Difference between revisions of "Hagedorn and Brown correlation"

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__TOC__
 
__TOC__
 
== Brief ==
 
== Brief ==
−
[[Hagedorn and Brown]] is an empirical two-phase flow correlation published in '''1965'''  <ref name="HB" />.
+
[[Hagedorn and Brown correlation |Hagedorn and Brown]] is an empirical two-phase flow correlation published in '''1965'''  <ref name=HB />.
  
 
It doesn't distinguish between the flow regimes.
 
It doesn't distinguish between the flow regimes.
  
−
The heart of the [[Hagedorn and Brown]] method is a correlation for the liquid holdup <math>H_L</math><ref name="Economides" />.
+
The heart of the [[Hagedorn and Brown correlation|Hagedorn and Brown]] method is a correlation for the liquid holdup H<sub>L</sub> <ref name=Economides />.
 +
 
 +
[[Hagedorn and Brown correlation|Hagedorn and Brown]]  is the default [[VLP]] correlation for the '''oil wells''' in the [[:Category:PQplot|PQplot]].
 +
 
 +
[[File: Hagedorn and Brown.png|thumb|500px|link=https://www.pengtools.com/pqPlot?paramsToken=57e2ad9dd84d56fb56b7515b2ef312bd|Hagedorn and Brown in PQplot Vs Prosper & Kappa |right]]
  
 
== Math & Physics ==
 
== Math & Physics ==
−
Following the law of conservation of energy the basic steady state flow equation is <ref name="HB" />:
+
Following the law of conservation of energy the basic steady state flow equation is:
−
:<math> 144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar \rho_m} + \bar \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}</math>
+
:<math> 144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar \rho_m} + \bar \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}</math><ref name="HB" />
 
where
 
where
−
:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math>
+
:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="HB" />
  
−
Colebrook–White equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:
+
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>
+
:<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 two phase number <ref name="HB" />:
+
Reynolds two phase number:
−
:<math> Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}</math>
+
:<math> Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}</math><ref name="HB" />
  
 
== Discussion  ==
 
== Discussion  ==
 +
 +
Why [[Hagedorn and Brown correlation| Hagedorn and Brown]]?
 +
 +
{{Quote| text = One of the consistently best correlations ... | source = Michael Economides et al<ref name=Economides />}}
 +
 +
== Demo ==
 +
 +
[[Hagedorn and Brown]] correlation overview video:
 +
 +
[[File:Hagedorn and Brown demo.png|400px|https://www.youtube.com/watch?v=DpSv3kWPsIk | Watch on youtube]]
 +
 +
[[Media:Hagedorn and Brown ppt.pdf|Download presentation (pdf)]]
 +
 +
In this video it's shown:
 +
*What the Hagedorn and Brown correlation is
 +
*History and practical application
 +
*Math & Physics
 +
*Flow diagram to get the VLP curve
 +
*Workflow to find HL
  
 
== Flow Diagram ==
 
== Flow Diagram ==
Line 25: Line 48:
 
[[File: HB Block Diagram.png|400px|HB Block Diagram]]
 
[[File: HB Block Diagram.png|400px|HB Block Diagram]]
  
−
== Workflow  ==
+
== Workflow  H<sub>L</sub> ==
  
−
To find <math>H_L</math> calculate:
+
:<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> M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR</math>
+
:<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_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>
+
:<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math><ref name= Lyons/>
  
−
:<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math>
+
:<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>
  
−
:<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math>
+
:<math> \sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>
  
−
:<math> \sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}</math>
+
:<math> N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}</math><ref name= HB/>
  
−
:<math> N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}</math>
+
:<math> CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 </math><ref name= Economides/>
  
−
:<math> CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 </math>
+
:<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_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )</math>
+
:<math> v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_R}{520}\ \frac{z}{1}</math><ref name= Lyons/>
  
−
:<math> v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}</math>
+
:<math> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
  
−
:<math> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math>
+
:<math> N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
  
−
:<math> N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math>
+
:<math> N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
  
−
:<math> N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} </math>
+
:<math> H = \frac{N_{LV}}{N_{GV}^{0.575}}\ \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D} </math><ref name= Economides/>
  
−
:<math> H = \frac{N_{LV}}{N_{GV}^{0.575}}\  \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D} </math>
+
:<math> \frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}} </math><ref name= Trina/>
  
−
:<math> \frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}} </math>
+
:<math> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math><ref name= Economides/>
−
 
 
−
:<math> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math>
 
  
 
:<math> \psi = \begin{cases}  
 
:<math> \psi = \begin{cases}  
−
27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999,  &\mbox{if } B <= 0.025 \\
+
27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999,  &\mbox{if } B \le 0.025 \\
 
-533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\
 
-533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\
 
2.5714 B +1.5962, & \mbox{if }B > 0.055
 
2.5714 B +1.5962, & \mbox{if }B > 0.055
−
\end{cases} </math>
+
\end{cases} </math><ref name= Trina/>
  
−
:<math> H_L = \frac{H_L}{\psi} \times \psi</math>
+
:<math> H_L = \frac{H_L}{\psi} \times \psi</math><ref name= HB/>
  
−
== Nomenclature ==
+
== Modifications ==
  
−
== References ==
+
1. Use the no-slip holdup when the original empirical correlation predicts a liquid holdup H<sub>L</sub> less than the no-slip holdup <ref name = Economides/>.
  
