Large-Scale Conditional Simulation: Domain and Matrix Decomposition and the Moving Template Model

by
Luiz Cavalcante de Lima, Ph.D.

University of Texas at Austin, 1995
Supervisors: Larry W. Lake
Kamy Sepehrnoori

The main objective of this research is the development of algorithms for large-scale conditional simulation of reservoir properties. The conditional step of these simulation algorithms uses ordinary Kriging (OK), the basic estimator of geostatistics. Since OK has an important shortcoming when Kriging data is aligned along a finite string, this research also studies this limitation, as well as how to determine the OK neighborhood.

This work first derives the conditions for the onset of artificial overweighing of the data at the ends of a string and then proposes a simplification to be used when these conditions apply. This proposed simplification completely avoids the artificial overweighing and produces a large reduction in computer time since the Kriging data-data correlation matrix is reduced to an identity matrix.

The determination of the Kriging neighborhood is important for the quality of the Kriging estimates. This work studies and proposes a new method of determining the Kriging neighborhood based on a three-term partition of the ordinary Kriging weights, as well as a criterion that defines how good the geometric configuration of the data is. Furthermore, a procedure to estimate the minimum number of data that are necessary to add to achieve a pre-specified variance of the estimated mean is presented.

With respect to large-scale conditional simulation, this work proposes and tests a new scheme that performs domain decomposition at a reservoir scale. This technique can be considered the core of a parallel domain decomposition algorithm for the conditional simulation of reservoir property fields that can be represented by Guassian bivariate statistics. The main advantage of combining domain decomposition at a reservoir scale and the matrix decomposition method are simplicity, flexibility, and efficiency. In large-scale conditional simulations, these advantages become more evident because of the significant decrease in computer time and storage necessary to decompose the covariances matrices.

This work also develops a new conditional simulation technique, the moving template simulation algorithm (MTSA), to statistically generate the spatial distribution of reservoir properties. The MTSA is simple, easy to implement, applicable in most practical situations and is 6 to 10 times faster than sequential Gaussian simulation (SGS). Also, the new algorithm has fewer fluctuations of the semi-variogram realizations, and the corresponding realization statistics tend more readily toward the input statistics for a small field, when compared with SGS. Thus, this work presents equations to explain how MTSA works, how it avoids matrix inversion, and defines the conditions where MTSA can be applied. Finally, this work presents a method that expands the applications of the MTSA, which, in turn, can lead to areas for new research.

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