Geoscience Reference
In-Depth Information
3.4.1 Kriging
3.4
Modelling Property
Distributions
Kriging is a fundamental spatial estimation tech-
nique related to statistical regression. The
approach was first developed by Matheron
( 1967 ) and named after his student Daniel
Krige who first applied the method for estimating
average gold grades at the Witwatersrand gold-
bearing reef complex in South Africa. To gain a
basic appreciation of Kriging, take the simple
case of an area we want to map given a few
data points, such as wells which intersect the
reservoir layer (Fig. 3.21 ).
We want to estimate a property, Z* at an
unmeasured location, o, based on known values
of Z i at locations x i . Kriging uses an interpolation
function:
Assuming we have a geological model with cer-
tain defined components (zones, model
elements), how should we go about distributing
properties within those volumes? There are a
number of widely used methods. We will first
summarize these methods and then discuss the
choice of method and input parameters.
The basic input for modelling spatial
petrophysical distribution in a given volume
requires the following:
￿ Mean and deviation for each parameter
(porosity, permeability, etc.);
￿ Cross-correlation between properties (e.g.
how well does porosity correlate with
permeability);
￿ Spatial correlation of the properties (i.e. how
rapidly does the property vary with position in
the reservoir);
￿ Vertical or lateral trends (how does the mean
value vary with position):
￿ Conditioning points (known data values at the
wells).
The question is “How should we use these
input data sensibly?” Commercial reservoir
modelling packages offer a wide range of
options, usually based on two or three underlying
geostatistical methods (e.g. Hohn 1999 ; Deutsch
2002 ). Our purpose is to understand what these
methods do and how to use them wisely in build-
ing a reservoir model.
X
n
Z ¼
1 ˉ i Z i
ð
3
:
25
Þ
i
¼
where
ˉ i are the weights, and employs an objec-
tive function for minimization of variance.
That is to say a set of weights are found to
obtain a minimum expected variance given the
available known data points.
The algorithm finds values for
such that the
objective function is honoured. The correlation
function ensures gradual changes, and Kriging
will tend to give a smooth function which is
close to the local mean. Mathematically there
are several ways of Kriging, depending on the
assumptions made. Simple Kriging is mathemat-
ically the simplest, but assumes that the mean
ˉ
z
x 1
z
z
o
x 3
z
x 2
Fig. 3.21 Illustration of
the Kriging method                   Search WWH ::

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