Difference between revisions of "Beggs and Brill correlation"

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[[Beggs and Brill correlation |Beggs and Brill]] is an empirical two-phase flow correlation published in '''1972'''  <ref name=BB />.
 
[[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.
+
It distinguish between 4 flow regimes.
  
 
[[Beggs and Brill correlation |Beggs and Brill]] is the default [[VLP]] correlation in [[:Category:sPipe|sPipe]].
 
[[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]]
+
[[File: Beggs and Brill.png|thumb|500px|link=https://www.pengtools.com/sPipe?paramsToken=cc8af4bdd85a3d7da86119d5367742e2|Beggs and Brill in sPipe Vs GAP |right]]
  
 
== Math & Physics ==
 
== Math & Physics ==
 
=== Fluid flow energy balance ===
 
=== 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\ \frac{p}{144}}}</math><ref name="BB" />
−
:<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
 
where
 
:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="BB" />
 
:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="BB" />
  
−
=== Flow pattern limits ===
+
=== Friction factor ===
−
SEGREGATED:
+
No slip Reynolds two phase number:
−
:<math> C_L < 0.01 AND N_{FR}< L_1</math><ref name="BB1991" />
+
:<math> Re = 1488 \times \frac {\rho_{m,ns} v_m D} { \mu_L\ C_L + \mu_g\ (1-C_L) } </math><ref name="BB1991" />
−
OR
 
−
:<math> C_L>=0.01 AND N_{FR}<L_2</math><ref name="BB1991" />
 
  
−
TRANSITION:
+
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/>
  
−
INTERMITTENT:
+
Corrected two phase friction factor:
 +
:<math> f' = f \times  e^S</math><ref name="BB1991" />
  
−
DISRIBUTED
+
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" />
  
−
=== HL  ===
+
and
 +
:<math> y = \frac{ C_L} { {H_L}^2 } </math><ref name="BB1991" />
  
−
Reynolds two phase number:
+
with constraint:
−
:<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" />
 
  
−
=== Friction factor  ===
+
:<math> S = ln (2.2\ y - 1.2), when\ 1 \le y \le 1.2 </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/>
 
  
 
== Discussion  ==
 
== Discussion  ==
  
−
Why [[Hagedorn and Brown correlation| Hagedorn and Brown]]?
+
Why [[Beggs and Brill correlation| Beggs and Brill]]?
  
−
{{Quote| text = One of the consistently best correlations ... | source = Michael Economides et al<ref name=Economides />}}
+
{{Quote| text = The best correlation for the horizontal flow. | source = pengtools.com}}
  
 
== Flow Diagram ==
 
== Flow Diagram ==
Line 48: Line 46:
 
[[File: HB Block Diagram.png|400px|HB Block Diagram]]
 
[[File: HB Block Diagram.png|400px|HB Block Diagram]]
  
−
== Workflow H<sub>L</sub> ==
+
== Workflow H<sub>L</sub> ==
 +
 
 +
:<math> \rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} (1-WCUT) + \frac{62.4\ SG_w}{B_w}\ WCUT</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 (1-WCUT) + \mu_w\ WCUT</math><ref name= Lyons/>
 +
 
 +
:<math> \sigma_L = \sigma_o (1-WCUT) + \sigma_w\ WCUT</math><ref name= Lyons/>
 +
 
 +
:<math> v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o (1-WCUT) + B_w\ WCUT \right )</math><ref name= Lyons/>
 +
 
 +
:<math> v_{SG} = \frac{q_g}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}</math><ref name= Lyons/>
 +
 
 +
:<math> G_m = \rho_L \times v_{SL} + \rho_g \times  v_{SG}</math><ref name= Lyons/>
 +
 
 +
:<math> v_m = v_{SL} +  v_{SG}</math>
 +
 
 +
:<math> C_L = \frac{v_{SL}}{v_m}</math>
 +
 
 +
:<math> L_1 = 316\ {C_L}^{0.302}</math><ref name= BB1991/>
 +
:<math> L_2 = 0.0009252\ {C_L}^{-2.4684}</math><ref name= BB1991/>
 +
:<math> L_3 = 0.1\ {C_L}^{-1.4516}</math><ref name= BB1991/>
 +
:<math> L_4 = 0.5\ {C_L}^{-6.738}</math><ref name= BB1991/>
  
−
:<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> N_{FR} = \frac{{v_m}^2}{g_c\ D}</math><ref name="BB" />
  
−
:<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/>
+
'''Determine the flow pattern:'''
  
−
:<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math><ref name= Lyons/>
+
*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" />
 +
*DISTRIBUTED: <math> (C_L < 0.4\ \&\ N_{FR} \ge L_1)\ or\ (C_L\ge0.4\ \&\ N_{FR}>L_4)</math><ref name="BB1991" />
  
