# Gray correlation

## Brief

Gray is an empirical two-phase flow correlation published in 1974 [1].

Gray is the default VLP correlation for the gas wells in the PQplot.

Gray in PQplot Vs Prosper & Kappa

## Math & Physics

Following the law of conservation of energy the basic steady state flow equation is:

$144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}$

where

$\bar \rho_m = \rho_L (1-H_g) + \rho_g H_g$, slip mixture density [1].
$\rho_m = \rho_L C_L + \rho_g (1-C_L)$, no-slip mixture density [1].

Colebrook–White [2] equation for the Darcy's friction factor:

$\frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon'} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)$[3]

The pseudo wall roughness:

$\varepsilon' = \begin{cases} \frac{28.5}{453.592} \frac{\sigma_L}{\rho_m v_m^2}, &\mbox{if } R \geqslant 0.007 \\ \varepsilon + R \frac{\varepsilon'-\varepsilon}{0.007}, & \mbox{if } R < 0.007 \end{cases}$, with the limit $\varepsilon' \geqslant 2.77 \times 10^{-5}$[1]

Reynolds two phase number:

$Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{C_L} \mu_g^{(1-C_L)}}$[4]

## Discussion

Why Gray correlation?

The Gray correlation was found to be the best of several initially tested ...
— Nitesh Kumar l[5]

## Workflow Hg & CL

$M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR$[4]
$\rho_L= 62.4\ SG_o \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}$
$\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R}$[6]
$v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )$[6]
$v_{SG} = \frac{q_g \times 10^6}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}$
$C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}$
$v_m = v_{SL} + v_{SG}$
$\rho_m = \rho_L C_L + \rho_g (1-C_L)$
$\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}$[6]
$\sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w}$ [1]
$N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)}$[1]
$N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L }$[1]
$R = \frac{v_{SL}}{v_{SG}}$[1]
$B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right )$[1]
$A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B$[1]
$H_g = \frac{1-e^A}{R+1}$[1]

## Modifications

1. Use Fanning correlation for the dry gas (WGR=0 and OGR=0 case).

2. Use watercut instead of WOR to account for the OGR=0 case.

3. If the relative roughness: $\frac{\varepsilon'}{D} > 0.05$ use 0.05 in the Moody Diagram [3].

4. If HL can't be calculated then HL = CL.

## Nomenclature

$A$ = correlation group, dimensionless
$A_p$ = flow area, ft2
$B$ = correlation group, dimensionless
$B$ = formation factor, bbl/stb
$C$ = no-slip holdup factor, dimensionless
$D$ = pipe diameter, ft
$h$ = depth, ft
$H$ = holdup factor, dimensionless
$f$ = friction factor, dimensionless
$GLR$ = gas-liquid ratio, scf/bbl
$M$ = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lbm/bbl
$N_D$ = pipe diameter number, dimensionless
$N_V$ = velocity number, dimensionless
$p$ = pressure, psia
$q_c$ = conversion constant equal to 32.174049, lbmft / lbfsec2
$q$ = production rate, bbl/d
$R$ = superficial liquid to gas ratio, dimensionless
$Re$ = Reynolds number, dimensionless
$SG$ = specific gravity, dimensionless
$T$ = temperature, °R or °K, follow the subscript
$v$ = velocity, ft/sec
$WOR$ = water-oil ratio, bbl/bbl
$z$ = gas compressibility factor, dimensionless

### Greek symbols

$\varepsilon$ = absolute roughness, ft
$\varepsilon'$ = pseudo wall roughness, ft
$\mu$ = viscosity, cp
$\rho$ = density, lbm/ft3
$\bar \rho$ = slip density, lbm/ft2
$\sigma$ = surface tension of liquid-air interface, dynes/cm

### Subscripts

g = gas
K = °K
L = liquid
m = gas/liquid mixture
o = oil
R = °R
SL = superficial liquid
SG = superficial gas
w = water

## References

1. Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API.
2. Colebrook, C. F. (1938–1939). . Journal of the Institution of Civil Engineers. London, England. 11: 133–156.
3. Moody, L. F. (1944). . Transactions of the ASME. 66 (8): 671–684.
4. 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.
5. Kumar, N.; Lea, J. F. (January 1, 2005). (SPE-92049-MS).
6. Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.