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Multivariate Geostatistical Grid-Free Simulation of Natural Phenomena Open Access


Other title
Intrinsic Cokriging
Grid-Free Simulation
Point-Scale Block Value Representation
Multivariate Geostatistical Simulation
Block Matrix Inversion
Type of item
Degree grantor
University of Alberta
Author or creator
Zagayevskiy, Yevgeniy
Supervisor and department
Deutsch, Clayton V. (Department of Civil and Environmental Engineering)
Examining committee member and department
Marcotte, Denis (Department of Civil, Geological and Mining Engineering, Ecole Polytechnique de Montreal)
Boisvert, Jeffery B. (Department of Civil and Environmental Engineering)
Van Der Baan, Mirko (Department of Physics)
Pourrahimian, Yashar (Department of Civil and Environmental Engineering)
Hall, Robert (Department of Civil and Environmental Engineering)
Department of Civil and Environmental Engineering
Mining Engineering
Date accepted
Graduation date
Doctor of Philosophy
Degree level
Geostatistical modeling of natural phenomena is an essential step in the development petroleum reservoirs and mineral deposits. This numerical modeling involves simulation of the geological attributes conditional to available relevant data sampled at various scales. The simulation is usually performed at a point scale on a grid of regularly spaced nodes. The simulated results are non-reproducible and order dependent when another simulation is performed on a closer spacing in some areas or for an expanded study area. A grid-free geostatistical simulation (GFS) method is proposed and developed in this thesis, where the properties of natural phenomena are modeled as a function of the coordinates of the simulation locations. The resulting realizations are conditioned to the data values, preserve the spatial structure of the modeled system, and the relationship between system's variables. Simulation is performed at the point scale and can be upscaled to larger volumes to establish block-scale realizations. The conditioning data can be expressed as a set of scattered point-scale data values or a set of regularly sampled block-scale data values. Gridded block-scale data are converted to pseudo point-scale values using a point-scale block value representation technique to avoid artifacts in the simulated values. The grid-free simulation proceeds as follows. The 1-D unconditional stochastic processes are generated in a grid-free manner with a proposed Fourier series simulation (FSS), where the target covariance function is decomposed with a Fourier series, and simulation is reconstructed with linear model of regionalization as a weighted sum of the Fourier coefficients and periodic stochastic cosine functions expressed as a function of the coordinates and a random phase. 2-D and 3-D unconditional stochastic processes are simulated grid-free using the modified turning bands concept, where a set of 1-D line processes with covariance functions related to the target 2-D/3-D covariance function is linearly combined to obtain realizations in the higher dimensional space. In the modified turning bands method, the bands are replaced with the points and line process simulation is carried out with the FSS as a function the projected simulation location on a line. The anisotropy is addressed by affine transformation of the simulation space. The conditioning is performed using kriging in a dual form to reduce computational time. Modeling of multivariate systems is possible with the linear model of coregionalization. The weighted random factors are summed to obtain valid multivariate realizations. The assimilation of the secondary gridded data is performed with intrinsic cokriging, where the secondary data at a simulation location and all primary data locations are used in the conditioning. When the secondary data do not cover the entire simulation domain, the secondary data is projected to the simulation location and weighted appropriately to avoid edge artifacts. The simulation of the nugget effect component is expressed as a function of the coordinates of the gridded space at the lowest possible refinement level. The grid-free simulation methodology is implemented in Fortran code. Applicability and efficacy of the proposed method are shown with numerous 2-D and 3-D synthetic examples and real case study based on the Firebag oil sands project located in northern Alberta, Canada. Implementation aspects are documented and optimal simulation parameters are defined. Computational time is reduced through the adoption of the turning bands simulation of higher dimensional systems, conditioning with the dual cokriging, intrinsic data assimilation, and block matrix inversion in presence of the exhaustively sampled gridded data at the data assimilation step.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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