Page 3 - Laplace transform numerical inversion

Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 - p 1/18
Laplace transform numerical inversion
Bruno Josso - Leif Larsen
1 Laplace transform numerical inversion issue
When running an analytical liquid rate simulation on a bounded reservoir an artefact due to
Laplace transform numerical inversion algorithm can be noticed. This issue can be illustrated
with a simple example: liquid rate simulation on a closed circular homogeneous reservoir with a
fracture. The details of the simulation are graphically given in Figure 1(a).The simulation results
can be seen in Figure 1(b).
At first, results seem coherent with what can be expected, but if one checks the mean pres-
sure from closer in Figure 1(b) (the yellow curve in the graph at the bottom), one can see that
whereas this pressure should monotonously decrease towards a fixed value (around 1000 psia ) as
the reservoir is depleted, it starts to increase slowly between years 2016 and 2017. In Figure 2(a),
one can follow the problem from even closer and see that the simulated rate, instead of exponen-
tially decrease towards 0 as expected, becomes negative at the end of 2015. Such results would
mean that one swaps from production to injection mode after this date, which is not the case.
The simulation artefact can also be spotted on the pressure LogLog plot. Hence, it can be seen
in Figure 2(b),that at late time, a break appears on the pressure plot (in white). Unfortunately,
according to the simulation parameters, one should see a unity slope straight line without any
distortion.
In the present case, both simulation and theory do not match. The conclusion is that the simu-
lation gives a wrong result and what is observed here is not a normal behaviour. The simulation
is the result of a numerical inversion of a function defined in the Laplace domain. The function
is relatively complex here, and we thought at first that the problem was caused either by a bug
in the calculation kernel, or by another issue hidden somewhere in Topaze. But it appears that a
very similar result can be obtained when inverting an exponential function with a Gaver-Stehfest
algorithm taken from an independent implementation.
Indeed: the problem does directly come from the numerical inversion algorithm.
After reminding a few definitions regarding the Laplace transform, we will simplify the problem and
study several algorithms behaviour when applied on a very simple example. We will eventually
see what can be done in the context of reservoir engineering.