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and distribution are known and that they are

statistically stationary (i.e. a global mean).
Ordi-

nary Kriging
is more commonly used because it

assumes slightly weaker constraints, namely that

the mean is unknown but constant (i.e. a local

mean) and that the variogram function is known.

Fuller discussion of the Kriging method can be

found in many textbooks; Jensen et al. (
2000
)

and Leuangthong et al. (
2011
) give very accessi-

ble accounts for non-specialists.

fact close to the variance of the population. The

range (equivalent to the correlation length)

describes the 'separation distance' at which this

occurs. A theoretical semi-variogram has a

smooth function rising up towards the variance

while measured/observed semi-variogram often

has oscillations and complex variations due to,

for example, cyclicity in rock architecture.

The most common functions for the semi-

variogram are spherical, Gaussian and exponen-

tial - each giving a different rate of rise towards

the sill value (ref. Fig.
2.24
). Note that for a

specific situation (second order stationarity) the

semi-variogram is the inverse of the covariance

(de Marsilly
1986
, p. 292). Jensen et al. (
1995
)

and Jensen et al. (
2000
) give a more extensive

discussion on the application of the semi-

variogram to permeability data in the petroleum

reservoirs.

3.4.2 The Variogram

The variogram function describes the expected

spatial variation of a property. In Chap.
2
we

discussed the ability of the variogram to capture

element architecture. Here we employ the same

function as a modelling tool to estimate spatial

property variations within that architecture. We

recall that the semi-variance is half the expected

value of

3.4.3 Gaussian Simulation

the squared differences

in values

separated by h:

Gaussian Simulation covers a number of related

approaches for estimating reservoir properties

away from known points (well observations).

The Sequential Gaussian Simulation (SGS)

method can be summarized by the following

steps (Jensen et al.
2000
):

1. Transform the sampled data to be Gaussian;

2. Assign unconditioned cells

n

o

1

2
EZx

2

ʳ

ðÞ
¼

h

½

ð

þ

h

Þ

Zx

ðÞ

ð

3

:

26

Þ

The semi-variogram (Fig.
3.22
) has several

important features. The lag (h) is the separation

between two points. Two adjacent points will

tend to be similar and have a

(h) of close to

zero. A positive value at zero lag is known as the

nugget. As the lag increases the chance of two

points having a similar value decreases, and at

some distance a sill is reached where the average

difference between two points is large, and in

ʳ

¼

(inter-well)

conditioned cells (wells);

3. Define a random path to visit each cell;

4. For each cell locate a specified number of

conditioning data (the neighbourhood);

5. Perform Kriging in the neighbourhood to

determine the local mean and variance

(using the variogram as a constraint);

6. Draw a random number to sample the Gauss-

ian distribution (from step 5);

7. Add the new simulated value to the “known”

data. Repeat step 4.

Repeating steps 4-7 gives one realisation of a

Gaussian random distribution conditioned to

known points. Repeating step 3 gives a new

realisation. The average of a large number of

realisations will approach the kriged result. In

this way we can use Gaussian simulation to

g
(h)

Sill

Nugget

Range

Lag

Fig. 3.22
Sketch of the semi-variogram (
blue
¼
theoret-

ical function;
red
¼
function through observed data

points)