Difference between revisions of "Hagedorn and Brown correlation"
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== Brief == | == Brief == | ||
− | [[Hagedorn and Brown]] is an empirical two-phase flow correlation published in '''1965'''. | + | [[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 | + | 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: | 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 = | + | :<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="HB" /> |
− | Colebrook–White equation for the 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: | 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 == | ||
+ | |||
+ | [[File: HB Block Diagram.png|400px|HB Block Diagram]] | ||
− | == Workflow == | + | == 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> | + | :<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> | + | :<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> | + | :<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> | + | :<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/> |
− | :<math> \ | + | :<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> | + | :<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> | + | :<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} </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_{SG} </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> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\ | + | :<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}{\ | + | :<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}{\ | + | :<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> | + | :<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> | + | :<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> | + | :<math> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math><ref name= Economides/> |
:<math> \psi = \begin{cases} | :<math> \psi = \begin{cases} | ||
− | 27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999, & \mbox{if }B\ | + | 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 | + | -533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\ |
− | 2.5714 B +1.5962, & \mbox{if }B | + | 2.5714 B +1.5962, & \mbox{if }B > 0.055 |
− | \end{cases} </math> | + | \end{cases} </math><ref name= Trina/> |
− | :<math> H_L </math> | + | :<math> H_L = \frac{H_L}{\psi} \times \psi</math><ref name= HB/> |
+ | |||
+ | == 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 == | == Nomenclature == | ||
− | :<math> | + | :<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=== | ||
+ | |||
+ | :<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 | ||
+ | |||
+ | ===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 == | ||
+ | <references> | ||
+ | |||
+ | <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 | ||
+ | |number=SPE-940-PA | ||
+ | |date=1965 | ||
+ | |volume=17(04) | ||
+ | |pages=475-484 | ||
+ | |url=https://www.onepetro.org/journal-paper/SPE-940-PA | ||
+ | |url-access=registration | ||
+ | }}</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> | ||
+ | |||
+ | [[Category:pengtools]] | ||
+ | [[Category:PQplot]] | ||
− | + | {{#seo: | |
− | + | |title=Hagedorn and Brown correlation | |
+ | |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
Contents
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.
Math & Physics
Following the law of conservation of energy the basic steady state flow equation is:
where
Colebrook–White [3] equation for the Darcy's friction factor:
Reynolds two phase number:
Discussion
Why Hagedorn and Brown?
One of the consistently best correlations ...— Michael Economides et al[2]
Demo
Hagedorn and Brown correlation overview video:
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
Workflow HL
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
- = flow area, ft2
- = correlation group, dimensionless
- = formation factor, bbl/stb
- = coefficient for liquid viscosity number, dimensionless
- = pipe diameter, ft
- = depth, ft
- = correlation group, dimensionless
- = liquid holdup factor, fraction
- = friction factor, dimensionless
- = gas-liquid ratio, scf/bbl
- = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lbm/bbl
- = pipe diameter number, dimensionless
- = gas velocity number, dimensionless
- = liquid viscosity number, dimensionless
- = liquid velocity number, dimensionless
- = pressure, psia
- = conversion constant equal to 32.174049, lbmft / lbfsec2
- = total liquid production rate, bbl/d
- = Reynolds number, dimensionless
- = solution gas-oil ratio, scf/stb
- = specific gravity, dimensionless
- = temperature, °R or °K, follow the subscript
- = velocity, ft/sec
- = water-oil ratio, bbl/bbl
- = gas compressibility factor, dimensionless
Greek symbols
- = absolute roughness, ft
- = viscosity, cp
- = density, lbm/ft3
- = integrated average density at flowing conditions, lbm/ft3
- = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
- = 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
- ↑ 1.0 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". Journal of Petroleum Technology. 17(04) (SPE-940-PA): 475–484.
- ↑ 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.
- ↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws". Journal of the Institution of Civil Engineers. London, England. 11: 133–156.
- ↑ Moody, L. F. (1944). "Friction factors for pipe flow". Transactions of the ASME. 66 (8): 671–684.
- ↑ 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.
- ↑ 6.0 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.