Neural Vector Quantization for Multivariate Upscaling
Authors: Chawathe A. , Ye M. 
Published: December 2007

Abstract
Stochastic reservoir models generate large grids, sometimes in excess of millions of cells. These models need to be upscaled to reduce the number of grid blocks for practical flow simulations. As a consequence, the volumes of the upscaled simulation grids are represented by corresponding coarser values. This may result in a detrimental correlation structure and variance which may be different from that obtained from the fine grid. Various upscaling algorithms have been proposed, ranging from simple averaging to conservation of fluid flux, and more recently, wavelet transforms. These algorithms are principally focused towards upscaling tensorial reservoir properties such as permeability. Scalar properties, such as porosity, are generally volume-averaged. This approach of alienating porosity from permeability for upscaling purposes questions the geological underpinnings of the depositional/diagenetic model being upscaled. In this paper, we propose a completely new perspective to address the issue of upscaling: Neural Vector Quantization. This new approach offers opportunities such as multivariate upscaling where many reservoir properties can be upscaled simultaneously. Neural Vector Quantization (NVQ) is used to achieve data compression in vector spaces. In NVQ upscaling the components of the input vector are the different reservoir properties (permeability, porosity, water saturation) to be upscaled simultaneously. The purpose of vector quantization is to categorize a given set, or a distribution of input vectors, into several clusters called Voronoi tessellations. Input vectors in the same Voronoi tessellation are considered to be similar and fall into the same cluster. The vector corresponding to the centroid of the tessellation is the globally averaged value of all the input vectors in that Voronoi tessellation. This vector is called the reference vector. The reference vector yields the upscaled values of the physical properties of interest. In this study, we achieve vector quantization using a new self-organizing, competitive-clustering, adaptive-structure neural network algorithm. Since we have specifically designed this algorithm for petroleum engineering applications, it performs clustering on highly-disjointed datasets. We call it the Heterogeneous Space Classifier (HSC) algorithm.
We tested HSC on a 25x25x4 three-phase, black-oil model. The model contained one injector and one producer. We compared the reservoir performance for the upscaled and fine grids by evaluating the average reservoir pressure, cumulative production, and fluid flowrates for 10 years. We upscaled the model to a 10x10x4 grid without compromising the flow behavior of the system.

Introduction
Constrained models have dominated recent efforts in reservoir modeling. Some of the initial studies concentrated on automatic history-matching algorithms, which result in pseudoization of reservoir properties, especially the relative permeabilities. Such methods were severely criticized by petrophysicists and geologists since the final "pseudo" property did not honor the experimentally measured value. In the past decade, the emphasis shifted to building spatially-descriptive models by honoring the point-data as well as the inherent correlation structures observed between the point-located data. These geostatistical techniques, constrained to 3D seismic or well-test data, can yield geologically consistent models. Some of these models have fine geological descriptions built into the model through iterative interaction between the geologist, the geophysicist, and the reservoir engineer. Nowadays, models containing geological entities, such as braided channels and alluvial fans, are commonly encountered in reservoir simulation literature. These beautifully detailed models capture the depositional and diagenetic nuances of the reservoir, but from a simulation standpoint, they demand a prohibitive number of simulation cells. Constrained, stochastic models can frequently exceed a million cells.
Although impressive, current computing technology limits us in simulating million-cell models. In fact, the routine demands simulation of grids that are one order lower in magnitude. This requires a translation of the detailed grids to a coarser, albeit a more manageable, level without sacrificing the anticipated reservoir performance. This translation is commonly known as upscaling.
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