−
:<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>
+
'''Calculate <math> H_{L(0)}:</math>'''
 +
*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" />
 +
*DISTRIBUTED: <math> H_{L(0)} = 1.065\frac{ {C_L}^{0.5824} } { {N_{FR}}^{0.0609}}</math><ref name="BB1991" />
  
−
:<math> \sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>
+
:with the constraint <math> H_L \ge C_L</math><ref name="BB1991" />
  
−
:<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> \psi = 1 + C\ (sin(1.8\ \theta) - 0.333\ (sin(1.8\ \theta))^3)</math><ref name="BB1991" />
  
−
:<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> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= BB1991/>
  
−
:<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/>
+
'''C 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" />
 +
*DISTRIBUTED: <math> C=0</math><ref name="BB1991" />
  
−
:<math> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
+
'''C 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" />
  
−
:<math> N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
+
:with the restriction <math> C \ge 0 </math><ref name="BB1991" />
  
−
:<math> N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
+
'''Finally:'''
  
−
:<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/>
+
*SEGREGATED, INTERMITTENT, DISTRIBUTED:
  
−
:<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> H_L = H_{L(0)} \times \psi </math><ref name="BB1991" />
  
−
:<math> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math><ref name= Economides/>
+
*TRANSITION:  
  
−
:<math> \psi = \begin{cases}  
+
:<math> H_L = A \times H_{L(segregated)} + (1-A) \times {H_{L(intermittent)}} </math><ref name="BB1991" />
−
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/>
+
where:
 +
:<math> A = \frac{L_3-N_{FR}}{L_3-L_2}</math><ref name="BB1991" />
  
 
== Modifications  ==
 
== 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/>.
+
1. Force approach gas at low C<sub>L</sub>. If C<sub>L</sub><0.001 Then f'=f.  
  
−
2. Use the [[Griffith correlation]] to define the bubble flow regime<ref name = Economides/> and calculate H<sub>L</sub>.
+
2. Force approach to single phase fluid. If H<sub>L</sub>>1 Then H<sub>L</sub>=1.  
  
−
3. Use [[WCUT| watercut]] instead of [[WOR]] to account for the watercut = 100%.
+
3. Use calculated water density instead of the constant value of 62.4 lbm/ft3.
  
 
== Nomenclature  ==
 
== Nomenclature  ==
  
 +
:<math> A </math> = correlation variable, dimensionless
 
:<math> A_p </math> = flow area, ft2
 
:<math> A_p </math> = flow area, ft2
−
:<math> B </math> = correlation group, dimensionless
 
 
:<math> B </math> = formation factor, bbl/stb
 
:<math> B </math> = formation factor, bbl/stb
−
:<math> C </math> = coefficient for liquid viscosity number, dimensionless
+
:<math> C </math> = correlation variable, dimensionless
 +
:<math> C_L </math> = non-slip liquid holdup factor, dimensionless
 
:<math> D </math> = pipe diameter, ft
 
:<math> D </math> = pipe diameter, ft
 +
:<math> G </math> = total flux weight, lb<sub>m</sub>/ft<sup>2</sup>/sec
 
:<math> h </math> = depth, ft
 
:<math> h </math> = depth, ft
−
:<math> H </math> = correlation group, dimensionless
 
 
:<math> H_L </math> = liquid holdup factor, dimensionless
 
:<math> H_L </math> = liquid holdup factor, dimensionless
 +
:<math> H_{L(0)} </math> = liquid holdup factor when flow is horizontal, dimensionless
 
:<math> f </math> = friction factor, dimensionless
 
:<math> f </math> = friction factor, dimensionless
 +
:<math> f' </math> = corrected friction factor, dimensionless
 
:<math> GLR </math> = gas-liquid ratio, scf/bbl
 
:<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> L_1, L_2, L_3, L_4 </math> = correlation variables, dimensionless
−
:<math> N_D </math> = pipe diameter number, dimensionless
+
:<math> N_FR </math> = Froude 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> N_LV </math> = liquid velocity number, dimensionless
 
:<math> p </math> = pressure, psia
 
:<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_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> q </math> = flow rate, bbl/d - liquid, scf/d - gas
 
:<math> Re </math> = Reynolds number, dimensionless
 
:<math> Re </math> = Reynolds number, dimensionless
 
:<math> R_s </math> = solution gas-oil ratio, scf/stb
 
:<math> R_s </math> = solution gas-oil ratio, scf/stb
 +
:<math> S </math> = correlation variable, dimensionless
 
:<math> SG </math> = specific gravity, dimensionless
 
:<math> SG </math> = specific gravity, dimensionless
 
:<math> T </math> = temperature, °R or °K, follow the subscript
 
:<math> T </math> = temperature, °R or °K, follow the subscript
 
:<math> v </math> = velocity, ft/sec
 
:<math> v </math> = velocity, ft/sec
−
:<math> WOR </math> = water-oil ratio, bbl/bbl
+
:<math> WCUT </math> = watercut, fraction
 +
:<math> y </math> = correlation variable, dimensionless
 
