You must log in to edit PetroWiki. Help with editing

Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. More information


Bessel functions in transient analysis: Difference between revisions

PetroWiki
Jump to navigation Jump to search
No edit summary
No edit summary
 
Line 19: Line 19:
== 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">_</ref> 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
Line 75: 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><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>
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|RTENOTITLE]]....................(15)
[[File:Vol1 page 0084 eq 008.png|RTENOTITLE]]....................(15)
Line 199: Line 199:


[[PEH:Mathematics_of_Transient_Analysis]]
[[PEH:Mathematics_of_Transient_Analysis]]
[[Category:5.6.3 Pressure transient analysis]]
 
==Category==
 
[[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