Page 16 - Laplace transform numerical inversion
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Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 -
p 14
4.2 Algorithms with complex Laplace argument:
p
∈
C
None of the tested inversion algorithms based on real Laplace argument offers satisfying perfor-
mance. To go further, one now needs to test more general inversion algorithm for which functions
to be inverted can be defined in the complex plane.
Let us recall that reservoir models are, most of the time, only known in the Laplace domain for
p
the Laplace argument being a real number. They have not even always got a closed-form ex-
pression. As a way of consequence, algorithms presented hereafter cannot be straightforwardly
applied in our context. One makes the assumption that some kind of analytic continuation could
be found and applied to the models prior to performing inversion. The existence of such a pro-
cessing is not discussed in this paper.
Two general Fourier based inversion methods are presented in this section. The first one is directly
deduced from expression
. It belongs to the Dubner-Abate algorithm family
. The second
one, called Den Iseger's method, is more recent since it appeared in
.
4.2.1 Fourier algorithm
When considering the inversion relation
, one straightforwardly can think that direct Fourier
based methods are very strong candidates to efficiently invert Laplace transform as stated in
.
This is especially true when considering the very fast Fourier transform algorithms sample group
that is available nowadays. There is no need to know the Laplace function for all the values of
p
∈
C
in the whole complex plane. Indeed, inversion is often based on a simple integration along
a line in the complex plane, which is distorted to avoid singularities.
Fourier based methods usually give fairly good inversion results. The inverted function gently
converges towards the analytical solution without oscillating much as can be seen in Figure
Unfortunately, again, at some point, the inversion result becomes negative. Tweaking the algo-
rithm can allow one to delay this instant but only on a limited range. As a conclusion, even if
reservoir models were known in the complex plane, this algorithm would not be suitable for our
positive convergence issue.
2
3
4
5
6
-0.5
0
1
0
0.5
1
1.5
2
Fp_inv(t)
f(t)
0
1
2
3
4
5
6
-0.001
-0.0005
0
0.0005
0.001
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0
Fourier inverse Laplace transform
Figure 13: Inversion of
f
(
t
) =
e
−
µt
by Fourier algorithm.
4.2.2 Den Iseger algorithm
The Den Iseger algorithm, described in
, has recently become very popular in petroleum en-
gineering research, as can be seen in
. This algorithm is indeed very accurate. It can also be
categorised as a Fourier based algorithm and can only be applied on functions
ˆ
F
(
p
)
defined in the
complex plane with
p
∈
C
.
