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:

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

Flow Diagram

Workflow

To find $H_L$ calculate:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References

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fekete.com

wikipedia.org Darcy friction factor formulae

Economides Production Petroleum Systems

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), 475-484.

Lyons WC. 1996. Standard handbook of petroleum and natural gas engineering. Gulf Publishing Company, Houston, TX.

Guo B, Lyons WC, Chalambor A. 2007. Petroleum production engineering, A computer assisted approach. Elsevier Science & Technology Books

Trina S. 2010. An integrated horizontal and vertical flow simulation with application to wax precipitation. Master of Engineering Thesis, Memorial University of Newfoundland, Canada.

Haaland SE. 1983. Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow. Journal of Fluids Engineering. Vol. 105, pp. 89-90.