Geostatistics for geoscientists and geologist

1. Advanced Uncertainty and
Sensitivity Analysis in 3D
Geological Modeling
Basic Terms and Concepts
May 2022
Serdar Kaya,
serdar@grenergyllc.com
Mobile:+90 539 4772377

2. 2
Variability is how the data spread out or clustered in a
distribution around the central point.

4. 4
Measuring Variability
• Variability can be measured with
• the range
• the interquartile range
• the standard deviation/variance.

5. 5
The range is the total difference between minimum and maximum value
of a population

6. 6
The interquartile range is the distance covered by the middle 50% of
the distribution (the difference between Q1 and Q3).

8. 8
Standard deviation measures the standard distance between a score
and the mean.

9. 9
Deviation is the distance from the mean
Variance is the mean of squared all deviations
Standard deviation is the square root of the variance

11. 11
The Mean
• The mean 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 and Harmonic 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
• The process 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 of Measurement 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 of Measurement 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
• The goal 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 measurements obtained 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
• Descriptive statistics 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 statistics are 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
• The discrepancy 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 distribution is 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 bar graph is just like a histogram except that gaps or spaces are left
between adjacent bars.

33. 33
Relative frequency
Population distributions can be shown using relative frequency instead
of the absolute number of individuals for each category.

35. 35
Smooth curve
An interval or 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 Negatively Skewed 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 Ranks and 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 a mathematical process based on the assumption that the
scores and the percentages change in a regular, linear fashion

41. 41
Bimodal Distributions
• A distribution has more than one mode is called bimodal.
• "mode" is a peak in a distribution

42. 42
Probability is the likelihood of obtaining a specific sample

46. GEOSTATISTICS
Principles of Statistics
Spatial Relationships - Estimation &
Modelling
Estimation Techniques
Conditional Simulation Techniques
Grid-Based Simulation Methods
Object-Based Simulation Techniques
Scale-Up

47. Interpolation and Extrapolation
Spatial Distribution Analysis
Risk Analysis and Uncertainty Estimation
Use of Intercorrelated Attributes
APPLICATIONS

48. • Not empirical as 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. Porosity with Fitted Function - All wells
Coeff. Cor: above 0.9
w
w
w
w
w
w
w
w
w
w
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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 continuous update 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
Basic Statistics - 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
Theoretical Distribution Functions
Uniform distribution
Normal (Gaussian) distribution
Log Normal distribution
PRINCIPLES of STATISTICS…

54. Requirement of stationarity
Estimation of 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 result in 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 Spatial Relationships
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 unsampled locations 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) into a 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 A B 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 ss1 ss2 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)
Final RRT 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 Porosity Models
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 includes the 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 Permeability using 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 for each Reservoir Subzone and per RRT
HISTOGRAMS - Porosity & Permeability

86. Zone I
Transform to a
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