Multi-scale and multi-resolution stochastic modeling of subsurface heterogeneity by tree-indexed Markov chains

A new methodology is proposed to handle multi-scale heterogeneous structure s. It can be of importance in the field of hydrogeology and for petroleum e ngineers who are interested in characterizing subsurface heterogeneity at v arious scales. The framework of this methodology is based on a coarse to fi ne scale representation of the heterogeneous structures on trees. Different depths in the tree correspond to different spatial scales in representing the heterogeneous structures on trees. On these trees a Markov chain is use d to describe scale to scale transitions and to account for the uncertainty in the stochastically generated images. We focus in this work on the description and application of the methodology to synthetic data that are geologically realistic. The methodology is flex ible. Conditioning on field data and measurements is straightforward. Non-s tationary and stationary fields, compound and nested structures can be addr essed.