Fanning correlation

Brief

The Fanning is the name used to refer to the calculation of the hydrostatic pressure difference and the friction pressure loss for the dry gas.

Fanning is the default VLP correlation for the dry gas wells in the PQplot.

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 .

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

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

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

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 $\varepsilon' \geqslant 2.77 \times 10^{-5}$

Reynolds two phase number:

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

Discussion

Why Gray correlation?

The Gray correlation was found to be the best of several initially tested ...

— Nitesh Kumar l

Workflow H_g & C_L

$M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR$

$\rho_L= 62.4\ SG_o \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}$

$\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R}$

$v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )$

$v_{SG} = \frac{q_g \times 10^6}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}$

$C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}$

$v_m = v_{SL} + v_{SG}$

$\rho_m = \rho_L C_L + \rho_g (1-C_L)$

$\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}$

$\sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w}$

$N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)}$

$N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L }$

$R = \frac{v_{SL}}{v_{SG}}$

$B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right )$

$A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B$

$H_g = \frac{1-e^A}{R+1}$

Modifications

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

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

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, lb_m/bbl

$N_D$ = pipe diameter number, dimensionless

$N_V$ = velocity number, dimensionless

$p$ = pressure, psia

$q_c$ = conversion constant equal to 32.174049, lb_mft / lb_fsec²

$q$ = production rate, bbl/d

$R$ = superficial liquid to gas ratio, dimensionless

$Re$ = Reynolds number, dimensionless

$R_s$ = solution gas-oil ratio, scf/stb

$SG$ = 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, lb_m/ft³

$\bar \rho$ = slip density, lb_m/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