Practical Geostatistics - An Armchair Overview for Petroleum Reservoir Engineers
Authors: Yarus Jeffrey M. , Chambers Richard L. 
Published: June 2014

Distinguished Author Series articles are general, descriptive representations that summarize the state of the art in an area of technology by describing recent developments for readers who are not specialists in the topics discussed. Written by individuals recognized as experts in the area, these articles provide key references to more definitive work and present specific details only to illustrate the technology. Purpose: to inform the general readership of recent advances in various areas of petroleum engineering.

Abstract
Some engineers are skeptical of statistical, let alone geostatistical, methods. Geostatistical analysis in reservoir characterization necessitates an understanding of a new and often unintuitive vocabulary. Statistical approaches for measuring uncertainty in reservoirs is indeed a rapidly growing part of the best-practice set of methodologies for many companies. For those already familiar with the basic concepts of geostatistics, it is hoped that this overview will be a useful refresher and perhaps clarify some concepts. For others, this overview is intended to provide a basic understanding and a new level of comfort with a technology that may be useful to them in the very near future.

Introduction
Geoscientists and geological engineers have been making maps of the subsurface since the late 18th century. The evolution of our ability to predict structure beneath the surface of the Earth has been a complex interaction between quantitative analysis and qualitative judgment. Geostatistics combines the empirical conceptual ideas that are implicitly subject to degrees of uncertainty with the rigor of mathematics and formal statistical analysis. It has found its way into the field of reservoir characterization and dynamic flow simulation for a variety of reasons including its ability to successfully analyze and integrate different types of data, provide meaningful results for model building, and quantitatively assess uncertainty for risk management. Additionally, from a management point of view, its methodologies are applicable for both geoscientists and engineers, thereby lending itself to a shared Earth model and a multidisciplinary workforce.

Why Geostatistics?
Fig. 1 depicts two images of hypothetical 2D distribution patterns of porosity. Fig. 1a shows a random distribution of porosity values, while Fig. 1b is highly organized, showing a preferred northwest/southeast direction of continuity. While this difference is obvious to the eye, the classical descriptive-summary statistics suggest that the two images are the same. That is, the number of red, green, yellow, and blue pixels in each image is the same, as are the univariate statistical summaries such as the mean, median, mode, variance, and standard deviation (Fig. 1c). Intuitively, as scientists and engineers dealing with Earth properties, we know that the geological features of reservoirs are not randomly distributed in a spatial context. The reservoirs are heterogeneous and have directions of continuity in both 2D and 3D space and are products of specific depositional, structural, and diagenetic histories. Strangely, that these two images would appear identical in a classical statistical analysis is the basis of a fundamental problem inherent in all sciences dealing with spatially organized data. Classical statistical analysis inadequately describes phenomena that are both spatially continuous and heterogeneous. Thus, use of classical statistical descriptors alone to help characterize petroleum reservoirs often will result in an unsatisfactory model.