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Basic geostatistics
This document provides an overview of basic statistical concepts and terms used in 3D geological modeling and uncertainty analysis, including: 1. It defines key terms like variability, measures of variability (range, interquartile range, standard deviation), and types of data (discrete, continuous). 2. It describes statistical measures like the mean, median, mode, and different types of averages (arithmetic, geometric, harmonic). 3. It covers types of variables, measurement scales, correlational studies, descriptive and inferential statistics, and sampling error. 4. It discusses probability, probability distributions, and introduces basic geostatistical concepts like variograms and kriging which are used in 3D geological modeling
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Basic geostatistics
- 1. Advanced Uncertainty and SensitivityAnalysis in 3D Geological Modeling Basic Terms and Concepts May 2022 Serdar Kaya, serdar@grenergyllc.com Mobile:+90 539 4772377
- 2. 2 Variability is howthe data spread out or clustered in a distribution around the central point.
- 4. 4 Measuring Variability • Variabilitycan be measured with • the range • the interquartile range • the standard deviation/variance.
- 5. 5 The range isthe total difference between minimum and maximum value of a population
- 6. 6 The interquartile rangeis the distance covered by the middle 50% of the distribution (the difference between Q1 and Q3).
- 8. 8 Standard deviation measuresthe standard distance between a score and the mean.
- 9. 9 Deviation is thedistance from the mean Variance is the mean of squared all deviations Standard deviation is the square root of the variance
- 11. 11 The Mean • Themean is obtained by computing the sum, or total, for the entire set of scores, then dividing this sum by the number of scores. • Computation of the mean requires scores that are numerical values measured on an interval or ratio scale.
- 15. Arithmetic, Geometric andHarmonic Mean 15
- 16. 16 Types of Variables •Discrete variables (such as class size) consist of indivisible categories • continuous variables (such as time or weight) are infinitely divisible into whatever units a researcher may choose.
- 17. 17 Measuring Variables • Theprocess of measuring a variable requires a set of categories called a scale of measurement and a process that classifies each individual into one category.
- 18. 18 4 Types ofMeasurement Scales 1. A nominal scale is an unordered set of categories identified only by name. Nominal measurements only permit you to determine whether two individuals are the same or different. 2. An ordinal scale is an ordered set of categories. Ordinal measurements tell you the direction of difference between two individuals.
- 19. 19 4 Types ofMeasurement Scales… 3. An interval scale is an ordered series of equal-sized categories. Interval measurements identify the direction and magnitude of a difference. The zero point is located arbitrarily on an interval scale. 4. A ratio scale is an interval scale where a value of zero indicates none of the variable. Ratio measurements identify the direction and magnitude of differences and allow ratio comparisons of measurements.
- 20. 20 Correlational Studies • Thegoal of a correlational study is to determine whether there is a relationship between two variables and to describe the relationship. • A correlational study simply observes the two variables as they exist naturally.
- 22. 22 Data • The measurementsobtained in a research study are called the data. • The goal of statistics is to help researchers organize and interpret the data.
- 23. 23 Descriptive Statistics • Descriptivestatistics are methods for organizing and summarizing data. • A descriptive value for a population is called a parameter • a descriptive value for a sample is called a statistic.
- 24. 24 Inferential Statistics Inferential statisticsare methods for using sample data to make general conclusions (inferences) about populations. • Because a sample is typically only a part of the whole population, • sample data provide only limited information about the population. • sample statistics are generally imperfect representatives of the corresponding population parameters.
- 25. 25 Sampling Error • Thediscrepancy between a sample statistic and its population parameter is called sampling error. • Defining and measuring sampling error is a large part of inferential statistics.
- 26. 26 A frequency distributionis an organized tabulation showing exactly how many individuals are located in each category on the scale of measurement
- 27. 27 Histograms In a histogram,a bar is centered above each score (or class interval) so that the height of the bar corresponds to the frequency and the width extends to the real limits, so that adjacent bars touch.
- 29. 29 In a polygon,a dot is centered above each score so that the height of the dot corresponds to the frequency.
- 31. 31 Bar graphs A bargraph is just like a histogram except that gaps or spaces are left between adjacent bars.
- 33. 33 Relative frequency Population distributionscan be shown using relative frequency instead of the absolute number of individuals for each category.
- 35. 35 Smooth curve An intervalor ratio scale is presented as a smooth curve rather than a jagged histogram or polygon.
- 36. 36 Frequency distribution graphs •Frequency distribution graphs show the entire set of scores. • The highest score, the lowest score, and where the scores are centered. • Data is clustered or scattered
- 37. 37 Positively and NegativelySkewed Distributions • In a positively skewed distribution, the scores tend to pile up on the left side of the distribution • In a negatively skewed distribution, the scores tend to pile up on the right side
- 39. 39 Percentiles, Percentile Ranksand Interpolation • The relative location of individual scores within a distribution can be described by percentiles and percentile ranks. • The percentile rank for a particular X value is the percentage of individuals with scores equal to or less than that X value. • When an X value is described by its rank, it is called a percentile.
