Download - 9. Universal Kriking 11 Dan 13 Nov 2013
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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
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Cokriging
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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):
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Cross Variograms
Estimation of cross variograms
LuYuYLuXuXELc
2
1
LuyuyLuxuxLn
L ii
Ln
i
iic
12
1
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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
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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
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Cross variogram and cross covariance for data in
Numerical Example 3.7
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Cokriging (Doyen, 1988)
Kriged porosity
Porosity by regression from
1/(acoustic impedance)
Porosity by
cokriging
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Universal Kriging
Conventional Estimation Techniques
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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
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Stationarity
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Example Of Stationary And Nonstationary
Surfaces
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Beware: Stationarity Is A Matter Of Scale!!!
Porosity log of well M-10x in the Dan Field (Almeida and Frykman, 1994)
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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?
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The Stationarity Assumption
Mean and variance are independent
of location:
Covariance between values at two
locations only depends on distance:
mhxZExZE
2 hxZVarxZVar
hChxZxZCov ,
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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
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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
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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.
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Location of reservoir top well data
Field Example 4.8 Universal Kriging To Estimate Top of Reservoir
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Reservoir top running east/west Reservoir top running north/south.
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Cross-validation estimates for reservoir top using universal,
simple, and ordinary kriging
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Map of reservoir top generated with universal kriging