Difference between revisions of "18.41 derivation"

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(Created page with "== Brief== '''18.41''' is the well know constant which is used for converting from the Darcy's law units to the metric units in the well's inflow equations. For example...")
 
 
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'''18.41''' is the well know constant which is used for converting from the [[Darcy's law]] units to the metric units in the well's inflow equations.
 
'''18.41''' is the well know constant which is used for converting from the [[Darcy's law]] units to the metric units in the well's inflow equations.
  
−
For example [[Darcy's law]] for the single-phase flow is as follows<ref name=KermitBrown/>:
+
For example [[Darcy's law]] for the single-phase flow is as follows<ref name=DW/>:
  
 
:<math> q_o = \frac{1}{18.41} \times \frac{k_oh}{B_o\mu_o} \times \Delta P \times J_D</math>
 
:<math> q_o = \frac{1}{18.41} \times \frac{k_oh}{B_o\mu_o} \times \Delta P \times J_D</math>
  
−
The derivation of the 141.2 constant is given below.
+
The derivation of the 18.41 constant is given below.
  
 
== Math and Physics ==
 
== Math and Physics ==
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:<math> \frac{[cm^3]}{[sec]} = - \frac{[D][cm^2]}{[cP]} \frac{[atm]}{[cm]}</math>
 
:<math> \frac{[cm^3]}{[sec]} = - \frac{[D][cm^2]}{[cP]} \frac{[atm]}{[cm]}</math>
  
−
Converting to the field units:
+
Converting to the metric units:
  
−
:<math> \frac{[cm^3] \frac{[ft^3]}{[30.48^3 cm^3]} \frac{[bbl]}{[5.61458333 ft^3]} }{[sec] \frac{[day]}{[86400 sec]}} = - \frac{[D] \frac{[1000 mD]}{[D]}[cm^2] \frac{[ft^2]}{[30.48^2 cm]}}{[cP]} \frac{[atm] \frac{[14.695950253959 psia]}{[atm]}}{[cm] \frac{[ft]}{[30.48 cm]}}</math>
+
:<math> \frac{[cm^3] \frac{[m^3]}{[1000000 cm^3]}}{[sec] \frac{[day]}{[86400 sec]}} = - \frac{[D] \frac{[1000 mD]}{[D]}[cm^2] \frac{[m^2]}{[10000 cm^2]}}{[cP]} \frac{[atm]}{[cm] \frac{[m]}{[100 cm]}}</math>
  
 
So:
 
So:
  
−
:<math> \frac{[bbl]}{[day]} \frac{86400}{30.48^3\ 5.61458333} = - \frac{[mD][ft^2]}{[cP]} \frac{[psia]}{[ft]} \frac{1000\ 14.695950253959}{30.48}</math>
+
:<math> \frac{[m^3]}{[day]} \frac{86400}{1000000} = - \frac{[mD][m^2]}{[cP]} \frac{[atm]}{[m]} \frac{1000 \times 100}{10000}</math>
  
 
And:
 
And:
  
−
:<math> \frac{[bbl]}{[day]} = - C_{LF} \frac{[mD][ft^2]}{[cP]} \frac{[psia]}{[ft]}</math>
+
:<math> \frac{[m^3]}{[day]} = - C_{LF} \frac{[mD][m^2]}{[cP]} \frac{[atm]}{[m]}</math>
  
 
where  
 
where  
  
−
:<math> C_{LF} = \frac{1000\ 14.695950253959}{30.48} \frac{30.48^3\ 5.61458333}{86400} = \frac{1000\ 14.695950253959\ 30.48^2\ 5.61458333}{86400} = 887.2201322</math>
+
:<math> C_{LF} = \frac{1000 \times 100}{10000} \frac{1000000}{86400} = \frac{10000000}{86400} = 115.7407407</math>
  
 
For the radial flow:
 
For the radial flow:
  
−
:<math> C_{RF} =  \frac{C_{LF}}{2\pi} = \frac{887.2201322}{2\pi} = 141.20546964</math>
+
:<math> C_{RF} =  \frac{C_{LF}}{2\pi} = \frac{115.7407407}{2\pi} = 18.42 </math>
  
−
One can be familiar with the inverse of the '''141.2''' constant:
+
Note that the published<ref name=DW/> and commonly used constant in '''18.41'''.
−
 
 
−
:<math> \frac{1}{C_{RF}} =  \frac{1}{141.20546964} = 7.08E-03 </math>
 
−
 
 
−
==See Also==
 
−
[[Darcy's law]]<BR/>
 
−
[[141.2 derivation]]
 
−
[[18.41 derivation]]
 
  
 
== Nomenclature  ==
 
== Nomenclature  ==
  
 
:<math> A </math> = [[Darcy's law]] cross-sectional area, cm<sup>2</sup>
 
:<math> A </math> = [[Darcy's law]] cross-sectional area, cm<sup>2</sup>
−
:<math> B_o </math> = oil formation volume factor, bbl/stb
+
:<math> B_o </math> = oil formation volume factor, m^3/m^3
 
:<math> C_{LF} </math> = linear flow units conversion constant  
 
:<math> C_{LF} </math> = linear flow units conversion constant  
 
:<math> C_{RF} </math> = radial flow units conversion constant
 
:<math> C_{RF} </math> = radial flow units conversion constant
−
:<math> h</math> = effective feet of oil pay, ft
+
:<math> h</math> = effective feet of oil pay, m
 
