Transcript GEOSTATISTICS FOR SEISMIC DATA INTEGRATION IN EARTH …

INTRODUCTION
Outline
•
A Brief Historical Perspective
•
The interaction between 3D Earth Modeling and Geostatistics
•
Basic Probability and Statistics Reminders
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RANDOM VARIABLES
A random variable takes certain values with certain probabilities.
Example:
Z = sum of two dice
FREQUENCY
NORMALIZED)
FREQUENCY (NOT
PROBABILITYHISTOGRAM
DENSITY FUNCTION
7
6
5
4
Série1
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
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SUM OF TWO DICE
SUM OF TWO DICE
Each value, for instance 4, is a realization
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1-12
THE IMPACT OF AVERAGING (2)
HISTOGRAMS
1x1
1x1
Scale
27x27
9x9
3x3
1x1
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Count
100
900
8100
72900
9x9
9x9
Minimum
13.55%
9.43%
6.12%
4.80%
Maximum
40.73%
53.47%
75.58%
98.87%
27x27
27x27
Mean
24.42%
24.42%
24.42%
24.42%
Std. Dev.
6.49%
8.27%
9.89%
10.34%
Correlation
0.72
0.90
0.99
1.00
P. Delfiner/X. Freulon
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THE SUPPORT EFFECT
(FRYKMAN AND DEUTSCH, 2002)
Impact on Cut-off
Histogram
of core F
Variance is
volume-dependent!
Histogram
of log F
Well log
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NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, s =5)
95%
15
35
CONFIDENCE INTERVAL:
 ( x  m) 2 
1
f ( x) 
exp
also called N (m, s )
 m-2
2
95%
of
values
fall
between
2s  s and m+2s
s 2

Porosity Uncertainty: Df2s
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1-26
INTRODUCTION
Lessons Learned
•
Geostatistics role in geosciences still evolving
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Geostatistics more and more closely integrated with earth modeling
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Probability and statistics help quantify degree of knowledge
•
Support effect : decrease of variance as volume of support increases
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Confidence interval closely related to mean and standard deviation
for normal distribution
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The correlation coefficient quantifies linear relationships
•
Trend surface analysis is a useful model, but too simple
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NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, s =5)
95%
15
35
CONFIDENCE INTERVAL:
 ( x  m) 2 
1
f ( x) 
exp
also called N (m, s )
 m-2
2
95%
of
values
fall
between
2s  s and m+2s
s 2

Porosity Uncertainty: Df2s
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THE COVARIANCE AND THE VARIOGRAM
Outline
•
Stationarity
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How geostatistics sees the world. The model.
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How to calculate a variogram
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A gallery of variogram models
•
Examples
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STATIONARITY OF THE MEAN
Stationary
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Nonstationary
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STATIONARITY OF THE VARIANCE (1)
A spatial phenomenon can be modeled using 2 terms:
• a low-frequency trend
• a residual
Constant trend: stationary variable Quadratic trend + stationary residual
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P. Delfiner/X. Freulon
2-1
STATIONARITY OF THE VARIANCE (2)
The residual should have a constant variance
A variable with
A variable with
• constant trend and
• quadratic trend and
• residual with varying variance
• residual with varying variance
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P. Delfiner/X. Freulon
2-2
WHAT TO DO WHEN NOT ENOUGH DATA ARE
AVAILABLE?
Vertical variograms
Vertical Wells
Variance gives sill of
horizontal variograms
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Horizontal Wells
Horizontal variograms
A priori
geological
knowledge
Behavior at origin and
nugget effect
Seismic data
Horizontal anisotropy ratios
and ranges
2-39
THE COVARIANCE AND THE VARIOGRAM
Lessons Learned
•
The model: low frequency trend + higher frequency residual +noise
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Variogram model more general than stationary covariances
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Meaning of the various parameters of the variogram model
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Relationship between fractals and geostatistics, covariance and
spectral density
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KRIGING AND COKRIGING
Outline
•
What is kriging
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How noise is handled by kriging. Error Cokriging
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Factorial Kriging for removing acquisition footprints
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Combining seismic and well information
– External Drift
– Collocated Cokriging
•
Kriging versus other interpolating functions
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NUGGET EFFECT VS NYQUIST FREQUENCY
x
d
x
x
x
x
x
x
x
x
x
x
x
Distance between data=d
g(h)
Minimum detectable variogram range = d
Minimum detectable wavelength = 2d
h
0
d
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Maximum detectable spatial frequency = 1/(2d)
(m/s)2
(m/s)2
THE FACTORIAL KRIGING MODEL
MARINE EXAMPLE: HORIZON-CONSISTENT VSTACK (3)
in-line effect
(4) Spherical
Final model
 (D1)
300 m (D2)
450 (m/s)2
geophysicist effect
(3) Spherical
1600 m (D1)
100 m (D2)
100 (m/s)2
Geological signal
(1) Spherical 7500 m, 1000 (m/s)2
(2) Spherical 1600 m, 300 (m/s)2
m
m
artefacts
(1) Linear 1000 (m/s)2
(2) Spherical 300 (m/s)2
(3) Spherical 100 (m/s)2
(4) Spherical 450 (m/s)2
(5) Nugget 400 (m/s)2
m
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J.L. Piazza and L. Sandjivy
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INTRODUCING EXTERNAL DRIFT
AND COLLOCATED COKRIGING
The situation
• Scattered well data giving exact measurements of one
parameter (depth, average velocity, porosity, thickness of a
lithology…)
• 2D or 3D seismic data giving information about the
variations of this parameter away from the wells (time,
stacking velocity, inverted impedance, seismic attribute…)
The problem
• How to combine well and seismic information properly, in such a
way that the parameter measured at the well is interpolated away
from the well using the seismic information?
