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SEG/EAGE DISC 2003 INTRODUCTION Outline A Brief Historical Perspective The interaction between 3D Earth Modeling and Geostatistics Basic Probability and Statistics Reminders

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SEG/EAGE DISC 2003 A random variable takes certain values with certain probabilities. Example: Z = sum of two dice RANDOM VARIABLES Each value, for instance 4, is a realization 1-12 PROBABILITY DENSITY FUNCTION SUM OF TWO DICE FREQUENCY (NOT NORMALIZED)

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SEG/EAGE DISC 2003 THE IMPACT OF AVERAGING (2) HISTOGRAMS 1-18 P. Delfiner/X. Freulon 9x927x27 1x1 9x927x27

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SEG/EAGE DISC 2003 THE SUPPORT EFFECT (FRYKMAN AND DEUTSCH, 2002) Well log Histogram of core Histogram of log 2-31 Impact on Cut-off Variance is volume-dependent!

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SEG/EAGE DISC 2003 NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, =5) 1-26 CONFIDENCE INTERVAL: 95% of values fall between m- 2 and m+2 Porosity Uncertainty: %

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC 2003 NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, =5) 1-26 CONFIDENCE INTERVAL: 95% of values fall between m- 2 and m+2 Porosity Uncertainty: %

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC 2003 STATIONARITY OF THE MEAN 2-3 Nonstationary Stationary

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SEG/EAGE DISC 2003 A spatial phenomenon can be modeled using 2 terms: a low-frequency trend a residual Constant trend: stationary variable Quadratic trend + stationary residual STATIONARITY OF THE VARIANCE (1) 2-1 P. Delfiner/X. Freulon

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SEG/EAGE DISC 2003 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 STATIONARITY OF THE VARIANCE (2) 2-2 P. Delfiner/X. Freulon

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SEG/EAGE DISC 2003 WHAT TO DO WHEN NOT ENOUGH DATA ARE AVAILABLE? Vertical Wells Vertical variograms Variance gives sill of horizontal variograms A priori geological knowledge Behavior at origin and nugget effect 2-39 Seismic data Horizontal anisotropy ratios and ranges Horizontal Wells Horizontal variograms

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC 2003 NUGGET EFFECT VS NYQUIST FREQUENCY (h) 0 Minimum detectable variogram range = d h d Minimum detectable wavelength = 2d Maximum detectable spatial frequency = 1/(2d) Distance between data=d x x x x x x x x x x x x d

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SEG/EAGE DISC 2003 THE FACTORIAL KRIGING MODEL MARINE EXAMPLE: HORIZON-CONSISTENT VSTACK (3) 3-39 J.L. Piazza and L. Sandjivy m (m/s) 2 in-line effect (4) Spherical (D1) 300 m (D2) 450 (m/s) 2 geophysicist effect (3) Spherical 1600 m (D1) 100 m (D2) 100 (m/s) 2 m (m/s) 2 Geological signal (1) Spherical 7500 m, 1000 (m/s) 2 (2) Spherical 1600 m, 300 (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

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SEG/EAGE DISC 2003 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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SEG/EAGE DISC 2003 V. Bigault de Cazanove 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) 3-56 Deterministic external-drift Random residual

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SEG/EAGE DISC 2003 COKRIGING Two variables Z 1 (x) and Z 2 (x) (such as porosity & acoustic impedance) Use of Z 1 and Z 2 data to get a better interpolation of Z Porosity estimation by cokriging Porosity data at wells Acoustic impedance data from seismic

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SEG/EAGE DISC 2003 COLLOCATED COKRIGING COKRIGING COLLOCATED COKRIGING Complicated system of equations Requires variograms of Z 1, Z 2, cross-variograms of Z 1 and Z 2

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SEG/EAGE DISC 2003 COLLOCATED COKRIGING (JEFFERY ET AL., 1996) 3-70 Just the variance of residual gravity is used, not the whole variogram! WELL CONTROL DEPTHING VELOCITY ISOTROPIC 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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SEG/EAGE DISC 2003 EXTERNAL DRIFT OR COLLOCATED COKRIGING? Collocated CokrigingExternal Drift 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 Interaction between variogram model and correlation coeff Applications Equal to linear transform of seismic beyond variogram range Construction of structural model Interpolation of petrophysical parameters

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC = THE THREE PROSPECTS m 1 =75 1 =15m 2 =100 2 =25 m 3 =200 3 =40 m=375 Independence assumption: conclusion obtained by Monte-Carlo simulation (or by properly combining variances) =50 Full dependence assumption: conclusion obtained by simply adding min and max of prospects =80

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SEG/EAGE DISC 2003 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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SEG/EAGE DISC 2003 A KRIGING EXAMPLE IN 3D (LAMY ET AL., 1998b) 49 AI km.g / s.cm 3 N 4-10 Why should the reservoir be smooth precisely away from the data points? Total UK Geoscience Research Centre

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SEG/EAGE DISC 2003 KRIGING OR CONDITIONAL SIMULATION? KrigingConditional simulation Output 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 Use Heterogeneity Modeling, Uncertainty quantification Mapping

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SEG/EAGE DISC 2003 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.

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SEG/EAGE DISC 2003 GEOSTATISTICAL INVERSION Outline What is geostatistical inversion Examples of geostatistical inversion Using geostatistical inversion results to predict other petrophysical parameters and lithotypes

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SEG/EAGE DISC 2003 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.

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SEG/EAGE DISC 2003 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

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SEG/EAGE DISC 2003 EARTH MODELLING AND QUANTIFICATION OF RESERVOIR UNCERTAINTIES Geometry Static properties Dynamic properties 6-4 Impact on GRV! Impact on OIP! Impact on Reserves!

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SEG/EAGE DISC 2003 QUANTIFICATION OF STRUCTURAL UNCERTAINTIES THE APPROACH Estimation of uncertainties Estimation of uncertainties Identify uncertainties in the interpretation workflow, Identify uncertainties in the interpretation workflow, Quantify their magnitude (Confidence interval) Quantify their magnitude (Confidence interval) Interpreter s input Geostatistican s input Measure of their impact on the results (GRV,OIP...) Measure of their impact on the results (GRV,OIP...) Geostatistical Simulation Geostatistical Simulation Statistical Analysis Statistical Analysis

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SEG/EAGE DISC 2003 NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION CASE STUDY (ABRAHAMSEN ET AL., 2000) GRV (Mm 3 ) pdf Base case = 652 Mm 3

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SEG/EAGE DISC 2003 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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SEG/EAGE DISC 2003 Generation of 3D heterogeneity models Integration of seismic data in reservoir models Uncertainty quantification 3 AREAS WHERE GEOSTATISTICS IS CRUCIAL 7-2

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SEG/EAGE DISC ekofisk.stanford.edu/SCRFweb/index.html WEBSITES ABOUT PETROLEUM GEOSTATISTICS 7-3

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SEG/EAGE DISC 2003 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 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 BOOKS AND PAPERS TO READ 7-3

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