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