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The [[Laplace transformation for solving transient flow problems|Laplace transform]] of the [[Source function solutions of the diffusion equation|diffusion equation]] in radial coordinates yields a modified Bessel’s equation, and its solutions are obtained in terms of modified Bessel functions. This page introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems.
The [[Laplace_transformation_for_solving_transient_flow_problems|Laplace transform]] of the [[Source_function_solutions_of_the_diffusion_equation|diffusion equation]] in radial coordinates yields a modified Bessel’s equation, and its solutions are obtained in terms of modified Bessel functions. This page introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems.
 
== Preliminary definitions ==


==Preliminary definitions==
A differential equation of the type
A differential equation of the type


[[File:Vol1 page 0083 eq 001.png]]....................(1)
[[File:Vol1 page 0083 eq 001.png|RTENOTITLE]]....................(1)


is called a Bessel’s equation of order ''v''. A solution of Bessel’s equation of order ''v'' is called a Bessel function of order ''v''. A differential equation of the type
is called a Bessel’s equation of order ''v''. A solution of Bessel’s equation of order ''v'' is called a Bessel function of order ''v''. A differential equation of the type


[[File:Vol1 page 0083 eq 002.png]]....................(2)
[[File:Vol1 page 0083 eq 002.png|RTENOTITLE]]....................(2)


is called a modified Bessel’s equation of order ''v''. '''Eq. 2''' is obtained by substituting ''λz'' for ''z'' in '''Eq. 1'''. Of particular interest is the case in which ''λ''=''ki'' so that '''Eq. 2''' becomes
is called a modified Bessel’s equation of order ''v''. '''Eq. 2''' is obtained by substituting ''λz'' for ''z'' in '''Eq. 1'''. Of particular interest is the case in which ''λ''=''ki'' so that '''Eq. 2''' becomes


[[File:Vol1 page 0083 eq 003.png]]....................(3)
[[File:Vol1 page 0083 eq 003.png|RTENOTITLE]]....................(3)


'''Eq. 3''' is called the modified Bessel’s equation of order ''v''. A solution of the modified Bessel’s equation of order ''v'' is called a modified Bessel function of order ''v''.
'''Eq. 3''' is called the modified Bessel’s equation of order ''v''. A solution of the modified Bessel’s equation of order ''v'' is called a modified Bessel function of order ''v''.


==Solutions of Bessel’s equations and Bessel functions==
== Solutions of Bessel’s equations and Bessel functions ==
There are many methods of obtaining or constructing Bessel functions.<ref name="r1" /> Only the final form of the Bessel functions that are of interest are presented here.
 
There are many methods of obtaining or constructing Bessel functions.<ref name="r1">Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press.</ref> Only the final form of the Bessel functions that are of interest are presented here.


If ''v'' is not a positive integer, then the general solution of Bessel’s equation of order ''v'' ('''Eq. 1''') is given by
If ''v'' is not a positive integer, then the general solution of Bessel’s equation of order ''v'' ('''Eq. 1''') is given by


[[File:Vol1 page 0083 eq 004.png]]....................(4)
[[File:Vol1 page 0083 eq 004.png|RTENOTITLE]]....................(4)


where ''A'' and ''B'' are arbitrary constants, and ''J''<sub>''v''</sub>(''z'') is the Bessel function of order ''v'' of the first kind given by
where ''A'' and ''B'' are arbitrary constants, and ''J''<sub>''v''</sub>(''z'') is the Bessel function of order ''v'' of the first kind given by


[[File:Vol1 page 0083 eq 005.png]]....................(5)
[[File:Vol1 page 0083 eq 005.png|RTENOTITLE]]....................(5)


In '''Eq. 5''', Γ(''x'') is the gamma function defined by
In '''Eq. 5''', Γ(''x'') is the gamma function defined by


[[File:Vol1 page 0083 eq 006.png]]....................(6)
[[File:Vol1 page 0083 eq 006.png|RTENOTITLE]]....................(6)


If ''v'' is a positive integer, ''n'', then ''J''<sub>''v''</sub> and ''J''<sub>''-v''</sub> are linearly dependent, and the solution of '''Eq. 1''' is written as
If ''v'' is a positive integer, ''n'', then ''J''<sub>''v''</sub> and ''J''<sub>''-v''</sub> are linearly dependent, and the solution of '''Eq. 1''' is written as


