# An investigation of the properties of geological simulation techniques based on orthogonal decompositions

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Geological modeling is an important aspect of reservoir exploration and field development planning in which data obtained from the reservoir is interpolated to the locations in the field where the values of a given property are unknown. This is accomplished by statistically characterizing the pattern of variability exhibited by the data, and then using this characterization to estimate the values of a given location where the actual value is unknown. There have been many algorithms developed to model geological properties. The conventional geological simulation methods involve solving a system of equations for each point where an interpolated value is required. This method is somewhat inefficient, as simulation nodes are visited in a sequential fashion, and this results in increased computation time for processes such as history matching. Moreover, the sequential simulation approach is hampered by statistical constraints such as ergodicity that imply that the generated models reflect the target statistics only in an average sense and any one realization may deviate significantly from the target. In order to reduce the computational time that it takes generate these simulations, a new set of modeling methods based on orthogonal decompositions have been developed. In an orthogonal decomposition, an image is transformed from the original data space to a space defined by a set of orthogonal bases. These bases will have different levels of significance; the less significant bases can be ignored, allow the remaining bases to provide a reasonable representation of the data, while reducing the number of degrees of freedom. In this research, we investigate properties of decomposition based simulations. First we demonstrate a generalized method to condition models constructed using decomposition methods to known data points. Next, we show how these decomposition-based methods can be used to analyze the heterogeneity of a reservoir. Then, we combine the concept of dimensionality reduction with that of conditional simulation using an orthogonal bases and demonstrate the properties shown by the resulting models. There are two orthogonal decomposition based methods that are developed for this thesis. The first method is a novel method, based on a “rotational bases”, where a simulation is performed using somewhat arbitrary basis images generated using a rotational transformation. The second method is based on principal component analysis. These methods are analyzed to provide some insight into the characteristics of proper orthogonal decompositions.