The
reader must grasp a number of basic principles to understand the strengths
and weaknesses of geostatistics. In this section, principles regarding
spatial modeling, sample support, and scale are discussed.
By their very nature, earth
science data have an inherent spatial structure. That is, variables we
measure, like elevation, lithology, or porosity, are tied to geographic
positions on the earth. Observations taken in close proximity will be more
strongly correlated than observations separated by more distance. Although
this relationship is an accepted norm to geologists, it is not the case in
most other sciences. Classical statistical analysis does not necessarily
account for spatial relationships, and its use in the earth sciences can
often produce misleading results. A prerequisite of any geostatistical
study is the development of the spatial model. R. A. Olea provides an
excellent overview of semivariograms, the tool used in geostatistics to
estimate and model spatial variability, in the chapter entitled
Fundamentals of Semivariogram
Estimation
, Modeling, and
Usage.
The sample support is the
area or volume over which a measurement is made. Understanding how one
measurement relates to another in close proximity is critical because
geostatistical methods are highly sensitive to the spatial controls on
variability. This is particularly true with tensor variables like
permeability, for which mathematical averaging over volumes of various
dimensions tends to be a problem. Y. Anguy et al. discuss how image
analytical data can be blended with physical measurements to provide
information for modeling permeability at scales greater than the
permeability plug in The Sample Support Problem for Permeability
Assessment in Sandstone Reservoirs.
In the nineteenth century,
Johannas Walther, a noted geologist, recognized the Law of Correlation of
facies. The law, as paraphrased by Krumbine and Sloss (1963, Stratigraphy
and Sedimentation, W. H. Freeman and Company), states that "in a given
sedimentary cycle, the same succession of facies that occurs laterally is
also present in vertical succession." Taken commutatively, geologists have
often attempted to predict the lateral variability of facies from a
vertical sequence of stratigraphy. In the chapter Theory and
Applications of Vertical Variability Measures from Markov Chain
Analysis, J. H. Doveton presents an interesting alternative to the
geostatistical simulation approach of reservoir facies. He points out that
Markovian statistics of vertical variability are applicable to selected
problems of lateral facies prediction and simulation.
The final chapter in this
section is A Review of Basic Upscaling Procedures: Advantages and
Disadvantages. Upscaling is the term applied to the process of
changing from a fine scale of resolution to a coarser scale of resolution.
The process is often necessary when translating a digital geostatistical
model consisting of perhaps millions of grid nodes to a model consisting
of tens of thousands of grid nodes, which is a more economically viable
size for reservoir simulation. All of the techniques presently available
cause a severe reduction in vertical and horizontal resolution, and thus
can present a problem for estimating reservoir performance and
maintenance. Rather than trying to provide an exhaustive review of all
practical details, J. Mansoori highlights permeability upscaling methods
and provides an annotated bibliography of pertinent reference
literature.