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THE EXTERNAL-DRIFT MODEL
Two variables Z(x) and S(x)
S(x) assumed to be known at each location x
S(x) defines the shape of Z(x)
Z ( x)  a0  a1S (x)  R(x)
Deterministic
external-drift
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Random
residual
V. Bigault de Cazanove
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COKRIGING
Two variables Z1(x) and Z2(x) (such as porosity & acoustic impedance)
Use of Z1 and Z2 data to get a better interpolation of Z1
Z1cok (x0 ) 
 Z (x )   Z (x )
i 1 i
i
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Porosity
estimation by
cokriging
j 2
j
j
Porosity
data at wells
Acoustic
impedance data
from seismic
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COLLOCATED COKRIGING
COKRIGING
Z1cok (x0 ) 
 Z (x )   Z (x )
i 1 i
i
j 2
j
j
Complicated system of equations
Requires variograms of Z1, Z2, cross-variograms of Z1 and Z2
COLLOCATED
COKRIGING
Z1ccok (x0 ) 
 Z (x )  Z (x )
i 1 i
i
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2
0
COLLOCATED COKRIGING
(JEFFERY ET AL., 1996)
WELL CONTROL
DEPTHING VELOCITY
ISOTROPIC VARIOGRAM
CORRELATION 0.76
RESIDUAL GRAVITY
ISOTROPIC VARIOGRAM
Just the
variance
ofimprovement
residual
Cross-validation
shows
25 %
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gravity is used, not the
(Mean absolute error
22 to 15.5 m/s)
wholefrom
variogram!
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EXTERNAL DRIFT OR COLLOCATED COKRIGING?
External Drift
Collocated Cokriging
Model
Seismic is low frequency term
Correlation coeff between
seismic & primary variable
Input
Seismic map and wells
Variogram of residuals
Seismic map and wells
Correlation coefficient
Variogram of primary variable
Variance of seismic data
Properties
Equal to linear transform of
seismic beyond variogram range
Interaction between variogram
model and correlation coeff
Applications
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Construction of structural model
Interpolation of
petrophysical parameters
KRIGING AND COKRIGING
Lessons Learned
•
Kriging a weighted average of surrounding data points
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Nugget effect can be interpreted as variance of random errors
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Factorial kriging can handle multiscale variogram models
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Two techniques are preferred for combining seismic and wells:
- External Drift
- Collocated Cokriging
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Kriging surface expression similar to that generated by splines
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CONDITIONAL SIMULATION
Outline
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Monte-Carlo simulation reminders
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Conditional simulation versus kriging
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How are conditional simulations realisations produced?
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Multivariate conditional simulations
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Conditional simulation of lithotypes
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Constraining conditional simulations of lithotypes by seismic
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Generalized multi-scale geostatistical reservoir models
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THE THREE PROSPECTS
+
+
m1=75 s1=15
m3=200 s3=40
m2=100 s2=25
=
Independence
assumption:
conclusion
obtained by
Monte-Carlo
simulation (or
by properly
combining
variances)
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s=50
s=80
m=375
Full
dependence
assumption:
conclusion
obtained by
simply adding
min and max
of prospects
4-7
DEPENDENCE OR INDEPENDENCE?
1. Independence: Variances are added:
s 2  s12  s 22  s 32
2. Full Dependence: Confidence Intervals (or standard
deviations in the gaussian case) are added
s  s1  s 2  s 3
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A KRIGING EXAMPLE IN 3D (LAMY ET AL., 1998b)
Why should
the reservoir
be smooth
precisely away
from the data
points?
4
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AI
km.g / s.cm3
9
N
Total UK Geoscience Research Centre
4-10
KRIGING OR CONDITIONAL SIMULATION?
Conditional simulation
Kriging
Multiple realizations.
