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Bessel functions in transient analysis

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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

Vol1 page 0083 eq 001.png....................(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

Vol1 page 0083 eq 002.png....................(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

Vol1 page 0083 eq 003.png....................(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

Vol1 page 0083 eq 004.png....................(4)

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

Vol1 page 0083 eq 005.png....................(5)

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

Vol1 page 0083 eq 006.png....................(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

Vol1 page 0083 eq 007.png....................(7)

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

Vol1 page 0084 eq 001.png....................(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

Vol1 page 0084 eq 002.png....................(9)

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

Vol1 page 0084 eq 003.png....................(10)

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

Vol1 page 0084 eq 004.png....................(11)

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

Vol1 page 0084 eq 005.png....................(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:

Vol1 page 0084 eq 006.png....................(13)

and

Vol1 page 0084 eq 007.png....................(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]

Vol1 page 0084 eq 008.png....................(15)

Vol1 page 0085 eq 001.png....................(16)

Vol1 page 0085 eq 002.png....................(17)

where γ = 0.5772…, and

Vol1 page 0085 eq 003.png....................(18)

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

Vol1 page 0085 eq 004.png....................(19)

for |argz| < π / 2, and

Vol1 page 0085 eq 005.png....................(20)

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

Vol1 page 0086 eq 001.png....................(21)

Vol1 page 0086 eq 002.png....................(22)

Vol1 page 0086 eq 003.png....................(23)

Vol1 page 0086 eq 004.png....................(24)

Vol1 page 0086 eq 005.png....................(25)

Vol1 page 0086 eq 006.png....................(26)

Vol1 page 0086 eq 007.png....................(27)

Vol1 page 0086 eq 008.png....................(28)

and

Vol1 page 0086 eq 009.png....................(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.

Noteworthy papers in OnePetro

Abramowitz, M. and Stegun, I. A., eds. 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover.

Bowman, F. 1958. Introduction to Bessel Functions, Dover Publications, Inc. New York.

Carslaw, H.S. and Jaeger, J.C. 1986. Conduction of heat in solids, 2nd. Oxford Oxfordshire New York: Clarendon Press ; 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.

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