Geostatistics for Seismic Data Integration in Earth Models

Transcription

1 2003 Distinguished Instructor Short Course Distinguished Instructor Series, No. 6 sponsored by the Society of Exploration Geophysicists European Association of Geoscientists & Engineers SUB Gottingen iiiiiiiiiwi Geostatistics for Seismic Data Integration in Earth Models presented by Olivier Dubrule EUROPEAN ASSOCIATION OF GEOSCIENTISTS & ENGINEERS

2 Table of Contents 1 Introduction Historical Perspective The Role of Geostatistics at Different Steps of the Earth Modeling Workflow The Goal of This Course Basics of Univariate Statistics Random variables Mean, variance, standard deviation, and support effect Measures of central tendency and quantiles Two important distributions Two important theorems Basics of Bivariate Statistics Covariance and correlation coefficient Fitting a regression line The Multivariate Normal Distribution Trend Surface Analysis The Covariance and the Variogram Stationarity Versus Nonstationarity The Stationary Model Calculation of a Variogram Stationary Variogram Models Examples of Anisotropic Experimental Models Unbounded Variogram Models and Their Relationship with Fractals First Uses of the Variogram: Predicting the Support Effect Cross-covariance and the Variogram Practical Considerations about the Variogram 2-26 Covariance, Fractals, and Spectral Density Interpolation: Kriging, Cokriging, Factorial Kriging, and Splines Introduction Kriging, an Interpolation Technique Introduction 3-1 Distinguished Instructor Short Course v

3 Geostatistics for Seismic Data Integration in Earth Models Universal kriging Generalized covariances of order k Kriging considered as an interpolating function Cross-validation Conclusion on kriging Error Cokriging and Factorial Kriging to Distinguish Noise from Signal Error cokriging for V sta ck interpolation Factorial kriging Kriging with an External Drift The external-drift model Examples of time-to-depth conversion using the external-drift approach Bayesian Kriging, a Generalization of Kriging with an External Drift Cokriging and Collocated Cokriging Introduction A cokriging example Collocated cokriging 3-49 A Few Words on Bayesian Statistics Revisiting collocated cokriging Collocated cokriging versus external drift Factorial cokriging Some Relationships between the Kriging Techniques Kriging Versus Other Interpolation Techniques Introduction Splines Comparison among kriging, splines, and radial-basis functions Why this relationship between splines and kriging? Conditional Simulation for Heterogeneity Modeling and Uncertainty Quantification Introduction A Few Reminders on Monte-Carlo Simulation Conditional Simulations for Continuous Parameters Example Algorithms Cosimulation 4-22 vi Society of Exploration Geophysicists / European Association of Geoscientists & Engineers

4 4.4.1 Collocated simulation with seismic data as a secondary variable Joint simulation of two parameters Cascade or parallel conditional simulation of several parameters? Conditional Simulation for Geological Facies Modeling Introduction Pixel-based models Object-based models Facies models constrained by seismic information Hierarchical modeling of geology and petrophysical parameters 4-46 Geostatistics, Inverse of the Covariance, and Filtering Geostatistical Inversion Introduction Basics of Geostatistical Inversion Accounting for Faults A Variety of Methods A different sampling algorithm Geostatistical inversion based on fractals Analytical approach Emerging techniques A Generalized Downscaling Approach Going Further with Geostatistical Inversion Results Two-step approach: from seismic to impedance, from impedance to other properties Predicting porosity during the acoustic-impedance inversion process Predicting facies during the acoustic-impedance inversion process Stochastic Earth Modeling That Integrates All Subsurface Uncertainties Introduction Geometrical Uncertainties Two-step approach Bayesian kriging approach How many realizations? Static and Dynamic Model Uncertainties 6-13

5 Geostatistics for Seismic Data Integration in Earth Models Static model uncertainties Link with dynamic flow simulation Multirealization Uncertainty-quantification Approach: A Panacea? Approaches by scenarios Combining scenarios and geostatistical realizations Conclusion on Uncertainties: A Word of Caution Conclusions What We Have Learned Future Topics Websites and Software The Role of Geostatistics in Geophysics Exercises Exercise 1: Fitting a Variogram Model Exercise 2: Understanding the Kriging System Exercise 3: Generating a Nonconditional Simulation in ID Notation Acknowledgments References * 11-1 viii Society of Exploration Geophysicists / European Association of Geoscientists & Engineers