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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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)

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**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

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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? 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.

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**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

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**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

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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 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

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**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

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**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

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**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

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**+ + = 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

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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)**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION CASE STUDY (ABRAHAMSEN ET AL., 2000)**

pdf Base case = 652 Mm3 GRV (Mm3)

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**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

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**3 AREAS WHERE GEOSTATISTICS IS CRUCIAL**

Generation of 3D heterogeneity models Integration of seismic data in reservoir models Uncertainty quantification 7-2

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**WEBSITES ABOUT PETROLEUM GEOSTATISTICS**

ekofisk.stanford.edu/SCRFweb/index.html 7-3

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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 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

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Lecture 6: Point Interpolation

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