1. INTRODUCTION Outline A Brief Historical Perspective
The interaction between 3D Earth Modeling and Geostatistics
Basic Probability and Statistics Reminders
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

2. PROBABILITY DENSITY FUNCTION FREQUENCY (NOT NORMALIZED)
RANDOM VARIABLES
A random variable takes certain values with certain probabilities.
Example: Z = sum of two dice
PROBABILITY DENSITY FUNCTION
FREQUENCY (NOT NORMALIZED)
Now we will start with the second pillar of the course, that of probabilities, and we will just mention a few key results. A whole day would not be enough to cover this topic in enought detail, but hopefully I can remind you a few of the basic principles, without covering all the things that are in the book.
An interesting point with this slide is the fact that, although each throw of the two dice is of equal probability, it results in a histogram giving a higher probabilityto the value 7. We will see that the same happens when we run geostatistical realizations: although each of semm is strictly equiprobable, they will all result in value of oil-in-place or reserves that are not distributed in an equiprobable way.
The distribution here is triangular, and symmetrical, which means that mode, mean and median are all equal.
On peut rappeler si nécessaire que plus on aurait de dés plus on convergerait vers une distribution normale, et que c ’est d ’ailleurs comme cela que sont générés les réalisations de distributions normales. On pourrait aussi dire que la somme des pdfs est la convolution des deux cdfs de départ
Each value, for instance 4, is a realization
SUM OF TWO DICE
1-12

3. THE IMPACT OF AVERAGING (2) HISTOGRAMS
9x9
27x27
1x1
1x1
9x9
27x27
Obviously, as the size of the moving window increases, the scatter of the histogram decreases. The mean remains the same, but the standard deviation decreases. Correlation is between 1x1 cvalues and averaged values
Bien dire que malheureusement les 3 histogrammes ont l‘ air d ’avoir la même valeur du mode, mais qu ’on voit bien que ce n ’est pas le cas si on regarde bien l ’échelle.
This means that you cannnot talk of variance or standard deviation as an intrinsic property, since it depends on scale. Same applies to the coefficient of variation.
The mean remains constant butwould it be possible to predict how the variance decreases as a function of the number of averaged samples? We will see that the answer to this questions is one of the fundations of the geostatistical formalism.
Scale
Count
Minimum
Maximum
Mean
Std. Dev.
Correlation
27x27
100
13.55%
40.73%
24.42%
6.49%
0.72
9x9
900
9.43%
53.47%
8.27%
0.90
3x3
8100
6.12%
75.58%
9.89%
0.99
1x1
72900
4.80%
98.87%
10.34%
1.00
1-18
P. Delfiner/X. Freulon

4. Variance is volume-dependent!
THE SUPPORT EFFECT (FRYKMAN AND DEUTSCH, 2002)
Histogram of core F
Impact on Cut-off
Variance is volume-dependent!
Refer back to the example of Delfiner, and explain that the variogram can help us predict the change of support.
Histogram of log F
Well log
2-31

5. NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, s =5)15 35
95%
Another important definition is that of the confidence interval.
Thiswill prove very important when we dicuss uncertainty quantification. In the case of a normal distribution, we know that 90% of all the values are between mean plus two sigma, and mean minus two sigma.
This means that, if we know that a parameter follows a normal distribution, the best estimate is the mean and the true value of the parameter has a 95% chance to fall between the mean + twice the stddev and the mean minus twice the standdev.
This mean for instance that , in the case of a normal distribution, we can automatically translate a mean and a variance into a probability of being contained within a certain interval. On this example for instance, if the mean is 25 and the standard deviation is 5, we knwo that 95% of the values will be between 15 and 35.
The notion of confidence interval will be very important when we discuss uncertainty quantification. Being able to make a statement such as « I have a 95% chance that my true porosity value is between 15 and 35, this is an actual quantification of the uncertainty affecting porosity.
We know that the normal distribution is often used for porosity values.
The idea is uncertainty is that, when we draw a value in the distribution, we want to know which kind of SCATTER we should expect.
CONFIDENCE INTERVAL:
95% of values fall between m-2s and m+2s
Porosity Uncertainty: Df=2s
1-26