[[File:Vol1 page 0083 eq 007.png]]....................(7)
[[File:Vol1 page 0083 eq 007.png|RTENOTITLE]]....................(7)


In '''Eq. 7''', ''Y''<sub>''n''</sub>(''z'') is the Bessel function of order ''n'' of the second kind and is defined by
In '''Eq. 7''', ''Y''<sub>''n''</sub>(''z'') is the Bessel function of order ''n'' of the second kind and is defined by


[[File:Vol1 page 0084 eq 001.png]]....................(8)
[[File:Vol1 page 0084 eq 001.png|RTENOTITLE]]....................(8)


Similarly, if ''v'' is not a positive integer, the general solution of the modified Bessel’s equation of order ''v'' ('''Eq. 3''') is given by
Similarly, if ''v'' is not a positive integer, the general solution of the modified Bessel’s equation of order ''v'' ('''Eq. 3''') is given by


[[File:Vol1 page 0084 eq 002.png]]....................(9)
[[File:Vol1 page 0084 eq 002.png|RTENOTITLE]]....................(9)


where ''I''<sub>''v''</sub>(''z'') is the modified Bessel function of order ''v'' of the first kind defined by
where ''I''<sub>''v''</sub>(''z'') is the modified Bessel function of order ''v'' of the first kind defined by


[[File:Vol1 page 0084 eq 003.png]]....................(10)
[[File:Vol1 page 0084 eq 003.png|RTENOTITLE]]....................(10)


If ''v'' is a positive integer, ''n'', ''I''<sub>''v''</sub>, and ''I''<sub>''−v''</sub> are linearly dependent. The solution for this case is
If ''v'' is a positive integer, ''n'', ''I''<sub>''v''</sub>, and ''I''<sub>''−v''</sub> are linearly dependent. The solution for this case is


[[File:Vol1 page 0084 eq 004.png]]....................(11)
[[File:Vol1 page 0084 eq 004.png|RTENOTITLE]]....................(11)


where ''K''<sub>''n''</sub>(''z'') is the modified Bessel function of order ''n'' of the second kind and is defined by
where ''K''<sub>''n''</sub>(''z'') is the modified Bessel function of order ''n'' of the second kind and is defined by


[[File:Vol1 page 0084 eq 005.png]]....................(12)
[[File:Vol1 page 0084 eq 005.png|RTENOTITLE]]....................(12)


The modified Bessel functions of order zero and one are of special interest, and the section below discusses some of their special features.
The modified Bessel functions of order zero and one are of special interest, and the section below discusses some of their special features.


==Modified Bessel functions of order zero and one==
== Modified Bessel functions of order zero and one ==
 
Modified Bessel functions of order zero and one are related to each other by the following relations:<ref>Bowman, F. 1958. Introduction to Bessel Functions, Dover Publications, Inc. New York.</ref>
Modified Bessel functions of order zero and one are related to each other by the following relations:<ref>Bowman, F. 1958. Introduction to Bessel Functions, Dover Publications, Inc. New York.</ref>


[[File:Vol1 page 0084 eq 006.png]]....................(13)
[[File:Vol1 page 0084 eq 006.png|RTENOTITLE]]....................(13)


and
and


[[File:Vol1 page 0084 eq 007.png]]....................(14)
[[File:Vol1 page 0084 eq 007.png|RTENOTITLE]]....................(14)


'''Fig. 1''' shows these functions graphically.
'''Fig. 1''' shows these functions graphically.
Line 72: Line 75:
</gallery>
</gallery>


For small arguments, the following asymptotic expansions may be used for the modified Bessel functions of order zero and one:<ref name="r1" /><ref>Abramowitz, M. and Stegun, I. A., eds. 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover.</ref><ref>Carslaw, H.S. and Jaeger, J.C. 1986. Conduction of heat in solids, 2nd. Oxford Oxfordshire New York: Clarendon Press ; Oxford University Press. 85026963</ref>
For small arguments, the following asymptotic expansions may be used for the modified Bessel functions of order zero and one:<ref name="r1">Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press.</ref><ref>Abramowitz, M. and Stegun, I. A., eds. 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover.</ref><ref>Carslaw, H.S. and Jaeger, J.C. 1986. Conduction of heat in solids, 2nd. Oxford Oxfordshire New York: Clarendon Press ; Oxford University Press. 85026963</ref><ref>Spanier, J., Myland, J. and Oldham, K. B. 2009. An Atlas of Functions. Washington, DC: Hemisphere Publishing Corporation, Washington DC, Springer-Verlag, Berlin.</ref>


