Hagedorn and Brown correlation

Brief

Hagedorn and Brown is an empirical two-phase flow correlation published in 1965 .

It doesn't distinguish between the flow regimes.

The heart of the Hagedorn and Brown method is a correlation for the liquid holdup $H_L$.

Math & Physics

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

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

where

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

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

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

Reynolds two phase number:

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

Discussion

Why Hagedorn and Brown?

One of the consistently best correlations ...

— Michael Economides et al

... still one of the most highly regarded models, and its able to predict pressure, flowrate and liquid holdup easily

— Sameena Trina

Flow Diagram

Workflow

To find $H_L$ calculate:

$M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR$

$\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}$

$\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R}$

$\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}$

$\sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}$

$N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}$

$CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019$

$v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )$

$v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}$

$N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$

$N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$

$N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}}$

$H = \frac{N_{LV}}{N_{GV}^{0.575}}\ \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D}$

$\frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}}$

$B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}}$

$\psi = \begin{cases} 27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999, &\mbox{if } B <= 0.025 \\ -533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\ 2.5714 B +1.5962, & \mbox{if }B > 0.055 \end{cases}$

$H_L = \frac{H_L}{\psi} \times \psi$

Nomenclature

$B$ = 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, dimensionless

$f$ = 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, lb_m/bbl

$N_D$ = pipe diameter number 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 32.174, lb_mft / lb_fsec²

$q_L$ = 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

$\mu$ = oil viscosity, cp

$\rho$ = density, lb_m/ft²

$\bar \rho$ = integrated average density at flowing conditions, lb_m/ft²

$\sigma$ = surface tension of liquid-air interface, dynes/cm

$\psi$ = secondary correlation factor, dimensionless

Subscripts

g = gas
K = °K
L = liquid
o = oil
R = °R
SL = superficial liquid
SG = superficial gas
w = water