Hagedorn and Brown

General Description

The correlation for two-phase flow by Hagedorn and Brown (1965)¹ is based on experimental work on a 1500-ft vertical well with piping having 1-in, 1.25-in, and 1.5-in diameters. The authors suggested using four dimensionless numbers to calculate the liquid hold up. These four numbers are

$\begin{align} &N_{Lv}=v_{sL}\bigg(\frac{\rho_L}{g\sigma}\bigg)^{\frac{1}{4}}\\ &N_{gv}=v_{sg}\bigg(\frac{\rho_L}{g\sigma}\bigg)^{\frac{1}{4}}\\ &N_{d}=d_h\bigg(\frac{\rho_L g}{\sigma}\bigg)^{\frac{1}{2}}\\ &N_{\mu}=\mu_L\bigg(\frac{g}{\rho_L\sigma^3}\bigg)^{\frac{1}{4}} \end{align}$

Three plots were presented to estimate the liquid hold up, one to account for viscosity effects, one to estimate a preliminary liquid hold up, and one to correct the preliminary liquid hold up to obtain the final value. These three plots are shown in Fig. 1.

Fig.1—Plots used to calculate $H_L$

Once the parameters $CN_{\mu}$ and $\psi$ are estimated from the two top plots, one can estimate $H_L$ from the bottom plot.

Pressure Gradient Calculations

The correlation-specific properties in the pressure gradient is set to the following

Density in the gravity gradient $\rho_g =\rho_s$

Density in the friction gradient $\rho_f=\rho_m^2/\rho_s$

Density in the acceleration gradient $\rho_a=\rho_s$

Friction factor $f_D$ is calculated with $N_{Re}=\rho_sv_md_h/(\mu_L^{H_L}\mu_g^{1-H_L})$

Griffith Modification for Bubble Flow

A correction to the Hagedorn and Brown correlation has been suggested to better predict the liquid hold up in a bubble-flow regime. This correction is to replace the Hagedorn and Brown correlation to the one suggested by Griffith (REF). Bubble flow exists when $\begin{equation} 1-C_L < L_B \end{equation}$ where $L_B =\max(1.071 - 0.2218v_m^2/d_h, 0.13)$. When bubble flow is predicted, Griffith suggested calculating the liquid hold up from the definition of slip velocity $\begin{equation} H_L = 1-\frac{1}{2}\bigg[1+\frac{v_m}{v_s}-\sqrt{\bigg(1+\frac{v_m}{v_s}\bigg)^2-4\bigg(\frac{v_{sg}}{v_s}\bigg)}\bigg] \end{equation}$ where $v_s=0.8\text{ft/s}$. As bubble flow is a flow regime with a continuous liquid with dispersed bubbles, it was further suggested to change the following correlation-specific properties

Density in the friction gradient $\rho_f=\rho_L$

Velocity in the friction gradient $v_m = v_L$

Friction factor $f_D$ is calculated with $N_{Re}=\rho_Lv_Ld_h/\mu_L$

A.R. Hagedorn and K.E. Brown. Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits. Journal of Petroleum Technology, pages 475–484, 4 1965. SPE-940-PA. doi:10.2118/940-PA. ↩