[[File:Vol1 page 0084 eq 008.png]]....................(15)
[[File:Vol1 page 0084 eq 008.png|RTENOTITLE]]....................(15)


[[File:Vol1 page 0085 eq 001.png]]....................(16)
[[File:Vol1 page 0085 eq 001.png|RTENOTITLE]]....................(16)


[[File:Vol1 page 0085 eq 002.png]]....................(17)
[[File:Vol1 page 0085 eq 002.png|RTENOTITLE]]....................(17)


where ''γ'' = 0.5772…, and
where ''γ'' = 0.5772…, and


[[File:Vol1 page 0085 eq 003.png]]....................(18)
[[File:Vol1 page 0085 eq 003.png|RTENOTITLE]]....................(18)


Also, for large arguments, the following relations may be useful:
Also, for large arguments, the following relations may be useful:


[[File:Vol1 page 0085 eq 004.png]]....................(19)
[[File:Vol1 page 0085 eq 004.png|RTENOTITLE]]....................(19)


for |''arg'' ''z''| < ''π'' / 2, and
for |''arg'' ''z''| < ''π'' / 2, and


[[File:Vol1 page 0085 eq 005.png]]....................(20)
[[File:Vol1 page 0085 eq 005.png|RTENOTITLE]]....................(20)


for |''arg'' ''z''| < 3''π'' / 2. On the basis of the relations given by '''Eqs. 15''' through '''20''', the following limiting forms may be written:
for |''arg'' ''z''| < 3''π'' / 2. On the basis of the relations given by '''Eqs. 15''' through '''20''', the following limiting forms may be written:


[[File:Vol1 page 0086 eq 001.png]]....................(21)
[[File:Vol1 page 0086 eq 001.png|RTENOTITLE]]....................(21)


[[File:Vol1 page 0086 eq 002.png]]....................(22)
[[File:Vol1 page 0086 eq 002.png|RTENOTITLE]]....................(22)


[[File:Vol1 page 0086 eq 003.png]]....................(23)
[[File:Vol1 page 0086 eq 003.png|RTENOTITLE]]....................(23)


[[File:Vol1 page 0086 eq 004.png]]....................(24)
[[File:Vol1 page 0086 eq 004.png|RTENOTITLE]]....................(24)


[[File:Vol1 page 0086 eq 005.png]]....................(25)
[[File:Vol1 page 0086 eq 005.png|RTENOTITLE]]....................(25)


[[File:Vol1 page 0086 eq 006.png]]....................(26)
[[File:Vol1 page 0086 eq 006.png|RTENOTITLE]]....................(26)


[[File:Vol1 page 0086 eq 007.png]]....................(27)
[[File:Vol1 page 0086 eq 007.png|RTENOTITLE]]....................(27)


[[File:Vol1 page 0086 eq 008.png]]....................(28)
[[File:Vol1 page 0086 eq 008.png|RTENOTITLE]]....................(28)


and
and


[[File:Vol1 page 0086 eq 009.png]]....................(29)
[[File:Vol1 page 0086 eq 009.png|RTENOTITLE]]....................(29)


These relations are useful in the evaluation of the asymptotic behavior of transient pressure solutions.
These relations are useful in the evaluation of the asymptotic behavior of transient pressure solutions.


