SRM D3.2 Geostatistics Summarymdw-edited.pdf

Shell E&P RESTRICTED
GEOSTATISTICS FOR RESERVOIR MODELLING
– SUMMARY –
Yann C. Dexcote (IRM team, Houston)
with contributions from IRM Teams, Frans Floris, Chris
Townsend, Caroline Hern, Xavier le Varlet, Isatis, Gocad,
Earthworks, Statios & various projects.
1

A BIT OF HISTORY…
• The foundation for geostatistical techniques
was established by people like Kolmogorov,
Weiner, Matern, and Gandin in the early
1900's
• Geostatistics was started in the 1960's by
Krige and Sichel in South Africa and Matheron
in France. Two of Matheron's first students
(Journel and David) would start new centers of
teaching and research in the USA and Canada
Application became popular in mining and
meteorology.
• Now, these techniques are applied in many
fields – fisheries, forestry, environmental and…
extensively used by major oil companies!
Krige (?)
Matheron

Geostatistics is the statistics of spatially distributed data, e.g. depths, velocity, thickness
etc…
 Geostatistics provides tools to:
 Describe and quantify the spatial continuity of a set of samples.
 Estimate values at unsampled locations. In addition to a map of estimates, a map
of the uncertainty of those estimates is created at the same time.
 Create different equiprobable images of the subsurface (stochastic imaging or
simulation of multiple geostatistical realizations).
 Geostatistics is THE framework for integrating data from different sources and scales
such as well and seismic data.
 Geostatistics is a set of tools and models providing means to quantify the uncertainty
associated to our attempts to fill up the gaps between the data available.
 “Better that we create a subjective model of uncertainty rather than an illusion
of certainty” A.G. Journel, SCRF, Stanford University.
WHAT IS GEOSTATISTICS ?

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GEOSTATISTICS: ADVERTISING…

Geological concept
+
Facies, ntg, , K logs
data
+
Seismic
Data
Geology
Well logs
Statistical and
spatial analysis
Data analysis
Variograms
Data analysis
Zones subdivision +
Structural Framework
Correlation panel
Reservoir
Correlation &
Structural
modelling
Faults
Facies
Modelling
Facies model Porosity Model
Porosity
Modelling
Sw= f { K , H }
Saturation
Saturation model
Permeability model
Permeability
Modelling
THE GEOSTATISTICS IN THE S.M. WORKFLOW
5
!!! Never again
“5000 x 5000 x
10” (default
parameters) !!!
Should also
be used
here!

PURPOSE OF DATA ANALYSIS FOR GEOSTATISTICS
1. QA/QC your data
 Range is OK?
 No negative NTG ? No Por. above 43%?
2. Ensure that your data is stationary:
 Remove drift in data (called “trend” in statistics):
 Variogram analysis requires that mean & variance of the variables to model do
not vary in space.
3. Normalize:
 Gaussian distribution has some favorable
mathematical properties that makes the algorithms
tractable and fast. Mandatory for simulation.
Process is called “Data transformation”
6

Houston, TX:
Correlation of
elevation over 50 km
possible
Rockies, CO:
Correlation of
elevation over 50 km
pretty tricky
THE CONCEPT OF CHARACTERISTIC SPATIAL VARIABILITY

VARIOGRAM DEFINITION
h2
h2
h2 = 2 x h1
East
• The variogram is somewhat non-intuitive; an increase in the
semivariance represents a decrease in the correlation.
Overall variance of data (sill)
Correlation length
• At any distance larger than the correlation length, the semivariance
becomes equal to the overall variance of the data. This is the greatest
distance over which the value at a point is related to the value at
another point.
h1
h1
h1
h1
h1
h1
h1
h1
h1
• At a distance of zero, the correlation is perfect.
Make sure the spatial trend has been removed from the data !
Lag distance
Semivariance
(g)
The variogram is a measure of the semivariance, i.e.
average dissimilarity between samples measured at
increasing distances h apart (lag).  
 
N
x
z
h
x
z
h
i
i
2
)
(
)
(
N
1
i
2





g

Trend = simple tilt
Red line = average elevation of surface
Spread
of data (what
you want to
analyse)
Much bigger
spread of data
(and hence
bigger standard
deviation.
Dashed red line = new average elevation of surface
SPATIAL TREND (EXAMPLE!)
 No Trend
 With a Trend (vertical trend, etc…)
In the modelling this trend will “overprint” the native variability of the data!

