Handling non stationarity in Multiple-Point Statistics Simulation with a Hierarchical approach

Many approaches have been proposed to tackle the challenges of non-stationarity in multiple-point statistics (MPS) simulations, including the usage of “auxiliary variables” (AVs) maps. However, obtaining the additional information required to draw these AV maps can be challenging, and in many cases these maps are drawn with subjective ad hoc procedures. Recently, some authors proposed a hierarchical simulation procedure based on a tree-like frame of binary sequential indicator simulations (SIS), with a simulation tree-like frame based on the textural hierarchy of facies. In this work a similar approach is proposed by using MPS instead of SIS; in addition, this work explores the possibility of using a different tree-like frame based on stratigraphic hierarchy and relative chronology. The proposed approach is demonstrated by using outcrops of alluvial sediments to reconstruct a three-dimensional (3D) volume. First, the outcrops are analyzed to extract a tree-like frame describing the hierarchy of facies. Then, the frame is used to decompose the outcrop into multiple bi-dimensional (2D) training images (TIs), each of which represents the spatial distribution of a simplified interpretation of the outcrop, based on the given hierarchy of facies. Depending on the criteria used to build the tree-like frame, these 2D TIs are composed of a relatively low number of facies; it is therefore straightforward to use a sequence of 2D conditional simulations (s2Dcd approach) to build 3D TIs for each branch of the frame. Finally, the obtained 3D TIs are used to perform a sequence of MPS simulations, nested accordingly to the aforementioned tree-like frame, resulting in a 3D reconstruction of the spatial distribution of the alluvial sediments considered. On 2D test cases, the results obtained with the proposed approach are comparable with the results obtained by handling non-stationarity using AVs, with the advantage that the proposed approach does not require an AV map. In addition, the decomposition of the simulation problem into smaller groups of facies, allowed to have more control on the low-level reconstructions made with the s2Dcd approach to obtain the 3D TIs, and consequently to improve the final 3D reconstruction. In conclusion, with the additional effort required to conceptualize a hierarchy of facies, the proposed approach appears as a reliable alternative to obtain non-stationary MPS simulations without the need of additional information, as for example the one required by the use of AVs.