6. INTRODUCTION Lessons 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
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

7. NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, s =5)15 35
95%
Another important definition is that of the confidence interval.
Thiswill prove very important when we dicuss uncertainty quantification. In the case of a normal distribution, we know that 90% of all the values are between mean plus two sigma, and mean minus two sigma.
This means that, if we know that a parameter follows a normal distribution, the best estimate is the mean and the true value of the parameter has a 95% chance to fall between the mean + twice the stddev and the mean minus twice the standdev.
This mean for instance that , in the case of a normal distribution, we can automatically translate a mean and a variance into a probability of being contained within a certain interval. On this example for instance, if the mean is 25 and the standard deviation is 5, we knwo that 95% of the values will be between 15 and 35.
The notion of confidence interval will be very important when we discuss uncertainty quantification. Being able to make a statement such as « I have a 95% chance that my true porosity value is between 15 and 35, this is an actual quantification of the uncertainty affecting porosity.
We know that the normal distribution is often used for porosity values.
The idea is uncertainty is that, when we draw a value in the distribution, we want to know which kind of SCATTER we should expect.
CONFIDENCE INTERVAL:
95% of values fall between m-2s and m+2s
Porosity Uncertainty: Df=2s
1-26

8. THE COVARIANCE AND THE VARIOGRAM Outline
Stationarity
How geostatistics sees the world. The model.
How to calculate a variogram
A gallery of variogram models
Examples
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

9. STATIONARITY OF THE MEAN
Nonstationary
Stationary
Let us look at what stationarity means in 2DAgain, same remark, constant mean on the left (this is a synthetic case) but systematic trend
On the right (this is a filtered velocity field fromthe Handil field in Indonesia). Interpretation from MN Dufresne and Ph. Matheron. Handil est un gisement huile-gaz géant exploité par Total depuis les années 70, dans le delata de la Makaham..L ’étude avait été une intervention légère où l ’objectif était de fournir une image profondeur de l ’extension du gisement sur le flanc. Les puits dispiibles étaient situés en top, l ’extrapolation de l ’information de calage ou des vitesses apparentes est donc la préoccupation particulière et l ’utilistion d ’une dérive est un point sensible dans l ’estimation à distance des puits. La dérive linéaire semble bien portée par les données et son tutilisation est appuyée par les aspects d ’homogeneité et de cylindrité structurale du milieu. Les vitesses sont des viteesses de migration, d ’où leur aspect régulier et filtré.
2-3

10. STATIONARITY OF THE VARIANCE (1)
A spatial phenomenon can be modeled using 2 terms:
a low-frequency trend
a residual
Now we are going to leave this basic introduction to statistical concepts most of you probably already knew, and start discussion notions of spatial statistics. I would like to start with simple examples of what stationarity is all about.
A stationay phenomenon is one where at each location, the expected value is always the same,a non-stationary one is one where this means can be approximated by a deterministic function. We will see that this function is usually a polynomial of order zero, one or two, or sometimes a function that is closely related to the seismic data.
But when looking at stationarity, one should not only look at the variations of the mean, but also of the variance in space.
Constant trend: stationary variable
Quadratic trend + stationary residual
2-1
P. Delfiner/X. Freulon

11. STATIONARITY OF THE VARIANCE (2)
The residual should have a constant variance
On these two examples, you see that we have two datasets with a mean that can be approximated by a deterministic function, but the scatter around this means varies in space. For instance on the left-hand side picture, the scatter increases from left to tight.
A variable with
constant trend and
residual with varying variance
A variable with
quadratic trend and
residual with varying variance
2-2
P. Delfiner/X. Freulon

12. WHAT TO DO WHEN NOT ENOUGH DATA ARE AVAILABLE?
Vertical Wells
Vertical variograms
Variance gives sill of horizontal variograms
Horizontal Wells
Horizontal variograms
A priori geological knowledge
Behavior at origin and nugget effect
Seismic data
Horizontal anisotropy ratios and ranges
2-39

13. THE COVARIANCE AND THE VARIOGRAM Lessons 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
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

