Next Generation Geological Modeling for Hydrocarbon Reservoir Characterization - Advances in Data, Methods, Models and Their Applications in Geoscience - page 217

workflows for quantitative uncertainty assessment and risk management (Maučec and
Cullick, 2011) combining two critical steps of reservoir characterization: a) the reconciliation
of geomodels with well-production and seismic data, referred to as history-matching (HM)
(Oliver and Chen, 2011) and b) dynamic ranking and selection of representative model
realizations for reservoir production forecast.

2. Methods and techniques
2.1 Highlights of geostatistical analysis and modeling
To capture diversity of geologically-complex reservoirs, the next-generation of geological
modeling tools are relying increasingly on geostatistical methodologies (Isaaks and
Srivastava, 1989; Yarus and Chambers, 1994; Chambers et al. , 2000a; Chambers et al. , 2000b;
Deutsch, 2002; Yarus and Chambers, 2010). The in-depth explanation of geostatistical
terminology is available in Olea, 1991. Tailored to identify data limitations and provide
better representation of reservoir heterogeneity, geostatistical tools are particularly effective
when dealing with data sets with vastly different degrees of spatial density and diverse
vertical and horizontal resolution. Classical statistics methods are based on an underlying
assumption of data independence in randomly sampled measurements. These assumptions
are not true for geosciences data sets, where the data gathered are regionalized (map-able)
and demonstrate strong dependence on the distance and orientation. Geostatistical tools
provide the unique ability to integrate different types of data, with pronounced variation in
scales and direction of continuity. One of the fundamental tools in geostatistics is the
variogram , a measure of statistical dissimilarity between the pairs of data measurements. It
represents a model of spatial correlation and continuity that quantifies the directions and
scales of continuity. Variogram analysis can be applied to any regionalized variable X and is
used to compute the average square differences between data measurements based on
different separation intervals h , known as the lag interval.

n



2

( XX  )

i i  h

(1)

i  1

 () h 

2 n

where  (h) corresponds to the semivariogram (note that denominator 2n represents the
symmetry relation between data points X i and X i+h ) and index i runs over the number of
data pairs, n . We will in this document refer to semivariogram simply as a variogram.
Visualization of a generic variogram  (h) is given in Fig. 2. The left-hand panel corresponds
to three distinctive cases of geological continuity: the red curve describes the omni-
directional or isotopic variogram that assumes a single, average characteristic direction of
continuity, while blue and green curves, correspond to the minimum and maximum
direction of continuity, respectively; an anisotropic variogram. The curve of omni-
directional variogram will converge to the black line that represents the true variance of the
data and is usually referred to as sill . One additional parameter of the variogram, which is
not depicted in Fig. 2 is the nugget effect and corresponds to a discontinuity along the Y-axis
resulting in a vertical shift of the variogram curve at the origin. The distance at which the
variogram curve levels out at the sill corresponds to the data correlation range . The right-
hand panel of Fig. 2 is an example of the variogram polar plot, with the major ellipse axis,
corresponding to maximum and the (perpendicular) minor ellipse axis, to the minimum
direction of continuity.