View source for Gray correlation

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== Brief ==

[[Gray]] is an empirical two-phase flow correlation published in '''1974''' <ref name= Gray />.

[[Gray]] is the default [[VLP]] correlation for the '''gas wells''' in the [[:Category:PqPlot|PQplot]].

== Math & Physics ==
Following the law of conservation of energy the basic steady state flow equation is:
:<math> 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}</math><ref name="Gray" />

where
:<math> \bar \rho_m = \rho_L (1-H_g) + \rho_g H_g</math>,  slip mixture density

:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math>,  no-slip mixture density

Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:
:<math> \frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon'} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)</math><ref name = Moody1944/>

The pseudo wall roughness:
:<math> \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} </math>, with the limit <math> \varepsilon' \geqslant 2.77 \times 10^{-5}</math><ref name= Gray/>

Reynolds two phase number:
:<math> Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{C_L} \mu_g^{(1-C_L)}}</math><ref name= HB/>

== Discussion  ==

Why [[Gray]] correlation?

{{Quote| text = The Gray correlation was found to be the best of several initially tested ... | source = Nitesh Kumar l<ref name= Kumar />}}

== Workflow  H<sub>g</sub> & C<sub>L</sub>== 

:<math> M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR</math><ref name="HB" />

:<math> \rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}</math><ref name= Lyons/>

:<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math><ref name= Lyons/>

:<math> v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )</math><ref name= Lyons/>

:<math> v_{SG} = \frac{q_g \times 10^6}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}</math>

:<math> C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}</math>

:<math> v_m = v_{SL} +  v_{SG} </math>

:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math>

:<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>

:<math> \sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w}</math> <ref name= Gray/>

:<math> N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)} </math><ref name= Gray/>

:<math> N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L } </math><ref name= Gray/>

:<math> R = \frac{v_{SL}}{v_{SG}} </math><ref name= Gray/>

:<math> B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right )  \right ) </math><ref name= Gray/>

:<math> A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right )  \right )^B </math><ref name= Gray/>

:<math> H_g = \frac{1-e^A}{R+1}</math><ref name= Gray/>

== Modifications  ==

use watercut instead of WOR

== Nomenclature  ==

NV velocity number

== References ==
<references>

<ref name= Gray>{{cite journal
 |last1= Gray |first1=H. E.
 |title=Vertical Flow Correlation in Gas Wells
 |journal=User manual for API 14B, Subsurface controlled safety valve sizing computer program
 |publisher = API
 |date= 1974
}}</ref>

<ref name=Colebrook>{{cite journal
 |last1=Colebrook|first1=C. F.
 |title=Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws
 |journal=Journal of the Institution of Civil Engineers
 |date=1938–1939
 |volume=11
 |pages=133–156
 |location=London, England
 |url=https://www.scribd.com/doc/269398414/Colebrook-White-1939
 |url-access=subscription
}}</ref>

<ref name = Moody1944>{{cite journal 
 |first=L. F. 
 |last=Moody 
 |title=Friction factors for pipe flow 
 |journal=Transactions of the ASME 
 |volume=66 
 |issue=8 
 |pages=671–684 
 |year=1944 
 |url=https://www.onepetro.org/journal-paper/SPE-2198-PA
 |url-access=subscription 
}} </ref>

<ref name=HB>{{cite journal
 |last1=Hagedorn|first1=A. R.
 |last2= Brown |first2=K. E.
 |title=Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits
 |journal=Journal of Petroleum Technology
 |date=1965
 |volume=17(04)
 |pages=475-484
}}</ref>

<ref name= Lyons>{{cite book
 |last1= Lyons |first1=W.C.
 |title=Standard handbook of petroleum and natural gas engineering
 |date= 1996
 |volume=2
 |publisher=Gulf Professional Publishing
 |place=Houston, TX
 |isbn=0-88415-643-5
}}</ref>

<ref name=Kumar>{{cite journal
 |first1=N. |last1=Kumar 
 |first2=J. F. |last2=Lea 
 |title=Improvements for Flow Correlations for Gas Wells Experiencing Liquid Loading
 |journal=Journal of Petroleum Technology
 |number=SPE-92049-MS
 |date=January 1, 2005
 |url=https://www.onepetro.org/conference-paper/SPE-92049-MS
 |url-access= subscription 
}}</ref>

</references>

[[Category:pengtools]]
[[Category:pqPlot]]