View source for Fanning correlation

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== Brief ==

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

[[Fanning correlation]] is the default [[VLP]] correlation for the '''dry 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} =  \rho_g + \rho_g \frac{f v_g^2 }{2 g_c D} + \rho_g \frac{\Delta{(\frac{v_g^2}{2g_c}})}{\Delta h}</math>

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/>

Reynolds number:
:<math> Re = 1488 \frac {\rho_g v_g D}{\mu_g}</math>

== Discussion  ==

Why [[Gray correlation|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= 62.4\ SG_o \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}</math>

:<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  ==

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  ==

:<math> A </math> = correlation group, dimensionless
:<math> A_p </math> = flow area, ft2
:<math> B </math> = correlation group, dimensionless
:<math> B </math> = formation factor, bbl/stb
:<math> C </math> = no-slip holdup factor, dimensionless
:<math> D </math> = pipe diameter, ft
:<math> h </math> = depth, ft
:<math> H </math> = holdup factor, dimensionless
:<math> f </math> = friction factor, dimensionless
:<math> GLR </math> = gas-liquid ratio, scf/bbl
:<math> M </math> = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lb<sub>m</sub>/bbl
:<math> N_D </math> = pipe diameter number, dimensionless
:<math> N_V </math> = velocity number, dimensionless
:<math> p </math> = pressure, psia
:<math> q_c </math> = conversion constant equal to 32.174049, lb<sub>m</sub>ft / lb<sub>f</sub>sec<sup>2</sup>
:<math> q </math> = production rate, bbl/d
:<math> R </math> = superficial liquid to gas ratio, dimensionless
:<math> Re </math> = Reynolds number, dimensionless
:<math> R_s </math> = solution gas-oil ratio, scf/stb
:<math> SG </math> = specific gravity, dimensionless
:<math> T </math> = temperature, °R or °K, follow the subscript
:<math> v </math> = velocity, ft/sec
:<math> WOR </math> = water-oil ratio, bbl/bbl
:<math> z </math> = gas compressibility factor, dimensionless

===Greek symbols===

:<math> \varepsilon </math> = absolute roughness, ft
:<math> \varepsilon' </math> = pseudo wall roughness, ft
:<math> \mu </math> = viscosity, cp
:<math> \rho </math> = density, lb<sub>m</sub>/ft<sup>3</sup>
:<math> \bar \rho </math> = slip density, lb<sub>m</sub>/ft<sup>2</sup>
:<math> \sigma </math> = surface tension of liquid-air interface, dynes/cm

===Subscripts===

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

== 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
 |number=SPE-92049-MS
 |date=January 1, 2005
 |url=https://www.onepetro.org/conference-paper/SPE-92049-MS
 |url-access=registration
}}</ref>

</references>

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[[Category:pqPlot]]