- 40. 40 Interpolation Interpolation is amathematical process based on the assumption that the scores and the percentages change in a regular, linear fashion
- 41. 41 Bimodal Distributions • Adistribution has more than one mode is called bimodal. • "mode" is a peak in a distribution
- 42. 42 Probability is thelikelihood of obtaining a specific sample
- 46. GEOSTATISTICS Principles of Statistics SpatialRelationships - Estimation & Modelling Estimation Techniques Conditional Simulation Techniques Grid-Based Simulation Methods Object-Based Simulation Techniques Scale-Up
- 47. Interpolation and Extrapolation SpatialDistribution Analysis Risk Analysis and Uncertainty Estimation Use of Intercorrelated Attributes APPLICATIONS
- 48. • Not empiricalas the simple estimation methods • Allows the computation of the estimation error • Honors the sampled data • Allows the use of qualitative information • Allows the use of extensively sampled variables to estimate the values of other sparsely sampled variables ADVANTAGES
- 49. AI vs. Porositywith Fitted Function - All wells Coeff. Cor: above 0.9 w w w w w w w w w w w AI - POROSITY
- 50. Mean Porosity Map @Seismic Interval Contour Interval: 1 pu INVERTED MEAN POROSITY MAP Residual error at well location Well Map Por. Well Por. Error RA-1 18.1 15.2 2.9 RA-2 15.9 14.1 1.8 RA-3 13.7 14.6 -0.9 RA-4 11.7 13.3 -1.6 RA-5 9.9 13.2 -3.3 RA-6 13.7 14.1 -0.4 RA-7 11.3 12.6 -1.3 RA-10 20.6 16.7 3.9 RA-11 18.0 16.2 1.8 RA-12 14.0 15.0 -1.0 SN-1 15.3 16.2 -0.9 SN-2 12.1 14.4 -2.3 SN-3 15.2 13.3 1.9 Well Location POROSITY ESTIMATION
- 51. • Needs continuousupdate of parameters with additional data • Subjective decision maker • Different techniques/methods should be tested - project objective vs. time availability “DISADVANTAGES”
- 52. Univariate Statistics Frequency Distributions BasicStatistics - mean, median, mode, range, percentiles, variance, coefficient of variance, etc. Data Sets Sample declustering Moving window statistics Bivariate Statistics Conditional Frequency Distributions Proportion Curves Basic Statistics - covariance, correlation coefficient, linear regression PRINCIPLES of STATISTICS
- 53. Probability Random Variables Probability Functions TheoreticalDistribution Functions Uniform distribution Normal (Gaussian) distribution Log Normal distribution PRINCIPLES of STATISTICS…
- 54. Requirement of stationarity Estimationof variogram - number of pairs, lag tolerance Transforms - log transform, rank transform, power transform, normal score transform, indicator transform, Modelling of variogram - nugget effect, spherical model, exponential model, gaussian model, fractal model nested structures Identification of outliers Anisotropic Models Cross-variograms (estimation & modelling) SPATIAL RELATIONSHIPS - Estimation/Simulation
- 55. Kriging techniques resultin smooth models where small variabilities are eliminated and large-scale features are preserved Do not allow to capture of uncertainties - one single model Use a linear estimation procedure to estimate a value at unsampled locations. The estimated value is a weighted average of the neighboring values. Weights are defined by the variograms. Kriging - ordinary kriging, indicator kriging, kriging with external drift, co- kriging, collocated co-kriging ESTIMATION METHODS
- 56. Method Procedure SpatialRelationships Influence of Primary Data Influence of Secondary Data Advantages Disadvantages Possible Applications Cross Plot 1. Develop correlation between primary and secondary data 2. Estimate primary data from knowing the secondary data None Required none dominant simple Primary data estimation is completely controlled by secondary data Limited. Very strong relationship between primary and secondary variable is required. External Drift 1. Develop correlation between primary and secondary data 2. Overall primary trend is captured by secondary data 3. Primary variable is estimated using the trend plus surrounding primary data Primary variable variogram required limited very significantrelatively simple Large scale primary data estimation is completely controlled by secondary data. High frequency information captured by surrounding primary data Depth mapping based on two way travel time Co-kriging 1. Use both primary and secondary data to estimate primary variable 2. Limit the influence of secondary data by using different search neighborhoods and no of sample points within neighborhood Variograms for primary and secondary variables as well as cross variogram required Variable Variable Flexible Due to practical limitations, spatial continuities are often influenced by secondary data Porosity mapping using impedence data Collocated Co- kriging 1. Use primary variable continuity to guide estimation 2. Use secondary data only at the unsampled location Primary variable variogram required Significant Limited simple Due to limited data, primary variable spatial relationship is difficult to obtain. Secondary data are needed at each unsampled location Porosity mapping using impedence data Kriging 1. Use primary data only to estimate values at unsampled locations Primary variable variogram required Dominant none simple In the presence of limited primary data, the estimation may not be reliable. Extrapolation is very difficult without the secondary data Conventional kriging applications Integration of Seismic Data in Reservoir Description ESTIMATION METHODS…
