A Tree Representation of Plurigaussian Truncation Rules

Truncated Gaussian fields are a common way of modelling facies, where the correlation structure in the Gaussian field defines a spatial correlation structure for the facies. Plurigaussian simulation takes it further by using several underlying Gaussian fields. This allows more flexibility and makes it possible to model a wider range of geological settings, but conditioning can be difficult.

We present a fast and accurate implementation of conditional plurigaussian simulation. Our approach has two key elements. The first is to combine complex truncation rules with input facies probabilities. The truncation rule, which is a function from the Gaussian fields to a facies value, can be represented neatly as a binary truncation tree. This allows for a general representation that includes all the traditional 2D truncation masks. We show how to combine the use of such trees with facies probabilities, even in complicated cases with more than two Gaussian fields.

The second key element is correct conditioning to all facies observations, not just transitions, by treating them as inequality constraints on the Gaussian fields. We perform inequality Kriging by replacing these facies observations by synthetic observations of the underlying Gaussian fields. To generate synthetic observations that agree with the target posterior distribution, we use a Gibbs sampler. Since this is a quite slow algorithm, we take certain measures to make the calculations faster. Synthetic observations are then used in Kriging, improving the conditioning to facies logs from wells. We demonstrate the method with a synthetic case that combines a large number of observations with the use of a truncation tree tailored from a geological concept.