==Nomenclature==
== Nomenclature ==
 
{|
{|
|''d''
|=
|distance to a linear boundary, cm
|-
|-
|''I''<sub>''v''</sub>(''x'')
| ''d''
|=  
| =
|modified Bessel function of the first kind of order ''v''
| distance to a linear boundary, cm
|-
|-
|''J''<sub>''v''</sub>(''x'')  
| ''I''<sub>''v''</sub>(''x'')
|=  
| =
|Bessel function of the first kind of order ''v''
| modified Bessel function of the first kind of order ''v''
|-
|-
|''k''  
| ''J''<sub>''v''</sub>(''x'')
|=  
| =
|isotropic permeability, md
| Bessel function of the first kind of order ''v''
|-
|-
|''K''<sub>''n''</sub>(''x'')
| ''k''
|=  
| =
|modified Bessel function of the second kind of order ''n''
| isotropic permeability, md
|-
|-
|''m''  
| ''K''<sub>''n''</sub>(''x'')
|=  
| =
|pseudopressure, atm<sup>2</sup>/cp
| modified Bessel function of the second kind of order ''n''
|-
|-
|''t''  
| ''m''
|=  
| =
|time, s
| pseudopressure, atm<sup>2</sup>/cp
|-
|-
|''V''  
| ''t''
|=  
| =
|volume, cm<sup>3</sup>
| time, s
|-
|-
|''y''  
| ''V''
|=  
| =
|distance in ''y''-direction, cm
| volume, cm<sup>3</sup>
|-
|-
|''Y''<sub>''n''</sub>(''x'')
| ''y''
|=  
| =
|Bessel function of the second kind of order ''n''
| distance in ''y''-direction, cm
|-
|-
|''z''  
| ''Y''<sub>''n''</sub>(''x'')
|=  
| =
|distance in ''z''-direction, cm
| Bessel function of the second kind of order ''n''
|-
|-
|Γ
| ''z''
|=  
| =
|boundary surface, cm<sup>2</sup>
| distance in ''z''-direction, cm
|-
|-
(''x'')
| Γ
|=  
| =
|Gamma function
| boundary surface, cm<sup>2</sup>
|-
|-
| Γ(''x'')
| =
| Gamma function
|}
|}


==References==
== References ==
<references>
 
<ref name="r1">Watson, G. N. 1944. ''A Treatise on the Theory of Bessel Functions''. London: Cambridge University Press.</ref>
<references />
</references>


==Noteworthy papers in OnePetro==
== See also ==


Carslaw, H.S. and Jaeger, J.C. 1986. Conduction of heat in solids, 2nd. Oxford Oxfordshire New York: Clarendon Press ;
[[Transient_analysis_mathematics|Transient analysis mathematics]]
Oxford University Press. 85026963


Spanier, J., Myland, J. and Oldham, K. B.  2009. An Atlas of Functions. Washington, DC: Hemisphere Publishing Corporation, Washington DC, Springer-Verlag, Berlin.
[[Laplace_transformation_for_solving_transient_flow_problems|Laplace transformation for solving transient flow problems]]


==See also==
[[Green’s_function_for_solving_transient_flow_problems|Green’s function for solving transient flow problems]]
[[Transient analysis mathematics]]


[[Laplace transformation for solving transient flow problems]]
[[Source_function_solutions_of_the_diffusion_equation|Source function solutions of the diffusion equation]]


[[Green’s function for solving transient flow problems]]
[[Solving_unsteady_flow_problems_with_Green's_and_source_functions|Solving unsteady flow problems with Green's and source functions]]


[[Source function solutions of the diffusion equation]]
[[Solving_unsteady_flow_problems_with_Laplace_transform_and_source_functions|Solving unsteady flow problems with Laplace transform and source functions]]


[[Solving unsteady flow problems with Green's and source functions]]
[[Mathematics_of_fluid_flow|Mathematics of fluid flow]]


[[Solving unsteady flow problems with Laplace transform and source functions]]
[[Differential_calculus_refresher|Differential calculus refresher]]


[[Mathematics of fluid flow]]
[[PEH:Mathematics_of_Transient_Analysis]]


[[Differential calculus refresher]]
==Category==


[[PEH:Mathematics of Transient Analysis]]
[[Category:5.6.3 Pressure transient analysis]] [[Category:YR]]

Latest revision as of 12:44, 6 July 2015

The Laplace transform of the diffusion equation in radial coordinates yields a modified Bessel’s equation, and its solutions are obtained in terms of modified Bessel functions. This page introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems.

Preliminary definitions

A differential equation of the type

RTENOTITLE....................(1)

is called a Bessel’s equation of order v. A solution of Bessel’s equation of order v is called a Bessel function of order v. A differential equation of the type

RTENOTITLE....................(2)

is called a modified Bessel’s equation of order v. Eq. 2 is obtained by substituting λz for z in Eq. 1. Of particular interest is the case in which λ=ki so that Eq. 2 becomes

RTENOTITLE....................(3)

Eq. 3 is called the modified Bessel’s equation of order v. A solution of the modified Bessel’s equation of order v is called a modified Bessel function of order v.

