A 3D Property Modeling Primer
Table of ContentsAcknowledgments Overview
Part 1 – Statistical Concpts Part 2 – Modeling Concepts Part 3 – Modeling Methodologies Part 4 – Variograms
Subject: Part 1 – Statistical Concepts Section: Introduction
Chapter: Introduction Pages:
- Introduction
- Purpose of the Course
- Questions this Course Will Answer Section: Basic Concepts and Terminology
Chapter: Basic Concepts & Terminology Pages:
- Basic Concepts & Teriminology - Some Simple Examples
Chapter: Descriptive & Analytical Tools Pages:
- Preface - Histograms
- Typical Uses of Histograms - Univariate Statistics
- Typical Display & Uses of Univariate Statistics - Crossplot
- Crossplot Examples - H-Scatter Plots - Variogram Cloud
- Variogram Clous Usage Example - Variograms
- Why do We Need Variograms? - Variogram Maps
Subject: Part 2 – Modeling Concepts Section: Modeling Geometry
Chapter: 2D Grids Pages: - Basic Components - Grid Nomenclature - Grid Refinement - 2D Grid Examples Chapter: 3D Grids – Property Models
Pages:
- Components of a 3D Model - Example of a 3D Model Chapter: 3D Grids – Reservoir Models
Pages:
- Reservoir Models - 3D Simulation Grids - Structured Grids
- Unstructured Grid Geometry (FloGrid) Section: Data Transformations During Modeling Operations
Chapter: Data Transformations Pages: - Thresholding - Upscaling/Averaging - Segregation - Masking - Removing a Trend
- Normal Score Transforms - Weighting
Subject: Part 3- Modeling Methodologies Section: Gridding Methods
Chapter: Algorithm Classifications Pages:
- Introduction
- How Are Gridding Algorithms Classified? - Kriging vs. Non-Kriging Algorithms
- Algorithms for Discrete vs. Continuous Data - Deterministic Compared with Probabilistic Data Chapter: Variogram Basics
Pages:
- Variogram Roles in Gridding & Geostatistics - Facts to Remember About Variograms Chapter: Terminology Used During Algorithm Descriptions Pages
Chapter: Traditional Estimation Algorithms Pages:
- Mechanics of 3D Traditional Estimation Algorithms - A Survery of Traditional Methods
Chapter: Kriging Algorithms Pages:
- Kriging Workflow
- The Mechanics of 3D Kriging
- Difference Between the Search & Variogram Range Chapter: Anisotropy Pages: - What is Anisotropy? - An Example of Anisotropy Chapter: Survey Pages:
- A Survey of Some Kriging Algorithms Chapter: Simulation Algorithms
Pages:
- Simulation Algorithms
- Sequential Gaussian Simulation – Continuous - Sequential Indicator Simulation – Discrete - Truncated Gaussian Simulation – Discrete - Object Modeling
Section: Gridding Operations
Chapter: Gridding Guidelines Pages:
- An Overview of the Property Modeling Workflow Chapter: Algorithm Selection
Pages:
- How Do I Know Which Algorithms to Use With My Data?
- Selecting an Algorithm for Discrete or Facies Modeling - Selecting an Algorithm for Petrophysical Modeling Chapter: Quality Control Procedures
Pages:
- Quality Control During Modeling Chapter: Probability Options
Pages:
- How Probable Are My Models & Volumes? - Selecting a Best Realization
Chapter: Comparative Table of Gridding Algorithms Pages:
- Table of Gridding Algorithms Section: Congratulations
Chapter: Congratulations Pages:
- Congratulations - Further Study
Subject: Part 4 – Variograms
Section: Purpose of This Topic Section: Review
Chapter: Review Pages:
- Review of basic facts – experimental variogram - Review of basic facts – variogram model - Review of Anisotropy
- Review of Variogram Directions and Types Section: The Big Picture
Chapter: Generic Workflows Pages
- How do you make a variogram? - How do you determine anisotropy?
- How do you use variograms to determine layer distance?