−
dsafafadfadf <ref name="Moody1944" /><ref name="Colebrook" /><ref name="HB" /><ref name=Economides/>
+
2. Use the [[Griffith correlation]] to define the bubble flow regime<ref name = Economides/> and calculate H<sub>L</sub>.
  
−
[http://www.fekete.com/SAN/WebHelp/virtuwell/webhelp/c-te-pressdrop.htm#Introduction fekete.com]
+
3. Use [[WCUT| watercut]] instead of [[WOR]] to account for the watercut = 100%.
  
−
[https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae wikipedia.org Darcy friction factor formulae]
+
== Nomenclature  ==
  
−
Economides Production Petroleum Systems
+
:<math> A_p </math> = flow area, ft2
 +
:<math> B </math> = correlation group, dimensionless
 +
:<math> B </math> = formation factor, bbl/stb
 +
:<math> C </math> = coefficient for liquid viscosity number, dimensionless
 +
:<math> D </math> = pipe diameter, ft
 +
:<math> h </math> = depth, ft
 +
:<math> H </math> = correlation group, dimensionless
 +
:<math> H_L </math> = liquid holdup factor, fraction
 +
:<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_GV </math> = gas velocity number, dimensionless
 +
:<math> N_L </math> = liquid viscosity number, dimensionless
 +
:<math> N_LV </math> = liquid 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> = total liquid production rate, bbl/d
 +
:<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===
  
−
Lyons WC. 1996. Standard handbook of petroleum and natural gas engineering. Gulf Publishing Company, Houston, TX.
+
:<math> \varepsilon </math> = absolute roughness, ft
 +
:<math> \mu </math> = viscosity, cp
 +
:<math> \rho </math> = density, lb<sub>m</sub>/ft<sup>3</sup>
 +
:<math> \bar \rho </math> = integrated average density at flowing conditions, lb<sub>m</sub>/ft<sup>3</sup>
 +
:<math> \sigma </math> = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
 +
:<math> \psi </math> = secondary correlation factor, dimensionless
  
−
Guo B, Lyons WC, Chalambor A. 2007. Petroleum production engineering, A computer assisted approach. Elsevier Science & Technology Books
+
===Subscripts===
  
−
Trina S. 2010. An integrated horizontal and vertical flow simulation with application to wax precipitation. Master of Engineering Thesis, Memorial University of Newfoundland, Canada.
+
: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/>
  
−
Haaland SE. 1983. Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow. Journal of Fluids Engineering. Vol. 105, pp. 89-90.
+
== References ==
−
 
 
−
<ref>
 
−
{{cite journal|url=http://www.google.com |title=Search |url-access=subscription}}
 
−
</ref>
 
−
 
 
−
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.
 
  
 
<references>
 
<references>
Line 101: Line 154:
 
  |title=Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits
 
  |title=Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits
 
  |journal=Journal of Petroleum Technology
 
  |journal=Journal of Petroleum Technology
 +
|number=SPE-940-PA
 
  |date=1965
 
  |date=1965
 
  |volume=17(04)
 
  |volume=17(04)
 
  |pages=475-484
 
  |pages=475-484
 +
|url=https://www.onepetro.org/journal-paper/SPE-940-PA
 +
|url-access=registration
 
}}</ref>
 
}}</ref>
  
Line 112: Line 168:
 
  |last4= Zhu |first4=D.
 
  |last4= Zhu |first4=D.
 
  |title=Petroleum Production Systems
 
  |title=Petroleum Production Systems
 +
|edition=2
 
  |date=2013
 
  |date=2013
−
  |publisher=Pearson Education, Inc.
+
  |publisher=Prentice Hall
 +
|place=Westford, Massachusetts
 
  |isbn=978-0-13-703158-0
 
  |isbn=978-0-13-703158-0
 
}}</ref>
 
}}</ref>
Line 126: Line 184:
 
  |location=London, England
 
  |location=London, England
 
  |url=https://www.scribd.com/doc/269398414/Colebrook-White-1939
 
  |url=https://www.scribd.com/doc/269398414/Colebrook-White-1939
 +
|url-access=subscription
 