14. KRIGING AND COKRIGING Outline
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
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

15. NUGGET EFFECT VS NYQUIST FREQUENCY
Distance between data=d x d
g(h)
Minimum detectable variogram range = d h d
Minimum detectable wavelength = 2d
Pour Delfiner dans le rapport signal/bruit le signal ne comprend PAS le bruit.
Maximum detectable spatial frequency = 1/(2d)

16. THE FACTORIAL KRIGING MODEL MARINE EXAMPLE: HORIZON-CONSISTENT VSTACK (3)
(m/s)2
in-line effect
(4) Spherical  (D1)
300 m (D2)
450 (m/s)2
geophysicist effect
(3) Spherical m (D1) 100 m (D2) 100 (m/s)2
(m/s)2
Final model m
(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
artefacts
Geological signal
(1) Spherical 7500 m, 1000 (m/s)2
(2) Spherical 1600 m, 300 (m/s)2 m
En haut à droite, c ’est simplement le plot du variogramme dans chaque direction (D1 et D2, pas très clair pour moi pourquoi il y a un rouge et un jaune) correspondant à la compilation des deux artefacts. En enlevant les effets correspondant à ces deux variogrammes, on va faire deux choses, car il y a bien deux effets différents:
1. Enlever la continuité forcée due à l ’effet géophyisicien. Ceci edonne la réponse à l ’excellent question posée en Indonésie sur le pourquoi de ces lignes rouges le long des inlines dans les résidus filtrés.
2. Enlever l ’effet de pépite dans la direction N-S dùû au passage d ’une cross-line à l ’autre. Cet effet de pépite est contenu surtout dans le modèle à portée infini selon D1, mais aussi un peu dans le modèle à portée 1600 m selon D1, puisqu ’il y en a pour 100(m/s)2. C ’est un peu dommage car cela veut dire que l ’on est obligé de filtrer les deux effets en même temps, car si on ne filtre que l ’un des deux variogrammes on ne virerar qu ’une partie de l ’effet de pépite d ’une in-line à l ’autre. Evidemment l ’effet de pépite d ’une inline à l ’autre doit être modélisé par un modèle de portée inférieur à la distance entre in line (500m).
L ’effet de pépite affecte à la fois les quatre directions, comme on le voit sur le modèle ajusté à D1 sur la figure du bas. Cet effet de pépite représente le bruit blanc associé à l ’incertitude lié aux vitesses du stack.
Le modèle ajusté final en bas contient le modèle lié à la géologie plus l ’effet de pépite. Plus le modèle correspondant aux artefacts (celui qui est en haut à droite).
3-39
J.L. Piazza and L. Sandjivy

17. 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?
On préfère la méthode par FK qui ne demande pas de passer à une grille régulière.
Le tapering or padding est unpermet de faire du wrap-around.
D ’autre part on a unn ecart-type d ’estimation.

18. THE EXTERNAL-DRIFT MODEL Deterministic external-drift
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)
Voci le modèle.
Deterministic external-drift
Random residual
3-56
V. Bigault de Cazanove

19. Porosity estimation by cokriging Acoustic impedance data from seismic
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
Le cokrigeage comme généralsation simple du krigeage.
Evidemment, pour résoudre le système, il faudra utiliser les variogrames croisés déjà définis.
Porosity estimation by cokriging
Acoustic impedance data from seismic
Porosity data at wells
3-67

20. COLLOCATED COKRIGING COKRIGING Complicated system of equations
Requires variograms of Z1, Z2, cross-variograms of Z1 and Z2

21. COLLOCATED COKRIGING (JEFFERY ET AL., 1996)
WELL CONTROL DEPTHING VELOCITY ISOTROPIC VARIOGRAM
CORRELATION 0.76
Juste le variogramme des vitesses est la corrélation croisée est nécessaire, ce qui simplifie consdérablement les choses.
Grace au CCK, la cross-vlidation montre uné amélioration de 25% dans les estimations de vitesse.
RESIDUAL GRAVITY
ISOTROPIC VARIOGRAM
Just the variance of residual gravity is used, not the whole variogram!
Cross-validation shows 25 % improvement
(Mean absolute error from 22 to 15.5 m/s)
3-70

