9. Universal Kriking 11 Dan 13 Nov 2013

8/12/2019 9. Universal Kriking 11 Dan 13 Nov 2013

1/22

Others exerc ises for Ordinary K r ig ing

Field Example 4.4Ordinary

Kriging Cross Validation

Field Example 4.5Ordinary Kriging To Generate

Gross Thickness Maps.

Field Example 4.6Ordinary Kriging To

Estimates Original Oil in Place


2/22

Cokriging


3/22

Cokriging

21

1

2

1

10

ZZ Nj

jj

Ni

iicok xZxZxZ

000

2 xZxZVarxcokcok

0

2

0

20 xx kcok

Two variables Z1(x) and Z2(x) (such as porosity & acoustic impedance)

Use of Z1and Z2data to get a better interpolation of Z1

Thanks to the extra information coming from Z2(x):


4/22

Cross Variograms

Estimation of cross variograms

LuYuYLuXuXELc

2

1

LuyuyLuxuxLn

L ii

Ln

i

iic

12

1


5/22

LuYEuXELuYuXELCc

Cross covariance

Ln

i

i

Ln

i

iic uxLn

LuyuxLn

Lc11

11

Ln

i

i LuyLn 1

.1

LCCL ccc

0

n

i

i

n

i

i

n

i

iic uynux

nuyux

nc

111

1110


6/22

Numerical Example 3.7. Table 3.8

TABLE 3.8PERMEABILITY AND POROSITY DATA FORNUMERICAL EXAMPLE 3.7

Depth (ft) k(md)2,040 8.25 352,041 9.0 502,042

6.25

21

2,043 5.0 162,044 5.3 202,045 4.75 82,046 5.0 14


7/22

Cross variogram and cross covariance for data in

Numerical Example 3.7


8/22

Cokriging (Doyen, 1988)

Kriged porosity

Porosity by regression from

1/(acoustic impedance)

Porosity by

cokriging


9/22

Universal Kriging

Conventional Estimation Techniques


10/22

Stationarity

The residual should have a constant variance

A variable with

no trend and

a residual with varying dispersion

A variable with

a trend and

a residual with varying dispersion


11/22

Stationarity


12/22

Example Of Stationary And Nonstationary

Surfaces


13/22

Beware: Stationarity Is A Matter Of Scale!!!

Porosity log of well M-10x in the Dan Field (Almeida and Frykman, 1994)


14/22

The Basic Assumption Of Geostatistics

A geological variable z(x) is composed of a systematictrend and a random component.

z(x) is the realization of a random function Z(x)

equal to the sum of a trend m(x) and of a random

stationary residual R(x) of mean 0:

Z (x) = R(x) + m(x)

But how can we calculate statistics from just

one outcome?


15/22

The Stationarity Assumption

Mean and variance are independent

of location:

Covariance between values at two

locations only depends on distance:

mhxZExZE

2 hxZVarxZVar

hChxZxZCov ,


16/22

Universal Kriging

uRumuX

odneighborho

iuXum

umuXuR

L

i

ll ufaum0

uaaum 10

f0(u) =1 so that for a linear trend

2210 uauaaum

And for a quadratic trend

We define drift

Local mean is

Subtract local mean from individual samples, we obtain

Samples value


17/22

n

i

ii uXuX1

0

*

Begin with the ordinary Kriging

Application of the unbiased condition requares that

n

i

lili ufuf

1

0

for l= 0, , L

This imposes L+1 constraints

Applying the minimum variance,

0

00

0

01

0

1

1

010

01

110111

,

,

00

00

,,

,,

uf

uf

uuC

uuC

ufuf

ufuf

ufufuuCuuC

ufufuuCuuC

L

n

L

n

nLL

n

nLnnnn

Ln

n+L+1 Matrix


18/22

001

000

2,, ufuuCuuC

L

l

ll

n

i

iiE

The error variance is estimated by

010 uYaaum o

Another alternative uses the method of external drift. The trend in the data isProvided by the secondary variable Y.


19/22

Location of reservoir top well data

Field Example 4.8 Universal Kriging To Estimate Top of Reservoir


20/22

Reservoir top running east/west Reservoir top running north/south.


21/22

Cross-validation estimates for reservoir top using universal,

simple, and ordinary kriging


22/22

Map of reservoir top generated with universal kriging

Documents

9. Universal Kriking 11 Dan 13 Nov 2013