Abstract
Modern approaches for the spatial simulation of categorical variables are largely based on multi-point statistical methods, where a training image is used to derive complex spatial relationships using relevant patterns. In these approaches, simulated realizations are driven by the training image utilized, while the spatial statistics of the actual sample data are ignored. This paper presents a data-driven, high-order simulation approach based on the approximation of high-order spatial indicator moments. The high-order spatial statistics are expressed as functions of spatial distances that are similar to variogram models for two-point methods, while higher-order statistics are connected with lower-orders via boundary conditions. Using an advanced recursive B-spline approximation algorithm, the high-order statistics are reconstructed from the available data and are subsequently used for the construction of conditional distributions using Bayes’ rule. Random values are subsequently simulated for all unsampled grid nodes. The main advantages of the proposed technique are its ability to (a) simulate without a training image to reproduce the high-order statistics of the data, and (b) adapt the model’s complexity to the information available in the data. The practical intricacies and effectiveness of the proposed approach are demonstrated through applications at two copper deposits.
Highlights
Geostatistical simulations are often required in reservoir modeling, as well as in the quantification of geological uncertainty, pollutants in contaminated areas, and other spatially dependent geological and environmental phenomena
During the past few decades, geostatistical simulations of categorical variables, such as geological units with complex spatial geometries of mineral deposits and petroleum reservoirs, have largely been modeled within the framework of multiplepoint spatial simulation (MPS) methods that were introduced in the 1990s and have been further developed since (Guardiano and Srivastava 1993; Journel 1993; Strebelle 2002, 2021; Journel 2003; Zhang et al 2006; Chugunova and Hu 2008; Remy et al 2009; Mariethoz and Renard 2010; Straubhaar 2011; Stien and Kolbjørnsen 2011; Toftaker and Tjelmeland 2013; Strebelle and Cavelius 2014; Zhang et al 2017; Gómez-Hernández and Srivastava 2021, others)
The MPS framework is based on the use of training images (TI) or analogues of the attributes of interest being modeled and contains additional information about the complex spatial relations of the attributes to be simulated; the TIs are not conditioned to the available data and their spatial statistics
Summary
Geostatistical simulations are often required in reservoir modeling, as well as in the quantification of geological uncertainty, pollutants in contaminated areas, and other spatially dependent geological and environmental phenomena. Further developments in algorithmic performance (Yao et al 2018, 2020), generalization using splines (Minniakhmetov et al 2018), a high-order decorrelation method (Minniakhmetov and Dimitrakopoulos 2017a), and efficient block simulations (de Carvalho et al 2019), and training-image-free simulations (Yao et al.2021) have made the approach more practical These approaches are based on the approximation of a conditional distribution using Legendre polynomials, which are smooth functions and are incapable of an adequate approximation of the discrete distribution of categorical variables. It should be noted that, as shown in the subsequent sections, the proposed method works without a TI; additional information from a TI can be incorporated as a secondary condition, ensuring that the high-order spatial indicator moments are driven by the available data. A mathematical model for recursive approximation of high-order spatial indicator moments is presented, followed by the proposed high-order, data-driven, categorical simulation method.
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