22. EXTERNAL DRIFT OR COLLOCATED COKRIGING?
Correlation coeff between seismic & primary variable
Model
Seismic is low frequency term
Seismic map and wells Correlation coefficient Variogram of primary variable Variance of seismic data
Seismic map and wells
Variogram of residuals
Input
Equal to linear transform of seismic beyond variogram range
Interaction between variogram model and correlation coeff
Properties
Interpolation of petrophysical parameters
Applications
Construction of structural model

23. KRIGING AND COKRIGING Lessons 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
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

24. CONDITIONAL SIMULATION Outline
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
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

25. + + = THE THREE PROSPECTS
m1=75 s1=15
m2=100 s2=25
m3=200 s3=40 =
m=375
Independence assumption: conclusion obtained by Monte-Carlo simulation (or by properly combining variances)
s=50
Full dependence assumption: conclusion obtained by simply adding min and max of prospects
s=80
4-7

26. DEPENDENCE OR INDEPENDENCE?
1. Independence: Variances are added:
2. Full Dependence: Confidence Intervals (or standard deviations in the gaussian case) are added

27. A KRIGING EXAMPLE IN 3D (LAMY ET AL., 1998b)
Why should the reservoir be smooth precisely away from the data points?
Krigeage des impédances sur une portion du champ de Nelson. Bien indiquer ici que l ’on ne travaille qu ’à partir des puits, pas de la sismique. AI
km.g / s.cm3 N 4 9
4-10
Total UK Geoscience Research Centre

28. KRIGING OR 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

29. CONDITIONAL SIMULATION LESSONS 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.
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

30. GEOSTATISTICAL INVERSION Outline
What is geostatistical inversion
Examples of geostatistical inversion
Using geostatistical inversion results to predict other petrophysical parameters and lithotypes
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

31. GEOSTATISTICAL INVERSION Lessons 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.
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

32. QUANTIFYING UNCERTAINTIES Outline
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
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

33. EARTH MODELLING AND QUANTIFICATION OF RESERVOIR UNCERTAINTIES
Geometry
Impact on GRV!
Impact on OIP!
Impact on Reserves!
Static properties
Naturally, the ultimate goal of this whole exercise is production. This means we have to convert geology into a reservoir model, and we have to track and evaluate all the uncertainties in order to quantify their impact on production profiles.
This approach has now become standard within TotalFinaElf, thanks to our chain of in-house stochastic tools Alea-Jacta-Est, that can be used in combination in order to quantify the joint effects of structural, geological and dynamic uncertainties on production profiles.
A few minutes ago, I talked about a plane piloted by geologists. Well, the plane is now a jumbo jet, and the crew is a fully integrated team made up of geologists, geophysicists and reservoir engineers. Getting them to crew together can sometimes be quite a challenge in itself…
(slide V. de Feraudy)
Dynamic properties
6-4

34. QUANTIFICATION OF STRUCTURAL UNCERTAINTIES THE APPROACH
Estimation of uncertainties
Identify uncertainties in the interpretation workflow,
Quantify their magnitude (Confidence interval)
Interpreter ’s input
Geostatistican ’s input
Measure of their impact on the results (GRV,OIP...)
Geostatistical Simulation
Statistical Analysis

35. NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION CASE STUDY (ABRAHAMSEN ET AL., 2000)
pdf
Base case = 652 Mm3
GRV (Mm3)

36. 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.
Where did the main steps occured for seismic data integration
Integration of seismic data statred in the 70 ’s, with the work of Haas and others. It continued around the early nineties with the work of Doyen. In the mid-nineties, earth modeling software allowed geostatistical inversion

37. 3 AREAS WHERE GEOSTATISTICS IS CRUCIAL
Generation of 3D heterogeneity models
Integration of seismic data in reservoir models
Uncertainty quantification
7-2

38. WEBSITES ABOUT PETROLEUM GEOSTATISTICS
ekofisk.stanford.edu/SCRFweb/index.html
7-3

39. 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
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
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
7-3