INTRODUCTION Outline

1
INTRODUCTIONOutline

• 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
PROBABILITY DENSITY FUNCTION
FREQUENCY (NOT NORMALIZED)
SUM OF TWO DICE
1-12
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THE IMPACT OF AVERAGING (2)HISTOGRAMS
1x1
9x9
27x27
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P. Delfiner/X. Freulon
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THE SUPPORT EFFECT(FRYKMAN AND DEUTSCH, 2002)
Histogram of core F
Variance is volume-dependent!
Histogram of log F
Well log
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NORMAL (OR GAUSSIAN) DISTRIBUTION (m25, s 5)
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INTRODUCTIONLessons Learned

• Geostatistics role in geosciences still evolving
• Geostatistics more and more closely integrated
  with earth modeling
• Probability and statistics help quantify degree
  of knowledge
• Support effect decrease of variance as volume
  of support increases
• Confidence interval closely related to mean and
  standard deviation for normal distribution
• The correlation coefficient quantifies linear
  relationships
• Trend surface analysis is a useful model, but too
  simple

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NORMAL (OR GAUSSIAN) DISTRIBUTION (m25, s 5)
1-26
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THE COVARIANCE AND THE VARIOGRAMOutline

• Stationarity
• How geostatistics sees the world. The model.
• How to calculate a variogram
• A gallery of variogram models
• Examples

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STATIONARITY OF THE MEAN
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STATIONARITY OF THE VARIANCE (1)

• A spatial phenomenon can be modeled using 2
  terms
• a low-frequency trend
• a residual

Quadratic trend stationary residual
Constant trend stationary variable
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P. Delfiner/X. Freulon
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STATIONARITY OF THE VARIANCE (2)
The residual should have a constant variance

• A variable with
• constant trend and
• residual with varying variance

• A variable with
• quadratic trend and
• residual with varying variance

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P. Delfiner/X. Freulon
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WHAT TO DO WHEN NOT ENOUGH DATA ARE AVAILABLE?
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THE COVARIANCE AND THE VARIOGRAMLessons Learned

• The model low frequency trend higher frequency
  residual noise
• Variogram model more general than stationary
  covariances
• Meaning of the various parameters of the
  variogram model
• Relationship between fractals and geostatistics,
  covariance and spectral density

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KRIGING AND COKRIGINGOutline

• What is kriging
• How noise is handled by kriging. Error Cokriging
• Factorial Kriging for removing acquisition
  footprints
• Combining seismic and well information
• External Drift
• Collocated Cokriging
• Kriging versus other interpolating functions

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NUGGET EFFECT VS NYQUIST FREQUENCY
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THE FACTORIAL KRIGING MODELMARINE EXAMPLE
HORIZON-CONSISTENT VSTACK (3)
(m/s)2
Geological signal (1) Spherical 7500 m, 1000
(m/s)2 (2) Spherical 1600 m, 300 (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)
Deterministic external-drift
Randomresidual
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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
Porosity estimation by cokriging
Acoustic impedance datafrom seismic
Porosity data at wells
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COLLOCATED COKRIGING
COKRIGING
Complicated system of equations Requires
variograms of Z1, Z2, cross-variograms of Z1 and
Z2
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COLLOCATED COKRIGING (JEFFERY ET AL., 1996)
WELL CONTROL DEPTHING VELOCITYISOTROPIC
VARIOGRAM
CORRELATION 0.76
RESIDUAL GRAVITY ISOTROPIC VARIOGRAM
Cross-validation shows 25 improvement (Mean
absolute error from 22 to 15.5 m/s)
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EXTERNAL DRIFT OR COLLOCATED COKRIGING?
Collocated Cokriging
External Drift
Correlation coeff betweenseismic primary
variable
Model
Seismic is low frequency term
Seismic map and wellsCorrelation
coefficientVariogram of primary
variableVariance of seismic data
Seismic map and wells Variogram of residuals
Input
Interaction between variogram model and
correlation coeff
Equal to linear transform of seismic beyond
variogram range
Properties
Interpolation of petrophysical parameters
Construction of structural model
Applications
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KRIGING AND COKRIGINGLessons Learned

• Kriging a weighted average of surrounding data
  points
• Nugget effect can be interpreted as variance of
  random errors
• Factorial kriging can handle multiscale variogram
  models
• Two techniques are preferred for combining
  seismic and wells - External Drift -
  Collocated Cokriging
• Kriging surface expression similar to that
  generated by splines

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CONDITIONAL SIMULATIONOutline

• Monte-Carlo simulation reminders
• Conditional simulation versus kriging
• How are conditional simulations realisations
  produced?
• Multivariate conditional simulations
• Conditional simulation of lithotypes
• Constraining conditional simulations of
  lithotypes by seismic
• Generalized multi-scale geostatistical reservoir
  models

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THE THREE PROSPECTS


m3200 s340
m175 s115
m2100 s225

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DEPENDENCE OR INDEPENDENCE?
1. Independence Variances are added 2. Full
Dependence Confidence Intervals (or standard
deviations in the gaussian case) are added
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A KRIGING EXAMPLE IN 3D (LAMY ET AL., 1998b)
AI km.g / s.cm3
4
9
4-10
Total UK Geoscience Research Centre
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KRIGING OR CONDITIONAL SIMULATION?
Kriging
Conditional simulation
Output
Multiple realizations.
One deterministic model.
Honors wells, honors histogram,
variogram, spectral density.
Honors wells, minimizes error variance.
Properties
Noisy, especially if variogram model is noisy.
Smooth, especially if variogram model is noisy.
Image
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.
Data points
Heterogeneity Modeling,Uncertainty quantification
Mapping
Use
4-16
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CONDITIONAL SIMULATIONLESSONS LEARNED

• Conditional simulation generates representative
  heterogeneity models. Kriging does not.
• SGS and SIS most flexible simulation algorithms.
• Multivariate conditional simulation techniques
  can be used to account for correlations between
  various realizations.
• Bayesian-like techniques most suitable for
  constraining lithotype models by seismic data.
• Geostatistical conditional simulation provides
  toolkit for generating lithotype and
  petrophysical models at all scales.

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GEOSTATISTICAL INVERSIONOutline

• What is geostatistical inversion
• Examples of geostatistical inversion
• Using geostatistical inversion results to predict
  other petrophysical parameters and lithotypes

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GEOSTATISTICAL INVERSIONLessons Learned

• Geostatistical Inversion generates acoustic
  impedance models at higher frequency than the
  seismic data.
• Non-uniqueness quantified through multiple
  realizations.
• Geostatistical inversion still a tedious
  exercise, in terms of processing time and
  processing of multi-realizations.
• Emerging applications for predicting
  petrophysical parameters and lithotypes from
  acoustic impedance realizations.

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QUANTIFYING UNCERTAINTIESOutline

• Why should we quantify uncertainties
• Structural uncertainties. How to quantify them?
• Combining all uncertainties affecting the 3D
  earth model
• Multirealization vs scenario-based approaches
• Demystifying uncertainty quantification
  approaches

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EARTH MODELLING AND QUANTIFICATION OF RESERVOIR
UNCERTAINTIES
Geometry
Static properties
Dynamic properties
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QUANTIFICATION OF STRUCTURAL UNCERTAINTIESTHE
APPROACH
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NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION
CASE STUDY (ABRAHAMSEN ET AL., 2000)
pdf
GRV (Mm3)
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QUANTIFYING UNCERTAINTIESLessons 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/SC
RFweb/index.html www.math.ntnu.no/omre www.cg.en
smp.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 Users 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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