:<math> J_D </math> = dimensionless productivity index, dimensionless
 
:<math> J_D </math> = dimensionless productivity index, dimensionless
 
:<math> k</math> = [[Darcy's law]] permeability, d
 
:<math> k</math> = [[Darcy's law]] permeability, d
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:<math> L </math> = [[Darcy's law]] length, cm
 
:<math> L </math> = [[Darcy's law]] length, cm
 
:<math> P </math> = [[Darcy's law]] pressure, atm
 
:<math> P </math> = [[Darcy's law]] pressure, atm
−
:<math> \Delta P </math> = drawdown, psia
+
:<math> \Delta P </math> = drawdown, atm
 
:<math> q </math> = [[Darcy's law]] flow rate, cm<sup>3</sup>/sec
 
:<math> q </math> = [[Darcy's law]] flow rate, cm<sup>3</sup>/sec
−
:<math> q_o </math> = oil flow rate, stb/d
+
:<math> q_o </math> = oil flow rate, m^3/d
  
 
===Greek symbols===
 
===Greek symbols===
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:<math> \mu </math> = [[Darcy's law]] oil viscosity, cp
 
:<math> \mu </math> = [[Darcy's law]] oil viscosity, cp
 
:<math> \mu_o </math> = oil viscosity, cp
 
:<math> \mu_o </math> = oil viscosity, cp
 +
 +
==See Also==
 +
[[Darcy's law]]<BR/>
 +
[[141.2 derivation]]<BR/>
 +
[[18.41 derivation]]
  
 
==References==
 
==References==
  
 
<references>
 
<references>
−
 
+
<ref name=DW>
−
<ref name=KermitBrown>{{cite book
+
{{cite book
−
  |last1=Brown|first1= Kermit
+
  |last1= Wolcott |first1=Don
−
  |title=The Technology of Artificial Lift Methods. Volume 4. Production Optimization of Oil and Gas Wells by Nodal System Analysis
+
  |title=Applied Waterflood Field Development
−
  |publisher=PennWellBooks
+
  |date=2009
−
  |volume=4
+
  |publisher=Energy Tribune Publishing Inc
−
  |date=1984
+
  |place=Houston
−
  |place=Tulsa, Oklahoma
+
  |url=https://www.amazon.com/Applied-Waterflood-Field-Development-Wolcott/dp/0578023946/ref=sr_1_1?ie=UTF8&qid=1481788841&sr=8-1&keywords=Don+wolcott
 +
  |url-access=subscription
 
}}</ref>
 
}}</ref>
−
 
  
 
</references>
 
</references>
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|description=18.41 is the well know constant which is used for converting from the Darcy's law units to the metric units in the well's inflow equations.
 
|description=18.41 is the well know constant which is used for converting from the Darcy's law units to the metric units in the well's inflow equations.
 
}}
 
}}
 +
<div style='text-align: right;'>By Mikhail Tuzovskiy on {{REVISIONTIMESTAMP}}</div>

Latest revision as of 12:32, 12 July 2023

Brief

18.41 is the well know constant which is used for converting from the Darcy's law units to the metric units in the well's inflow equations.

For example Darcy's law for the single-phase flow is as follows[1]:

q_o = \frac{1}{18.41} \times \frac{k_oh}{B_o\mu_o} \times \Delta P \times J_D

The derivation of the 18.41 constant is given below.

Math and Physics

Darcy's law:

q = -\frac{kA}{\mu} \frac{dP}{dL}

In Darcy's units:


\frac{[cm^3]}{[sec]} = - \frac{[D][cm^2]}{[cP]} \frac{[atm]}{[cm]}

Converting to the metric units:

\frac{[cm^3] \frac{[m^3]}{[1000000 cm^3]}}{[sec] \frac{[day]}{[86400 sec]}} = - \frac{[D] \frac{[1000 mD]}{[D]}[cm^2] \frac{[m^2]}{[10000 cm^2]}}{[cP]} \frac{[atm]}{[cm] \frac{[m]}{[100 cm]}}

So:

\frac{[m^3]}{[day]} \frac{86400}{1000000} = - \frac{[mD][m^2]}{[cP]} \frac{[atm]}{[m]} \frac{1000 \times 100}{10000}

And:

\frac{[m^3]}{[day]} = - C_{LF} \frac{[mD][m^2]}{[cP]} \frac{[atm]}{[m]}

where

C_{LF} = \frac{1000 \times 100}{10000} \frac{1000000}{86400} = \frac{10000000}{86400} = 115.7407407

For the radial flow:

C_{RF} = \frac{C_{LF}}{2\pi} = \frac{115.7407407}{2\pi} = 18.42

Note that the published[1] and commonly used constant in 18.41.

Nomenclature

A = Darcy's law cross-sectional area, cm2
B_o = oil formation volume factor, m^3/m^3
C_{LF} = linear flow units conversion constant
C_{RF} = radial flow units conversion constant
h = effective feet of oil pay, m
J_D = dimensionless productivity index, dimensionless
k = Darcy's law permeability, d
k_o = effective permeability to oil, md
L = Darcy's law length, cm
P = Darcy's law pressure, atm
\Delta P = drawdown, atm
q = Darcy's law flow rate, cm3/sec
q_o = oil flow rate, m^3/d

Greek symbols

\mu = Darcy's law oil viscosity, cp
\mu_o = oil viscosity, cp

See Also

Darcy's law
141.2 derivation
18.41 derivation

References

  1. ↑ 1.0 1.1 Wolcott, Don (2009). Applied Waterflood Field DevelopmentPaid subscription required. Houston: Energy Tribune Publishing Inc. 
By Mikhail Tuzovskiy on 20230712123245
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