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A Spectral Approach to Conditional Simulation

Minniakhmetov, Ilnur *1; Pergament, Previous HitAnnaTop 2; Akhmetsafina, Alina 3
(1) Moscow Institute of Physics and Technology, Moscow, Russian Federation. (2) Keldysh Institute of Applied Mathematics, Moscow, Russian Federation. (3) ETH Zürich, Zürich, Switzerland.

The generation 2D and 3D grids of normally distributed random fields conditioned on well data is often required in reservoir modeling. Such fields can be obtained from application of three groups of methods: unconditional simulation with kriging interpolation (turning band or spectral methods), sequential gaussian simulation(SGS), Cholesky factorization of the covariance matrix. However, all methods have limitations. First, it is shown, that the second moment of the process conditionally simulated with the help of the kriging method are not identical to the target second moment (a priori known statistics). Second, SGS can't be calculated without limitation on number of neighbors. As the result, SGS is only asymptotically exact. Third approach, which has the advantage of being general and exact, is to use a Cholesky factorization of the covariance matrix representing grid points correlation. For fields of large size, however, the Cholesky factorization can be computationally prohibitive. Another approach is to use spectral function of full covariance matrix. In this work it is shown that, covariance of two arbitrary spectral components of conditional process could be represented as the product of functions. In this case the Cholesky factorization could considerably simplified. A feature of this approach is its computational simplicity and suitability to parallel implementation.

 

AAPG Search and Discovery Article #90141©2012, GEO-2012, 10th Middle East Geosciences Conference and Exhibition, 4-7 March 2012, Manama, Bahrain