Change-of-Support Models on Irregular Grids for Geostatistical Simulation
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Change-of-Support Models on Irregular Grids for Geostatistical Simulation Victor Zaytsev1,2 · Pierre Biver1 · Hans Wackernagel2 · Denis Allard3
Received: 5 May 2015 / Accepted: 6 August 2015 © International Association for Mathematical Geosciences 2015
Abstract In many domains, numerical models are initialized with inputs defined on irregular grids. In petroleum reservoir engineering, they consist of a great variety of grid cells of different size and shape to enable fine-scale modeling in the vicinity of the wells and coarse modeling in less important regions. Geostatistical simulation algorithms, which are used to populate the cells of unstructured grids, often have to address the problem of transition from the small-scale statistical data stemming from laboratory cores analysis and seismic processing to the multiple larger scale geological supports. The reasonable generalization of the above-mentioned problem is integrating the point-support data to simulations on irregular supports. Classical geostatistical simulation methods for generating realizations of a stationary Gaussian random function cannot be applied to unstructured grids directly, because of the uneven supports. This article provides a critical review of existing geostatistical simulation methodologies for unstructured grids, including fine-scale simulations with upscaling and direct sequential simulation algorithms, and presents two different generalizations of the discrete Gaussian model for this purpose, thereby discussing the theoretical assumptions and the accuracy when implementing these models. Keywords Unstructured grids · Geostatistics · Direct sequential simulation · Change of support · Discrete Gaussian model
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Victor Zaytsev [email protected]
1
Geostatistics and Uncertainties Service, Total S.A., Avenue Larribau, 64018 Pau Cedex, France
2
Equipe de Géostatistique, Centre de Géosciences, MINES ParisTech, PSL Research University, 35 rue Saint Honoré, 77305 Fontainebleau, France
3
Biostatistics and Spatial Processes (BioSP), INRA, Site Agroparc, Domaine St-Paul, 84914 Avignon Cedex 9, France
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Math Geosci
1 Introduction Unstructured reservoir grids constitute the basic building block of a new family of reservoir models which increase the accuracy of modeling of complicated geological features such as faults and enable performing a more detailed description of a reservoir through a computer model. Certain types of unstructured grids, such as Voronoï polygon grids, provide a more physical modeling of fluid flows as compared to classical regular grids, which is of high importance for the petroleum industry. Thus, unstructured grids enable incorporating local grid refinements for the accurate simulation of spherical flow around the wells in a consistent manner as well as operating with coarser grid blocks in less important areas to reduce the complexity of the numerical problem solved by the flow simulator. Grid blocks of an unstructured grid can be modeled with any convex polyhedron, the classical solution being the Voro
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