Stochastic simulation in the framework of variational-grid method of geological mapping

About the authors:

Andrey G. Plavnik, Dr. Sci. (Tech.), Chief Researcher, West Siberian Branch of Trofimuk Institute of Petroleum Geology and Geophysics of Siberian Branch Russian Academy of Sciences, Tyumen, Russia; Professor, Department of Oil and Gas Geology, Institute of Geology and Oil and Gas Production, Industrial University of Tyumen, Tyumen, Russia; plavnikag@ipgg.sbras.ru, https://orcid.org/0000-0001-8099-4874

Andrey N. Sidorov, Cand. Sci. (Geol.-Mineral.), Department Chief, Research and Analytical Center for the Rational Use of the Subsoil (Tyumen); andrey.sidorov21@gmail.com

Abstract:

The need for stochastic modeling of the geological objects properties is due to their significant heterogeneity and the limited amount of data. The existing simulation methods, in their formulation, are largely based on the stochastic representation of the model settings laid down and implemented in kriging. Within the framework of other mapping methods that use other model conditions, developing of novel approaches to the problem formulation and to implementation of stochastic simulations methods is necessary.

In this paper, we consider an approach based on the application of the variational-grid method of geological mapping. The method is based on minimizing the quadratic functional with ability taking into account a variety of heterogeneous data, including those of a stochastic nature. The direct stochastic simulation method is proposed and tested. It consists in application of the functionality, which includes three constituent elements responsible for: 1) the data approximation, 2) taking into account general spatial patterns, and 3) for the contribution of the random component to the model constructions. The main features of the method are as follows: 1) a small number of control parameters, 2) a predictable effect of their changing on the simulation results, 3) it provides an easy way to accurately mapping the mathematical expectation of the stochastic simulations options variety, and 4) it is applicable for modeling both continuous and categorical parameters.

The mathematical implementation of the approach allows reducing the problem to solving a system of linear algebraic equations with a symmetric and positive definite matrix. This determines the multioptional calculations’ computational efficiency due to a single execution of matrix factorization. The calculations are presented for two groups of data with significantly different both quantitative and model parameters, demonstrating the possibilities and features of the proposed approach implementation under different conditions. The calculations testify that the variograms’ parameters of the stochastic solutions and of the actual data could be coordinated.

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