Solutions of Bessel’s equations and Bessel functions

There are many methods of obtaining or constructing Bessel functions.[1] Only the final form of the Bessel functions that are of interest are presented here.

If v is not a positive integer, then the general solution of Bessel’s equation of order v (Eq. 1) is given by

RTENOTITLE....................(4)

where A and B are arbitrary constants, and Jv(z) is the Bessel function of order v of the first kind given by

RTENOTITLE....................(5)

In Eq. 5, Γ(x) is the gamma function defined by

RTENOTITLE....................(6)

If v is a positive integer, n, then Jv and J-v are linearly dependent, and the solution of Eq. 1 is written as

RTENOTITLE....................(7)

In Eq. 7, Yn(z) is the Bessel function of order n of the second kind and is defined by

RTENOTITLE....................(8)

Similarly, if v is not a positive integer, the general solution of the modified Bessel’s equation of order v (Eq. 3) is given by

RTENOTITLE....................(9)

where Iv(z) is the modified Bessel function of order v of the first kind defined by

RTENOTITLE....................(10)

If v is a positive integer, n, Iv, and I−v are linearly dependent. The solution for this case is

RTENOTITLE....................(11)

where Kn(z) is the modified Bessel function of order n of the second kind and is defined by

RTENOTITLE....................(12)

The modified Bessel functions of order zero and one are of special interest, and the section below discusses some of their special features.

Modified Bessel functions of order zero and one

Modified Bessel functions of order zero and one are related to each other by the following relations:[2]

RTENOTITLE....................(13)

and

RTENOTITLE....................(14)

Fig. 1 shows these functions graphically.

For small arguments, the following asymptotic expansions may be used for the modified Bessel functions of order zero and one:[1][3][4][5]

RTENOTITLE....................(15)

RTENOTITLE....................(16)

RTENOTITLE....................(17)

where γ = 0.5772…, and

RTENOTITLE....................(18)

Also, for large arguments, the following relations may be useful:

RTENOTITLE....................(19)

for |argz| < π / 2, and

RTENOTITLE....................(20)

for |argz| < 3π / 2. On the basis of the relations given by Eqs. 15 through 20, the following limiting forms may be written:

RTENOTITLE....................(21)

RTENOTITLE....................(22)

RTENOTITLE....................(23)

RTENOTITLE....................(24)

RTENOTITLE....................(25)

RTENOTITLE....................(26)

RTENOTITLE....................(27)

RTENOTITLE....................(28)

and

RTENOTITLE....................(29)

These relations are useful in the evaluation of the asymptotic behavior of transient pressure solutions.

Nomenclature

d = distance to a linear boundary, cm
Iv(x) = modified Bessel function of the first kind of order v
Jv(x) = Bessel function of the first kind of order v
k = isotropic permeability, md
Kn(x) = modified Bessel function of the second kind of order n
m = pseudopressure, atm2/cp
t = time, s
V = volume, cm3
y = distance in y-direction, cm
Yn(x) = Bessel function of the second kind of order n
z = distance in z-direction, cm
Γ = boundary surface, cm2
Γ(x) = Gamma function

References

  1. 1.0 1.1 Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press.
  2. Bowman, F. 1958. Introduction to Bessel Functions, Dover Publications, Inc. New York.
  3. Abramowitz, M. and Stegun, I. A., eds. 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover.
  4. Carslaw, H.S. and Jaeger, J.C. 1986. Conduction of heat in solids, 2nd. Oxford Oxfordshire New York: Clarendon Press ; Oxford University Press. 85026963
  5. Spanier, J., Myland, J. and Oldham, K. B. 2009. An Atlas of Functions. Washington, DC: Hemisphere Publishing Corporation, Washington DC, Springer-Verlag, Berlin.

See also

Transient analysis mathematics

Laplace transformation for solving transient flow problems

Green’s function for solving transient flow problems

Source function solutions of the diffusion equation

Solving unsteady flow problems with Green's and source functions

Solving unsteady flow problems with Laplace transform and source functions

Mathematics of fluid flow

Differential calculus refresher

PEH:Mathematics_of_Transient_Analysis

Category