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Petroleum geostatistics for nongeostaticians

Part 2

CGC, Broken Arrow, Oklahoma, U.S.

Jeffrey M. Yarus

Knowledge Reservoirs, Houston, Texas, U.S.

Kirk B. Hird

BP Amoco, Houston, Texas, U.S.

Corresponding author: geodoc547@aol.com

Editor's Note: Part 1 of this article (TLE, May 2000) defined geostatistics, examined its origins, and reviewed the concepts of the spatial model and the kriging interpolation algorithm. This article describes geostatistical conditional simulation, also known as stochastic modeling, and its use for generating realistic maps of reservoir heterogeneity, uncertainty analysis, and economic risk analysis. The Geologic Column, which appears monthly in TLE, is (1) produced cooperatively by the SEG Interpretation Committee and the AAPG Geophysical Integration Committee and (2) coordinated by M. Ray Thomasson and Lee Lawyer.

The first 20% of the full text of this article appears below.

For more than a decade, stochastic, or geostatistical, modeling methods have been increasingly used to "map" spatially correlated data. Recall that kriging is a deterministic method whose function has a unique solution and that does not attempt to represent actual variability of the studied attribute. Thus, the smoothing property of kriging dismisses local detail in place of a good average. However, often the geoscientist or reservoir engineer is more interested in fine-scale details captured by the estimation variance than a map of local estimates of the mean.

Geostatistical jargon is confusing. For example, many authors often use stochastic, probabilistic and conditional simulation interchangeably. We consider a stochastic model to be conditional when it honors the measured data and the spatial model (variogram or covariance). But for the sake of simplicity, we also consider these terms equivalent in this article.

To many, stochastic methods are analogous to tossing a coin. They are suspicious because it is well known that the natural processes responsible for creating reservoirs are not random. In light of this, stochastic methodologies are often rejected outright. Although it is true that reservoirs are not products of random processes, it is also true that they have attributes that cause them to behave as if they were random. For example, physical and chemical processes often change reservoir characteristics from their initial state, confounding our ability to make predictions even when the processes are understood. Such changes, though, result in behavior that can be captured using stochastic principles.

Additionally, like the traditional deterministic approach, stochastic modeling preserves "hard" data where known and interpretative "soft" data where informative. But unlike the deterministic approach, it provides geoscientists and reservoir engineers many equally probable models called realizations. The kriged solution is the average of numerous realizations, and the variability among different outcomes is . . . [Full Text of this Article]