:<math> z </math> = gas compressibility factor, dimensionless
 
:<math> z </math> = gas compressibility factor, dimensionless
  
Line 129: Line 159:
 
:<math> \mu </math> = viscosity, cp
 
:<math> \mu </math> = viscosity, cp
 
:<math> \rho </math> = density, lb<sub>m</sub>/ft<sup>3</sup>
 
:<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> \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> \sigma </math> = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
−
:<math> \psi </math> = secondary correlation factor, dimensionless
+
:<math> \psi </math> = inclination correction factor, dimensionless
 +
:<math> \theta </math> = inclination angle, ° from horizontal
  
 
===Subscripts===
 
===Subscripts===
Line 139: Line 170:
 
L = liquid<BR/>
 
L = liquid<BR/>
 
m = gas/liquid mixture<BR/>
 
m = gas/liquid mixture<BR/>
 +
ns = non-slip<BR/>
 
o = oil<BR/>
 
o = oil<BR/>
 
R = °R<BR/>
 
R = °R<BR/>
Line 163: Line 195:
 
}}</ref>
 
}}</ref>
  
−
<ref name=Economides>{{cite book
+
<ref name=BB1991>{{cite book
−
  |last1= Economides |first1=M.J.
+
  |last1= Brill |first1=J. P.
−
  |last2= Hill |first2=A.D.
+
  |last2=Beggs|first2=H. D.
−
|last3= Economides |first3=C.E.
+
  |title=Two-Phase Flow In Pipes
−
|last4= Zhu |first4=D.
+
  |edition=6
−
  |title=Petroleum Production Systems
+
  |date=1991
−
  |edition=2
+
  |publisher=U. of Tulsa Tulsa
−
  |date=2013
+
  |place=Oklahoma
−
  |publisher=Prentice Hall
+
  |url=https://www.scribd.com/document/130564301/Twophase-Flow-in-Pipes-Beggs-Amp-Brill
−
  |place=Westford, Massachusetts
+
|url-access=subscription
−
  |isbn=978-0-13-703158-0
 
 
}}</ref>
 
}}</ref>
  
Line 209: Line 240:
 
  |place=Houston, TX
 
  |place=Houston, TX
 
  |isbn=0-88415-643-5
 
  |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>
 
}}</ref>
  
Line 223: Line 245:
  
 
[[Category:pengtools]]
 
[[Category:pengtools]]
−
[[Category:PQplot]]
+
[[Category:sPipe]]
 +
 
 +
 
 +
{{#seo:
 +
|title=Beggs and Brill correlation
 +
|titlemode= replace
 +
|keywords=brill wiki, Beggs and Brill, correlation, equation, pipe pressure drop, pipeline sizing, flow rate, fluids flow, Reynolds number, liquid hold up
 +
|description=Beggs and Brill correlation used in pressure drop pipe calculator for pipeline sizing
 +
}}

Latest revision as of 18:10, 3 November 2018

Brief

Beggs and Brill is an empirical two-phase flow correlation published in 1972 [1].

It distinguish between 4 flow regimes.

Beggs and Brill is the default VLP correlation in sPipe.

Beggs and Brill in sPipe Vs GAP

Math & Physics

Fluid flow energy balance

-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\ \frac{p}{144}}}[1]

where

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

Friction factor

No slip Reynolds two phase number:

Re = 1488 \times \frac {\rho_{m,ns} v_m D} { \mu_L\ C_L + \mu_g\ (1-C_L) } [2]

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]

Corrected two phase friction factor:

f' = f \times e^S[2]

where

S = \frac{ ln(y)}{ -0.0523 + 3.182\ ln(y) - 0.8725\ ln(y)^{2} + 0.01853\ ln(y)^{4}}[2]

and

y = \frac{ C_L} { {H_L}^2 } [2]

with constraint:

S = ln (2.2\ y - 1.2), when\ 1 \le y \le 1.2 [2]

Discussion

Why Beggs and Brill?