VARIOGRAM MODEL
Behavior at origin (slope).
Privilege short distance!
Anisotropies. E.g. determine if
your property is directionally
dependent in the X, Y space.
Range. That’s a key parameter
for your modelling
Nugget effect. Tell you
something about your data…
Sill. Tells you about variance of
your data
10
THINK B.A.R.N.S
Important factors to
consider to interpret &
model variograms:

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.2
1.1
g
Separation
Distance
0 m 2000
1000 3000 4000 5000 6000 7000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.2
1.1
g
Separation
Distance
0 ft 4
2 6 10
8 12 14
Vertical Variogram Horizontal Variogram
Major axis
Minor axis
WHAT DO YOU SEE?

•Geologic cyclicity (hole effect): Geological
phenomena often occur repetitively over geologic
time leading to repetitive or cyclic variations in the
facies and petrophysical properties.
•Multiple scales of heterogeneity: Generally exist to a degree in most
depositional settings. Dual scales of variability are the most common. Long
correlation ranges in these variograms are related to large-scale variations
while shorter correlation ranges correspond to features which are
developed in some areas of the field.
Shuaiba Facies association – Algae
patches expected to have a shorter
scale facies variability than the full
ramp (Source: EP 2003-5272
report)
“THE” ANSWER…

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.2
1.1
g
Separation
Distance
0 ft 4
2 6 10
8 12 14
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.2
1.1
g
Separation
Distance
0 m 20
10 30 40 50 60 70
Vertical Variogram
Major axis
Minor axis
Horizontal Variogram
Vertical Variogram Vertical Variogram
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.2
1.1
g
Separation
Distance
0 ft 2
1 3 4 5 6 7
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.2
1.1
g
Separation
Distance
0 ft 2
1 3 4 5 6 7
ANOTHER EXAMPLE?

•Geological trends: Fining or coarsening upward or
the systematic decrease in reservoir quality from
proximal to distal portions of the depositional system. In
a fining upward sedimentary package, the high
porosity at the base of the unit is negatively correlated
with low porosity at the top.
•Areal property trends: For instance facies
progradation/retrogradation in a given direction,
reservoir property degradation towards the flank of an
anticline by differential compaction.
•Stratigraphic layering: There are often
stratigraphic layer-like features or vertical trends that
persist over the entire areal extent of the study area.
Fining upward sequence in
point-bar deposits (Source:
RMKB)
Conceptual model for
shoreface deposits
(Source: RMKB)
Yibal field cross-section – degradation or reservoir
properties towards the flank because of different
compaction & diagenesis (Source: Yibal VAR3 report)

REFERENCES
Shell Website
 http://sww-irm.shell.com/rmkb/clastics/geostats/index.htm  Geostats in the RMKB
External R&D
 Stanford Center for Reservoir Forecasting, U. Stanford (Consortium)
Deutsch, C.V., Geostatistical Reservoir Modeling, Applied Geostatistics Series, Oxford
U. Press, 2002
 Ecole des Mines de Paris http://cg.ensmp.fr/, Institut francais du Petrole http://www.ifp.fr,
 http://www.ai-geostats.org/
 Geovariances http://www.geovariances.fr
Lots of downloads,
links, training
material, etc…

PROPERTY MODELLING USING SEISMIC
Primary variable (i.e. porosity) is constrained by a secondary variable (i.e.
seismic attribute).
The barbarous name for the most currently used method is the “Sequential
Gaussian Simulation by co-located co-kriging with locally varying mean”
In other word you basically use the secondary variable to replace the mean in a
simple kriging system, using a correlation coefficient variable in space to
constrain your property
Secondary variable can be any seismic measurement/attribute etc… BUT the
Geophysicist should demonstrate that a linear relation between primary and
secondary data exists, with a reasonable level of confidence (i.e. correlation
factor above +/- 0.7)
Primary
Variable
(e.g. Porosity)
Secondary
Data
(e.g. seismic
impedance)
Co-located point
Correlation : r

High r (-1)
Low r (-0.01)
NB - In these 3 cases the variogram dimensions are the same
Tip: calculate vertical
variogram on well data, and
horizontal variogram on the
seismic.
A very low correlation coefficient
assigns more value to the logs
and little to the secondary
property, the acoustic impedance
A very high correlation coefficient
implies a high confidence in the
property distribution in the original
acoustic impedance cube
Malampaya Field
PROPERTY MOD. USING SEISMIC (EXAMPLE)

SRM D3.2 Geostatistics Summarymdw-edited.pdf

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