# Gray

## General Description

The two-phase flow correlation by Gray (1971)1 was originally presented in the manual for a computer program on sizing of surface controlled subsurface safety valves (SCSSV). The correlation is meant for lean gas-condensate wells with liquid as a dispersed phase (mist flow). The correlation includes an expression for the liquid hold up, and a modification of the pipe roughness to account for liquid adhesing to the pipe wall which increases friction forces.

The liquid hold up is calculated from

where with $$R=v_{sL}/v_{sg}$$, $$N_{v}=\rho_m^2v_{m}^2/(g\sigma(\rho_L-\rho_g))$$, and $$N_d=g(\rho_L-\rho_g)d_h^2/\sigma$$.

The modification to the pipe roughness $$k$$ is calculated by where $$k_0=28.5\sigma/(\rho_m v_m^2)$$, with the limit $$k_{eff}\geq 2.77\times 10^{-5}$$.

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

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

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

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

Friction factor $$f_D$$ is calculated with $$N_{Re}=\rho_m v_m d/\mu_m$$ and $$k_{eff}/d_h$$ instead of $$k/d_h$$.

1. H.E. Gray. Vertical Flow Correlation in Gas Wells. User’s Manual for API 14B Surface Controlled Subsurface Safety Valve Sizing Computer Program. American Petroleum Institute, 2 edition, 1978.