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Q U A N T I F Y I N G U N C E R T A I N T Y
Although most petrophysicists do not yet quantify uncertainty,
WellEval.com does so routinely using Monte-Carlo simulation and core
data (where available). Since software packages are now on the market
that allow Monte-Carlo simulation of petrophysical properties to be
carried out, our Principal Petrophysicist has written this brief to educate readers on some of the pitfalls with such modelling and how uncertainty is best quantified.
What Is Monte-Carlo Simulation?
When a petrophysical property such as porosity or water saturation is
calculated, an interpretation model is assumed. Usually, this model is
implemented in a mathematical equation. Petrophysicists typically
calculate properties using their most likely estimates of each input
required in said equation. In reality though, there is an uncertainty
range associated with the inputs used. Uncertainties in the inputs will
result in uncertainty in the output, although the magnitude of that
uncertainty can be difficult to quantify.
Monte-Carlo simulation provides a simple means by which uncertainties
in inputs can be translated into uncertainties in the required
petrophysical properties. This simulation is done by selecting a random
value from the distribution of likely values for each input parameter
required for the interprettaion model. Once a value has been selected
for all inputs, the interpretation equations are calculated and the
result stored. Then the process is repeated and all the results stored.
When the requested number of cycles has been completed, the results can
be sorted and histograms created allowing the derived petrophysical
property at any given probability level to be found.
Meaningful Simulation
It is interesting to note the number of simulations run in some of the
advertising for commercially available Petrophysics software
incorporating Monte-Carlo simulation. If the number of simulations is
small, then the results are unlikely to be meaningful. A minimum number
for Monte-carlo simulations to cover a meaningful range of inputs to
petrophysical equations is around 100. More than 1000 simulations is
even better, although it does take more time. It is this compromise
between time and number of simulations that has limited the use of
Monte-Carlo in Petrophysics until computing power has reached a speed
such that significant numbers of simulations can be carried out in a
reasonable time frame.
Interpretation Model Uncertainty
Since it's not touched upon anywhere else, it MUST be noted that
Monte-Carlo simulation only provides uncertainty estimates according to
the interpretation model used. It does not address the match between
the interpretation model and reality. In practice, this statement means
that if your interpretation model is wrong, then uncertainty analysis
using Monte-Carlo may not give results that include the "real" answer -
which is the objective.
One way to address this problem is to use multiple models to determine
the uncertainty. For example, if determining porosity, Monte-Carlo
could be run using the density, density-neuton and sonic porosities.
Routine & Special Core Analyses
A better way to quantify the uncertainty is to use the core data if
there is any available. The differences between the log and core
derived porosities can be used to determine the porosity uncertainty.
With water saturations, capillary pressure data (appropriately
corrected) can be used to compare with the wireline log derived water
saturations to determine uncertainty.
The Recommended Methodology
Of course the best methodology is to select the most appropriate
interpretation model and test the resulting petrophysical properties
against any core data. In this case, the uncertainties from the
comparison with the core data should be contained within the
Monte-carlo derived uncertainty distributions. If not, there may be a
systematic problem with either the interpretation model (most likely)
or the core measurements.
In the absence of core data, it's recommended to use an interpretation
model that has proven itself reliable in a similar Formation in a
nearby well. If no such data exists, then the most reliable porosities
are normally derived using the density log corrected for hydrocarbons
using the invaded zone resistivity, while water saturations estimated
using the Waxman-Smits or Dual Water equations are usually most
reliable.
When we first started Monte-Carlo modelling, there were no commercial
packages available to do such modelling for petrophysical deliverables
specifically. Consequently, WellEval.com have written our own
Monte-Carlo simulation software.
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