View source for Beggs and Brill correlation

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== Brief ==
[[Beggs and Brill correlation |Beggs and Brill]] is an empirical two-phase flow correlation published in '''1972'''  <ref name=BB />.

It does distinguish between the flow regimes.

[[Beggs and Brill correlation |Beggs and Brill]] is the default [[VLP]] correlation in [[:Category:sPipe|sPipe]].

[[File: Beggs and Brill.png|thumb|500px|link=https://www.pengtools.com|Beggs and Brill in sPipe Vs GAP |right]]

== Math & Physics ==
=== Fluid flow energy balance ===
Following the law of conservation of energy the basic steady state flow equation is:
:<math> -144 \frac{\Delta p}{\Delta z} = \frac{sin(\theta)\ \bar \rho_m + \frac{f'\ G_m\ v_m}{2\ g_c\ D}}{1- \bar \rho_m\ \frac{v_m\ v_{SG}}{g_c\ p}}</math><ref name="BB" />
where
:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="BB" />

=== Flow patterns  ===
Froude number:
<math> H_{FR} = \frac{{v_m}^2}{g_c\ D}</math>

SEGREGATED: <math> (C_L < 0.01\ \&\ N_{FR}< L_1)\ or\ (C_L\ge0.01\ \&\ N_{FR}<L_2)</math><ref name="BB1991" />

TRANSITION: <math> C_L \ge 0.01\ \&\ L_2 \le N_{FR} \le L_3</math><ref name="BB1991" />

INTERMITTENT: <math> (0.01 \le C_L <0.4\ \&\  L_3<N_{FR}\le L_1)\ or\ ( C_L\ge0.4\ \&\ L_3<N_{FR} \le L_4)</math><ref name="BB1991" />

DISRIBUTED: <math> (C_L < 0.4\ \&\ N_{FR} \ge L_1)\ or\ (C_L\ge0.4\ \&\ N_{FR}>L_4)</math><ref name="BB1991" />

=== Liquid Holdup H<sub>L</sub>  ===

SEGREGATED, INTERMITTENT, DISRIBUTED: <math> H_L = H_{L(0)} \times \psi </math><ref name="BB1991" />

TRANSITION: <math> H_L = A \times H_{L(segregated)} + (1-A) \times {H_{L(intermittent)}} </math><ref name="BB1991" />

where:

<math> A = \frac{L_3-N_{FR}}{L_3-L_2}</math><ref name="BB1991" />

=== Friction factor  ===
No slip Reynolds two phase number:
:<math> Re = 1488 \times \frac {\rho_{m,ns} v_m D} { {\mu_L}^{C_L} {\mu_g}^{1-C_L} } </math><ref name="BB1991" />

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/>

Corrected two phase friction factor:
:<math> f' = f \times  e^S</math><ref name="BB1991" />

where

:<math> S = \frac{ ln(y)}{ -0.0523 + 3.182\ ln(y) - 0.8725\ ln(y)^2 + 0.01853\ ln(y)^ 4}</math><ref name="BB1991" />

and
:<math> y = \frac{ C_L} { {H_L}^2 } </math><ref name="BB1991" />

with constraint:

:<math> S = ln (2.2\ y - 1.2), when\ 1 < y < 1.2 </math><ref name="BB1991" />

== 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 />}}

== Flow Diagram ==

[[File: HB Block Diagram.png|400px|HB Block Diagram]]

== Workflow  H<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> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_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><ref name= Lyons/>

:<math> N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}</math><ref name= HB/>

:<math> CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 </math><ref name= Economides/>

:<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_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><ref name= Lyons/>

:<math> N_{LV} = 1.938\ v_{SL}\ \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><ref name= HB/>

:<math> N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>

:<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> \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> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math><ref name= Economides/>

:<math> \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} </math><ref name= Trina/>

:<math> H_L = \frac{H_L}{\psi} \times \psi</math><ref name= HB/>


SEGREGATED: <math> H_{L(0)} = 0.98 \frac{ {C_L}^{0.4846} } { {N_{FR}}^{0.0868}} </math><ref name="BB1991" />

INTERMITTENT: <math> H_{L(0)} = 0.845 \frac{ {C_L}^{0.5351} } { {N_{FR}}^{0.0173}}</math><ref name="BB1991" />

DISRIBUTED: <math> H_{L(0)} = 1.065\frac{ {C_L}^{0.5824} } { {N_{FR}}^{0.0609}}</math><ref name="BB1991" />

with the constraint <math> H_L \ge C_L</math><ref name="BB1991" />

:<math> \psi = 1 + C\ (sin(1.8\ \theta) - 0.333\ (sin(1.8\ \theta))^3)</math><ref name="BB1991" />


Uphill

SEGREGATED: <math> C = (1-C_L)\ ln( 0.011 \times {N_{LV}}^{3.539}\times {C_L}^{-3.768}\times {F_{FR}}^{-1.614})</math><ref name="BB1991" />

INTERMITTENT: <math> C = (1-C_L)\ ln( 2.96\times {N_{LV}}^{-0.4473}\times {C_L}^{0.305}\times {F_{FR}}^{0.0978})</math><ref name="BB1991" />

DISRIBUTED: <math> C=0</math><ref name="BB1991" />

Downhill

ALL: <math> C = (1-C_L)\ ln( 4.7\times {N_{LV}}^{0.1244}\times {C_L}^{-0.3692}\times {F_{FR}}^{-0.5056})</math><ref name="BB1991" />

== Modifications  ==

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/>.

2. Use the [[Griffith correlation]] to define the bubble flow regime<ref name = Economides/> and calculate H<sub>L</sub>.

3. Use [[WCUT| watercut]] instead of [[WOR]] to account for the watercut = 100%.

== Nomenclature  ==

:<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, 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_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===

:<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>2</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

===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=BB>{{cite journal
 |last1=Beggs|first1=H. D.
 |last2= Brill |first2=J. P.
 |title=A Study of Two-Phase Flow in Inclined Pipes
 |journal=Journal of Petroleum Technology
 |date=May 1973
 |publisher=AIME
 |volume=255
 |number=SPE-4007-PA
 |url=https://www.onepetro.org/journal-paper/SPE-4007-PA
 |url-access=subscription

}}</ref>

<ref name=Economides>{{cite book
 |last1= Economides |first1=M.J.
 |last2= Hill |first2=A.D.
 |last3= Economides |first3=C.E. 
 |last4= Zhu |first4=D.
 |title=Petroleum Production Systems
 |edition=2
 |date=2013
 |publisher=Prentice Hall
 |place=Westford, Massachusetts
 |isbn=978-0-13-703158-0
}}</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= 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>

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