Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. A feature of the covariance model by which the experimental points that define the model do not appear to intersect at the origin. The nugget model shows constant variance at all ranges, but often is modeled as zero-variance at the control point (well location).
Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. A linear combination of two or more variogram (correlogram) models (e.g., a short-range exponential model combined with a longer-range spherical model).
An understanding of statistical concepts is important to many aspects of petroleum engineering, but especially reservoir modeling and simulation. The discussion below focuses on a range of statistical concepts that engineers may find valuable to understand. The focus here is classical statistics, but differences in the application for geostatistics are included. A quantitative approach requires more than a headlong rush into the data, armed with a computer.
We have seen that two data sets can have the same univariate statistics, yet have very different spatial properties (Figure 1). The complex attributes we deal with in the petroleum industry can be described by random functions that are combinations of regionalized and random variables. Regionalized variable theory is based on the statistics of the RV,    which differs from ordinary scalar random variables in its spatial continuity, yet still possesses the usual distribution statistics, such as mean and variance. The RV also differs in that it has a defined location.
Reservoir models are constructed by distributing petrophysical properties in 3D space with geologic models as a template. Geologic models are constructed by distributing facies within a sequence stratigraphic framework using the systematic distribution of facies within a depositional model as a guide. There are many types of facies, and facies selection is normally based on the question to be answered. Thus, numerous "depositional" facies are commonly described from core material. Once a sequence model is built, however, the problem is to convert the geologic model into a reservoir model by populating the geologic model with petrophysical data.
A reservoir characterization study is a part of the development of a reservoir model. This article describes each of the basic elements involved in a reservoir characterization study. The result of reservoir characterization is the creation of the shared-earth model. The shared-earth model provides for efficient updating of the critical information necessary for 3D modeling. At the basic interpretation stage, the discipline expert interprets the primary data, whereas the geologist and geophysicist collaborate on the structure model and sequence definition.
A pixel-based model assumes that the variable to be simulated is a realization of a continuous (Gaussian) random function. Using the spatial model, search ellipse, and control data, a pixel-based method simulates values grid node by grid node. Some of the most popular pixel-based algorithms are: turning bands, sequential Gaussian, sequential indicator, truncated Gaussian, and simulated annealing. Each method can produce a range of realizations that capture the uncertainty of an regionalized variable (RV), and so the method choice here will be based on the goals and on data types and their availability. The pixel-based method works best in the presence of facies associations that vary smoothly across the reservoir, as often is the case in deltaic or shallow marine reservoirs.