The best correlation for the horizontal flow.
— pengtools.com

Flow Diagram

HB Block Diagram

Workflow HL

\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} (1-WCUT) + \frac{62.4\ SG_w}{B_w}\ WCUT[5]
\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} [5]
\mu_L = \mu_o (1-WCUT) + \mu_w\ WCUT[5]
\sigma_L = \sigma_o (1-WCUT) + \sigma_w\ WCUT[5]
v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o (1-WCUT) + B_w\ WCUT \right )[5]
v_{SG} = \frac{q_g}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}[5]
G_m = \rho_L \times v_{SL} + \rho_g \times v_{SG}[5]
v_m = v_{SL} + v_{SG}
C_L = \frac{v_{SL}}{v_m}
L_1 = 316\ {C_L}^{0.302}[2]
L_2 = 0.0009252\ {C_L}^{-2.4684}[2]
L_3 = 0.1\ {C_L}^{-1.4516}[2]
L_4 = 0.5\ {C_L}^{-6.738}[2]
N_{FR} = \frac{{v_m}^2}{g_c\ D}[1]

Determine the flow pattern:

  • SEGREGATED: (C_L < 0.01\ \&\ N_{FR}< L_1)\ or\ (C_L\ge0.01\ \&\ N_{FR}<L_2)[2]
  • TRANSITION: C_L \ge 0.01\ \&\ L_2 \le N_{FR} \le L_3[2]
  • INTERMITTENT: (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)[2]
  • DISTRIBUTED: (C_L < 0.4\ \&\ N_{FR} \ge L_1)\ or\ (C_L\ge0.4\ \&\ N_{FR}>L_4)[2]

Calculate H_{L(0)}:

  • SEGREGATED: H_{L(0)} = 0.98 \frac{ {C_L}^{0.4846} } { {N_{FR}}^{0.0868}} [2]
  • INTERMITTENT: H_{L(0)} = 0.845 \frac{ {C_L}^{0.5351} } { {N_{FR}}^{0.0173}}[2]
  • DISTRIBUTED: H_{L(0)} = 1.065\frac{ {C_L}^{0.5824} } { {N_{FR}}^{0.0609}}[2]
with the constraint H_L \ge C_L[2]


\psi = 1 + C\ (sin(1.8\ \theta) - 0.333\ (sin(1.8\ \theta))^3)[2]
N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [2]

C Uphill:

  • SEGREGATED: C = (1-C_L)\ ln( 0.011 \times {N_{LV}}^{3.539}\times {C_L}^{-3.768}\times {F_{FR}}^{-1.614})[2]
  • INTERMITTENT: C = (1-C_L)\ ln( 2.96\times {N_{LV}}^{-0.4473}\times {C_L}^{0.305}\times {F_{FR}}^{0.0978})[2]
  • DISTRIBUTED: C=0[2]

C Downhill:

with the restriction C \ge 0 [2]

Finally:

  • SEGREGATED, INTERMITTENT, DISTRIBUTED:
H_L = H_{L(0)} \times \psi [2]
  • TRANSITION:
H_L = A \times H_{L(segregated)} + (1-A) \times {H_{L(intermittent)}} [2]

where:

A = \frac{L_3-N_{FR}}{L_3-L_2}[2]

Modifications

1. Force approach gas at low CL. If CL<0.001 Then f'=f.

2. Force approach to single phase fluid. If HL>1 Then HL=1.

3. Use calculated water density instead of the constant value of 62.4 lbm/ft3.

Nomenclature

A = correlation variable, dimensionless
A_p = flow area, ft2
B = formation factor, bbl/stb
C = correlation variable, dimensionless
C_L = non-slip liquid holdup factor, dimensionless
D = pipe diameter, ft
G = total flux weight, lbm/ft2/sec
h = depth, ft
H_L = liquid holdup factor, dimensionless
H_{L(0)} = liquid holdup factor when flow is horizontal, dimensionless
f = friction factor, dimensionless
f' = corrected friction factor, dimensionless
GLR = gas-liquid ratio, scf/bbl
L_1, L_2, L_3, L_4 = correlation variables, dimensionless
N_FR = Froude number, dimensionless
N_LV = liquid velocity number, dimensionless
p = pressure, psia
q_c = conversion constant equal to 32.174049, lbmft / lbfsec2
q = flow rate, bbl/d - liquid, scf/d - gas
Re = Reynolds number, dimensionless
R_s = solution gas-oil ratio, scf/stb
S = correlation variable, dimensionless
SG = specific gravity, dimensionless
T = temperature, °R or °K, follow the subscript
v = velocity, ft/sec
WCUT = watercut, fraction
y = correlation variable, dimensionless
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 = inclination correction factor, dimensionless
\theta = inclination angle, ° from horizontal

Subscripts

g = gas
K = °K
L = liquid
m = gas/liquid mixture
ns = non-slip
o = oil
R = °R
SL = superficial liquid
SG = superficial gas
w = water

References

  1. ↑ 1.0 1.1 1.2 1.3 Beggs, H. D.; Brill, J. P. (May 1973). "A Study of Two-Phase Flow in Inclined Pipes"Paid subscription required. Journal of Petroleum Technology. AIME. 255 (SPE-4007-PA). 
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 Brill, J. P.; Beggs, H. D. (1991). Two-Phase Flow In PipesPaid subscription required (6 ed.). Oklahoma: U. of Tulsa Tulsa. 
  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 5.6 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5. 


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