Abstract
Abstract. Multiple-point statistics (MPS) has shown promise in representing complicated subsurface structures. For a practical three-dimensional (3-D) application, however, one of the critical issues is the difficulty in obtaining a credible 3-D training image. However, bidimensional (2-D) training images are often available because established workflows exist to derive 2-D sections from scattered boreholes and/or other samples. In this work, we propose a locality-based MPS approach to reconstruct 3-D geological models on the basis of such 2-D cross sections (3DRCS), making 3-D training images unnecessary. Only several local training subsections closer to the central uninformed node are used in the MPS simulation. The main advantages of this partitioned search strategy are the high computational efficiency and a relaxation of the stationarity assumption. We embed this strategy into a standard MPS framework. Two probability aggregation formulas and their combinations are used to assemble the probability density functions (PDFs) from different subsections. Moreover, a novel strategy is adopted to capture more stable PDFs, where the distances between patterns and flexible neighborhoods are integrated on multiple grids. A series of sensitivity analyses demonstrate the stability of the proposed approach. Several hydrogeological 3-D application examples illustrate the applicability of the 3DRCS approach in reproducing complex geological features. The results, in comparison with previous MPS methods, show better performance in portraying anisotropy characteristics and in CPU cost.
Highlights
Background information2.1 Pattern distanceA pattern distance d{NX, NY } is an approximation of the dissimilarity between patterns, which is used to compare the neighborhood of a node currently simulated with a data event in the training image (Mariethoz et al, 2010)
The original cross sections are divided into many subsections according to their spatial relationships, and nonstationarity is reduced since it is restricted into a local cube consisting of six or fewer subsections
Because 3-D geological heterogeneities using 2-D cross sections (3DRCS) is sensitive to the number of input cross sections, we offer two and four sections in each direction respectively, and the computational efficiencies when increasing the total number of grid cells are shown in Fig. 14a and b
Summary
3-D characterization of geological architectures plays a crucial role in the quantification of subsurface water, oil and ore resources (Chen et al, 2017, 2018; Foged et al, 2014; Hoffman and Caers, 2007; Jackson et al, 2015; Kessler et al, 2013; Raiber et al, 2012; Wambeke and Benndorf, 2016). Chen et al.: Locality-based 3-D multiple-point statistics reconstruction using 2-D geological cross sections 6549 there are many ways to acquire low-dimensional data for reconstructing a full 3-D model These methods using real geological analogs or sections as training images still face significant nonstationarity due to the heterogeneity of geological phenomena and processes (Comunian et al, 2011; de Vries et al, 2009). Compared to previous MPS implementations relying on partial data, our proposal is to use only several local subsections closer to the simulated node as training images, rather than full planes perpendicular to the x, y and z directions (Comunian et al, 2012) or searching in the entire 3-D domain (Mariethoz and Renard, 2010). The final section contains some concluding remarks and ideas for future work
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