Page 7 - Laplace transform numerical inversion
Basic HTML Version
Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 -
p 5
In the present study, stained by our application field, one focuses on a subset of seven algorithms
or implementation alternatives.
In the context of reservoir engineering, models are often only known in the Laplace domain for
p
,
the Laplace argument, being a real number:
p
∈
R
. Under such a constraint, numerical inversion
algorithms that only work with complex Laplace argument (
p
∈
C
) cannot be directly considered.
Not being able to express reservoir models in the complex plane is the main constraint when
selecting a numerical inversion algorithm. Furthermore, one also wants algorithms to be stable,
accurate and fast.
The Gaver-Stehfest inversion method, described in
and
, fulfills these criteria in most of the
practical cases that are encountered. Indeed, it is highly accurate and stable for most pressure
solutions. For this reason, it is one of the most widely used in this domain.
Unfortunately, as could be seen in the first exemple in section
, Gaver-Stehfest inversion
method can also fail and give wrong results under certain circonstances. Indeed, one could see
that it suffers limitations for late-time rate data from boundary dominated models produced at
constant pressure: late-time exponential decline data cannot be obtained accurately from the
Gaver-Stehfest algorithm without some modification.
In this document, we come back to this algorithm, as a reference and a start point, to first see
how it can be tuned by using more terms, and second how accuracy can be increased by changing
the implementation method.
As mentioned before, when referring to the list of inversion method families given in
, one can
see that about
2
/
3
of them, the ones requiring solutions in complex form, are simply not suitable
in the current application field. Therefore, in this list only series expansion algorithms could be
retained as serious candidates. Three polynomials families have been tested, namely Legendre,
Chebyshev and Laguerre. These algorithms respect our main constraint and are based on another
principle than the Gaver-Stehfest inversion method.
Since complex numerical inversion methods, i.e. with
p
∈
C
, cannot be totally ignored, we have
also tested two Fourier transform based algorithms. The first one is directly deduced from ex-
pression
. It belongs to the Dubner-Abate algorithm family
. The second one, recently
pointed out and very successfully applied in
, is called Den Iseger's method
. Of course,
none of these two algorithms can
directly
be applied in the context of reservoir engineering when
solutions are not known in complex form. Nevertheless, if one considers that some kind of an-
alytic continuation can be applied on the reservoir models, it is also important to see how well
more general algorithms can perform.
A list of seven algorithms has been tested. They are referred to here as :
1. Gaver-Stehfest
(
p
∈
R
)
å
see section
2. Big number Gaver-Stehfest
(
p
∈
R
)
å
see section
3. Legendre polynomials
(
p
∈
R
)
å
see section
4. Chebyshev polynomials
(
p
∈
R
)
å
see section
5. Laguerre polynomials
(
p
∈
R
)
å
see section
6. Fourier
(
p
∈
C
)
å
see section
7. Den Iseger
(
p
∈
C
)
å
see section
