Efficient Simulation Techniques for Uncertainty Quantification on Continuous Variables

In the petroleum industry, the standard Monte Carlo technique applied on global parameters (rock volume, average petrophysics) is often used to evaluate hydrocarbon in place uncertainty. With the increasing power of computers, these methodologies have become old fashioned compared to geostatistics. However care must be taken in using the latter blindly; multiple geostatistical realisations do not cover a reasonable uncertainty domain because of undesirable effects. For instance: a net to gross map with a small variogram range tends to reproduce the prior mean over the domain of interest even if this prior mean is established on a small data set; uncertainty on the data themselves may produce values outside the prior distribution range; more over the alternative data set may be biased comparing to initial prior distribution. In this paper, we present a fully automatic process based on the classical normal score transformation which is able to handle, in a non-stationary model, the following characteristics:

- uncertainty on the data set (with systematic biases)

- uncertainty on the prior mean

- local uncertainty

- preservation of bounds defined on the prior model

An example is given on a real field case in the framework of net to gross modelling. The beta law is used in order to provide high frequencies observed at the bounds (0,1); the robustness of an automatic fit for this distribution type is highlighted in order to adjust a non-stationary model on the data set. All aspects described above have been handled successfully in this non-stationary context; and the ensemble of realisations reproduces rigorously the prior distribution. The balance between local uncertainty and global uncertainty is provided by the user; consequently the volumetrics distribution are easily controlled. A final comparison with the classical geostatistical workflow is provided.