = pressure, psia
$q_c$ = conversion constant equal to 32.174049, lbmft / lbfsec2
$q$ = total liquid production rate, bbl/d
$Re$ = Reynolds number, dimensionless
$R_s$ = solution gas-oil ratio, scf/stb
$SG$ = specific gravity, dimensionless
$T$ = temperature, °R or °K, follow the subscript
$v$ = velocity, ft/sec
$WOR$ = water-oil ratio, bbl/bbl
$z$ = gas compressibility factor, dimensionless

Greek symbols

$\varepsilon$ = absolute roughness, ft
$\mu$ = viscosity, cp
$\rho$ = density, lbm/ft3
$\bar \rho$ = integrated average density at flowing conditions, lbm/ft3
$\sigma$ = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
$\psi$ = 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. Hagedorn, A. R.; Brown, K. E. (1965). . Journal of Petroleum Technology. 17(04) (SPE-940-PA): 475–484.
2. 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.
3. Colebrook, C. F. (1938–1939). . Journal of the Institution of Civil Engineers. London, England. 11: 133–156.
4. Moody, L. F. (1944). . Transactions of the ASME. 66 (8): 671–684.
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. Trina, S. (2010). An integrated horizontal and vertical flow simulation with application to wax precipitation (Master of Engineering Thesis). Canada: Memorial University of Newfoundland.