A Tree Representation of Pluri-Gaussian Truncation Rules

Stochastic facies models based on truncated Gaussian random fields are known for being flexible and well suited to reproduce patterns and features from analogues or conceptual models. In pluri-Gaussian simulation, the number of random fields is theoretically unlimited, which adds flexibility and makes it possible to model a wider range of geological settings. However, the truncation map traditionally used to set up these models quickly becomes unclear when used for higher dimensions. Hence, in practical pluri-Gaussian applications, the number of fields is typically kept as low as two or three. We present a formulation of pluri-Gaussian simulation in which the truncation rule, the function that maps combinations of Gaussian random field values to facies categories, is represented as a particular binary tree. This is used to decouple the fields in the critical Gibbs sampling step of the conditioning process in such a way that we can use multiple lower-dimensional samples instead of a single higher-dimensional sample. The resulting conditioning algorithm scales excellently with the amount of conditioning data and the number of fields. The algorithm accepts a combination of trends and probabilities in the same model setup, which provides additional flexibility in representing varying depositional geometries. We demonstrate the hierarchical pluri-Gaussian simulation with two practical examples. One is based on real data from the Volve oil field in the North Sea. The other combines a large number of synthetic observations with a truncation tree tailored to a more complex geological concept. The choices made when building the truncation tree affect the features of the realizations, especially when it comes to which facies can be in contact and which can overprint each other. This aspect of tree building is discussed in light of the numerical examples given.