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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

 ....................(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

 ....................(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

 ....................(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

 ....................(4)

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

 ....................(5)

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

 ....................(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

 ....................(7)

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

 ....................(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

 ....................(9)

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

 ....................(10)

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

 ....................(11)

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

 ....................(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]

 ....................(13)

and

 ....................(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]

 ....................(15)

 ....................(16)

 ....................(17)

where γ = 0.5772…, and

 ....................(18)

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

 ....................(19)

for |argz| < π / 2, and

 ....................(20)

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

 ....................(21)

 ....................(22)

 ....................(23)

 ....................(24)

 ....................(25)

 ....................(26)

 ....................(27)

 ....................(28)

and

 ....................(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

Category