}}</ref>
 
}}</ref>
  
Line 137: Line 196:
 
  |pages=671–684  
 
  |pages=671–684  
 
  |year=1944  
 
  |year=1944  
−
  |url=https://www.scribd.com/doc/269398353/Friction-Factors-for-Pipe-Flow-MoodyLFpaper1944
+
  |url=https://www.onepetro.org/journal-paper/SPE-2198-PA
 +
|url-access=subscription
 
}} </ref>
 
}} </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= Trina>{{cite thesis
 +
|last= Trina |first=S.
 +
|title=An integrated horizontal and vertical flow simulation with application to wax precipitation
 +
|date= 2010
 +
|type=Master of Engineering Thesis
 +
|publisher=Memorial University of Newfoundland
 +
|place= Canada
 +
}}</ref>
  
 
</references>
 
</references>
  
 +
[[Category:pengtools]]
 +
[[Category:PQplot]]
  
−
 
+
{{#seo:
−
[[Category:PEngTools]]
+
|title=Hagedorn and Brown correlation
−
[[Category:pqPlot]]
+
|titlemode= replace
 +
|keywords=Hagedorn and Brown, correlation, equation, flow rate, fluids flow, Reynolds number, liquid hold up
 +
|description=Hagedorn and Brown correlation used to calculate reservoir inflow performance curve for nodal analysis
 +
}}

Latest revision as of 12:21, 1 November 2018

Brief

Hagedorn and Brown is an empirical two-phase flow correlation published in 1965 [1].

It doesn't distinguish between the flow regimes.

The heart of the Hagedorn and Brown method is a correlation for the liquid holdup HL [2].

Hagedorn and Brown is the default VLP correlation for the oil wells in the PQplot.

Hagedorn and Brown 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 + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar \rho_m} + \bar \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}[1]

where

\bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)[1]

Colebrook–White [3] 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)[4]

Reynolds two phase number:

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

Discussion

Why Hagedorn and Brown?

One of the consistently best correlations ...
— Michael Economides et al[2]

Demo

Hagedorn and Brown correlation overview video:

Watch on youtube

Download presentation (pdf)

In this video it's shown:

  • What the Hagedorn and Brown correlation is
  • History and practical application
  • Math & Physics
  • Flow diagram to get the VLP curve
  • Workflow to find HL

Flow Diagram

HB Block Diagram

Workflow HL

M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR[1]
\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}[5]
\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} [5]
\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}[5]
\sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}[5]
N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}[1]
CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 [2]
v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )[5]
v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_R}{520}\ \frac{z}{1}[5]
N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [1]
N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [1]
N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} [1]
H = \frac{N_{LV}}{N_{GV}^{0.575}}\ \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D} [2]
\frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}} [6]
B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} [2]
\psi = \begin{cases} 27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999, &\mbox{if } B \le 0.025 \\ -533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\ 2.5714 B +1.5962, & \mbox{if }B > 0.055 \end{cases} [6]
H_L = \frac{H_L}{\psi} \times \psi[1]

Modifications

1. Use the no-slip holdup when the original empirical correlation predicts a liquid holdup HL less than the no-slip holdup [2].

2. Use the Griffith correlation to define the bubble flow regime[2] and calculate HL.

3. Use watercut instead of WOR to account for the watercut = 100%.

Nomenclature

A_p = flow area, ft2
B = correlation group, dimensionless
B = formation factor, bbl/stb
C = coefficient for liquid viscosity number, dimensionless
D = pipe diameter, ft
h = depth, ft
H = correlation group, dimensionless
H_L = liquid holdup factor, fraction
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_GV = gas velocity number, dimensionless
N_L = liquid viscosity number, dimensionless
N_LV = liquid velocity number, dimensionless
p = pressure, psia
q_c = conversion constant equal to 32.174049, lbmft / lbfsec2
q = total liquid production rate, bbl/d
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
\mu = viscosity, cp
\rho = density, lbm/ft3
\bar \rho = integrated average density at flowing conditions, lbm/ft3
\sigma = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
\psi = secondary correlation factor, dimensionless

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. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Hagedorn, A. R.; Brown, K. E. (1965). "Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits"Free registration required. Journal of Petroleum Technology. 17(04) (SPE-940-PA): 475–484. 
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Economides, M.J.; Hill, A.D.; Economides, C.E.; Zhu, D. (2013). Petroleum Production Systems (2 ed.). Westford, Massachusetts: Prentice Hall. ISBN 978-0-13-703158-0. 
  3. ↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws"Paid subscription required. Journal of the Institution of Civil Engineers. London, England. 11: 133–156. 
  4. ↑ Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5. 
  6. ↑ 6.0 6.1 Trina, S. (2010). An integrated horizontal and vertical flow simulation with application to wax precipitation (Master of Engineering Thesis). Canada: Memorial University of Newfoundland. 
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