One “deterministic” model.
Properties
Honors wells,
honors histogram, variogram,
spectral density.
Honors wells,
minimizes error variance.
Image
Noisy, especially if variogram
model is noisy.
Smooth, especially if
variogram model is noisy.
Data
points
Image has same variability
everywhere. Data location
cannot be guessed from image.
Tendency to come back to
trend away from data. Data
location can be spotted.
Output
Use
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Heterogeneity Modeling,
Uncertainty quantification
Mapping
4-16
CONDITIONAL SIMULATION
LESSONS LEARNED
•
Conditional simulation generates representative heterogeneity
models. Kriging does not.
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SGS and SIS most flexible simulation algorithms.
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Multivariate conditional simulation techniques can be used to
account for correlations between various realizations.
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Bayesian-like techniques most suitable for constraining lithotype
models by seismic data.
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Geostatistical conditional simulation provides toolkit for generating
lithotype and petrophysical models at all scales.
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GEOSTATISTICAL INVERSION
Outline
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What is geostatistical inversion
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Examples of geostatistical inversion
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Using geostatistical inversion results to predict other
petrophysical parameters and lithotypes
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GEOSTATISTICAL INVERSION
Lessons Learned
•
Geostatistical Inversion generates acoustic impedance models at
higher frequency than the seismic data.
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Non-uniqueness quantified through multiple realizations.
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Geostatistical inversion still a tedious exercise, in terms of
processing time and processing of multi-realizations.
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Emerging applications for predicting petrophysical parameters and
lithotypes from acoustic impedance realizations.
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QUANTIFYING UNCERTAINTIES
Outline
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Why should we quantify uncertainties
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Structural uncertainties. How to quantify them?
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Combining all uncertainties affecting the 3D earth model
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Multirealization vs scenario-based approaches
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Demystifying uncertainty quantification approaches
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EARTH MODELLING AND
QUANTIFICATION OF RESERVOIR UNCERTAINTIES
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Geometry
Impact on GRV!
Static properties
Impact on OIP!
Dynamic properties
Impact on Reserves!
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QUANTIFICATION OF STRUCTURAL UNCERTAINTIES
THE APPROACH
• Identify uncertainties in the interpretation workflow,
• Quantify their magnitude (Confidence interval)
• Geostatistical Simulation
• Statistical Analysis
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Geostatistican ’s
input
• Measure of their impact on the results (GRV,OIP...)
Interpreter ’s
input
• Estimation of uncertainties
pdf
NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION
CASE STUDY (ABRAHAMSEN ET AL., 2000)
GRV (Mm3)
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Base case = 652 Mm3
QUANTIFYING UNCERTAINTIES
Lessons Learned
•
Geostatistical techniques can be used to quantify the combined
impact of uncertainties affecting the earth model.
•
Uncertainty-quantification nothing more than translating input
uncertainties into output uncertainties. Input is always subjective.
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3 AREAS WHERE GEOSTATISTICS IS CRUCIAL
• Generation of 3D heterogeneity models
• Integration of seismic data in reservoir models
• Uncertainty quantification
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WEBSITES ABOUT PETROLEUM GEOSTATISTICS
www.ualberta.ca/~cdeutsch/
ekofisk.stanford.edu/SCRFweb/index.html
www.math.ntnu.no/~omre
www.cg.ensmp.fr
www.tucrs.utulsa.edu/joint_industry_project.htm
www.ai-geostats.org
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BOOKS AND PAPERS TO READ
AAPG Computer Applications in Geology, No. 3, Stochastic Modeling and
Geostatistics, J.M. Yarus and R.L. Chambers eds
Chilès, J.P., and Delfiner, P., 1999, Geostatistics. Modeling Spatial Uncertainty,
Wiley Series in Probability and Statistics, Wiley & Sons, 695p.
Deutsch, C.V., and Journel, A.G., 1992, GSLIB, Geostatistical Software Library
and User’s Guide, New York, Oxford University Press, 340p.
Doyen, P.M., 1988, Porosity from Seismic Data: A Geostatistical Approach,
Geophysics, Vol. 53, No. 10, p. 1263-1275.
Isaaks, E.H., and Srivastava, R.M., 1989, Applied Geostatistics, New York,
Oxford University Press, 561p.
Lia, O., Omre, H., Tjelmeland, H., Holden, L., and Egeland, T., 1997,
Uncertainties in Reservoir Production Forecasts, AAPG Bulletin, Vol. 81, No.
5, May 1997, p. 775-802.
Thore, P., Shtuka, A., Lecour, M., Ait-Ettajer, T., and Cognot, R., 2002,
Structural Uncertainties: Determination, Management, and Applications,
Geophysics, Vol. 67, No. 3, May-June 2002, p. 840-852.
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