- 59. Values at unsampledlocations are simulated rather than estimated. The overall goal is to obtain several simulations of the reality rather than obtain a single picture of the reservoir which minimizes the error variance. Quantification of uncertainty - multiple realizations Includes Grid-based models (discretized blocks) and Object-based models (generation of properties assuming shapes and sizes of geological objects). Which method is more appropriate depends on the overall project goal and data availability. Main advantages over conventional estimation techniques: preservation of variability preservation of spatial relationships quantification of uncertainty through multiple images CONDITIONAL SIMULATION METHODS
- 60. Transform variable(s) intoa new domain Spatial Modelling in Transformed Domain Random Path Selection Estimation at the Unsampled Location Back transform Sequential Indicator Simulation (SIS) Sequential Gaussian Simulation (SGS) Sequential Co-Simulation (SISCOSIM, SGSCOSIM) - simulation of multiple attributes simultaneously Probability Field Simulation Cloud Transform Simulating Annealing GRID-BASED SIMULATION METHODS
- 61. Variance Distance to neighbours C AB D Probability Property Random number Simulated value The Volume is Populated by Simulated AI and Lithotype Random cell Neighbor cells Next cell Local PDF GAUSSIAN SIMULATION WORKFLOW
- 62. Cloud Transforms Sh ss1ss2 ss3 ss4 R1 R2 Facies Type Porosity 15 14 13 12 11 10 9 8 x106 Impedance Band Classes CLOUD TRANSFORM SIMULATION
- 63. Model Operations (or Co-kriging) FinalRRT Models RRT 5 Categorical Indicator Kriging RRT Probability Model AI Subzone Average Map + RRT @ Wells SIS w/ Collocated Co-kriging RRT Models RRT MODELLING with SEISMIC CONSTRAIN
- 64. Connectivity Analysis 3 PorosityModels Streamline Simulation Ranked Permeability Models SGS w/ Collocated Co-kriging low high average Porosity Models AI Subzone Average Map + RRT Model Porosity @ Wells Permeability Models Cloud Transform w/ P-Fields & Collocated Co-kriging POROSITY MODELLING without RRT CONSTRAIN
- 67. STOCHASTIC (when Cv >2) DETERMINISTIC (when Cv < 0.5) RESERVOIR MODELLING APPROACHES TECHNIQUES DRIVERS METHODS PIXEL BASED TECHNIQUE (for carbonates) OBJECT BASED TECHNIQUE (for clastics ) ESTIMATION TECHNIQUES (Kriging) INVERSE DISTANCE SQUARE FACIES DRIVEN NO FACIES GMPP Boolean Simulation GEOBODY SHAPES (channels) SEISMIC NO SEISMIC Block Kriging Point Kriging Co-kriging Collocated co-kriging SIS TGS SGS ATTRIBUTES •porosity •permeability •facies •saturations •porosity •permeability •saturations •tops •thicknesses •twt data •isochrones •velocities •faults OUTCOME multiple equi-probable realizations (models) one single model Reservoir Modelling Nomenclature (modified after Seifert & Jensen, 2000) OVERVIEW
- 68. Standard upscaling includesthe following techniques: 1 Arithmetic average - typically used to obtain average porosity and represents the upper bound permeability upscaled value. 2 Geometric average – produces a lower average than the arithmetic average. 3 Harmonic average – produces a lower average than the arithmetic and geometric average and represents the lower bound permeability upscaled value. 4 Power average – includes Arithmetic, Geometric, and Harmonic averages. 5 Summation – used to obtain upscaled values of additive properties (e.g. pore volume) 6 Combined directional averaging (used for permability) – combine arithmetic and harmonic averages in the different directions of permeability X, Y and Z. This method relates to the upscaling of discrete properties, such as facies or RRT. The method chooses dominant categorical value STANDARD UPSCALING
- 69. Upscaling of Permeabilityusing Flow-based methods is the most accurate method to re-scale permeability. The main techniques are as follow: 1 Diagonal Tensor 2 Full Tensor These options differ in the choice of pressure and flow boundary conditions. 3 Re-normalisation The outputs of these upscalers are given in X, Y and Z directions Documentation available for all theses issues: course manual, RMS & RC2 software packages manuals, other references PERMEABILITY FLOW BASED UPSCALING
- 79. Histograms Definition foreach Reservoir Subzone and per RRT HISTOGRAMS - Porosity & Permeability
- 86. Zone I Transform toa normal distribution AI Zone I AI Zone I Ø Histograms Definition for each Reservoir Subzone and per Lithotype HISTOGRAMS - Transforms
- 87. SUBZONE BII -MSZ 2 6.5’ 6400 m Variograms Definition for each Reservoir Subzone and per RRT vertical variogram horizontal variogram Porosity Variograms VARIOGRAMS - RRT, Porosity & Permeability