# 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

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.

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 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 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 $$N_{Re}=\rho_Lv_Ld_h/\mu_L$$

1. 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