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

$\begin{equation} H_L = 1 - \frac{1-\exp(A)}{R+1} \end{equation}$ where $\begin{align} &A=-2.314\bigg[N_{v}\bigg(1+\frac{205}{N_d}\bigg)\bigg]^{B}\\ &B=0.0814\bigg[1-0.0554\ln\bigg(1+730\frac{R}{1+R}\bigg)\bigg] \end{align}$ 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 $\begin{equation} k_{eff}= \begin{cases} k_0, &0.007\leq R\\ k+R\frac{k_0-k}{0.007}, &\text{otherwise} \end{cases} \end{equation}$ where $k_0=28.5\sigma/(\rho_m v_m^2)$, with the limit $k_{eff}\geq 2.77\times 10^{-5}$.

Pressure Gradient Calculations

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 $k/d_h$.

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. ↩