Chapter: Petrel-Specific Variogram Facilities Pages
- Overview
- Variogram facilities in an object’s Settings tab - Variogram facilities in the modeling dialogs - Variogram facilities in the Data Analysis tool
• Preparing for variogramming discrete data. • Preparing for variogramming continuous
properties
Chapter: Petrel Interactive Tools and Icons for Making Variograms Pages
- Lag, Azimuth, and Search Angle Icon - Variogram Display
Chapter: Modeling the Variogram in the Petrel Data Analysis dialog Pages
- Simple Petrel procedure - Using the Variogram
Acknowledgements:
This primer could not have been written without the help of and previous documents provided by: Sujit Kumar Doug Palkowsky Sanjay Paranji Leonid Shmaryan Lothar Schulte Drew Wharton Overview:
This subject is too large for a single computer-based training model, and is therefore subdivided into four major topics – Statistical Concepts, Modeling and Geometrical Concepts, Modeling Methodologies, and Variograms.
While variograms are discussed in general in the first three topics, the last topic provides an independent summary as well as detailed instructions and workflows which are Petrel-specific.
Subject: Part 1 – Statistical Concepts Section: Introduction
Chapter: Introduction
Page title: Introduction
Ever read a book about geostatistics?
Page title: Purpose of the Course The three primary purposes of this course are:
1. Provide the student with a common sense generic background in geostatistical concepts
• Use simple terminology, not mathematical notation • Use plain language, common sense analogies
• Focus on mechanics of how to reach specific goals, not on proving or demonstrating theorems
• Recommend texts for deeper understanding to lead to creativity and true expertise 2. Provide the student with the vocabulary to become quickly productive with the tools in any geostatistical software, including Modeling Office, Petrel, FloGrid, LPM, …
PRODUCTIVE MEANS BEING ABLE TO DO THESE THINGS:
• Understand the variety of statistical tools available for data analysis before and after modeling.
• Determine if there is a relationship between a property and a seismic attribute. • Understand the definition of a variogram and its uses in the grand scheme of
things.
• Understand the variety of data transforms used in geostatistics and modeling. • Determine if your property values are directionally biased.
• Visualize the grid geometries used in modeling from 2D gridding to simulation • Tell the difference between kriging and non-kriging algorithms
• Tell the difference between deterministic and probabilistic algorithms
• Understand the difference between facies modeling and petrophysical property modeling
• Understand the use of stochastic methods.
• Learn how to use seismic attribute grids as secondary input data
• Measure the QUALITY of geostatistical models by comparing statistics
3. Provide specific recommendations for modeling lithology and properties under various conditions
• Choose appropriate kriging and non-kriging estimation algorithms for modeling based on data and reservoir characteristics.
How?
Page Title: Questions This Course Will Answer
• What’s the advantage of using GeoStatistics? • Why are there so many kinds of kriging algorithms?
• What’s the difference between kriging and non-kriging algorithms? • What are “stochastic realizations”?
• What does a variogram actually show? • Is a variogram actually necessary?
• Why do I have to make a vertical and a horizontal variogram? • What is the meaning of “anisotropy”?
• What do I do if my data set has “anisotropy”? • How do I know if my computed model is OK?
• Is there a recommended geostatistical modeling workflow for each property like lithology, permeability, etc?
• How can geostatistical algorithms help in the determination of probabilities? Here are some specific questions you’ll get answered in this course
Now that you know what this course is about, we’ll move along to the first topic – Basic Concepts - Good Luck !
Subject: Part 1 – Statistical Concepts
Section: Basic Concepts and Terminology
Chapter: Basic Concepts and Terminology
Page Title: Basic Concepts and Terminology
To make progress, we all need the right words. Here are a few of the more important ones we’ll need for learning about geostatistics.
• Variance - a measurement of how different the members of a collection are from each other. (Measured in units of the collection). [Q 1]
• Correlation - a way to measure whether two separate collections are related. (Measured in percent). [Q 2]
• Anisotropy - a way to measure whether variance within a collection of data is determined by direction. (Measured in azimuth and percent eccentricity). [Q 3]
• Probability – a measurement of the likelihood of an event. (Measured in percent). [Q 4]
Stationarity –is simply an ASSUMPTION which is made regarding the rules for behavior of the properties which we analyze, study, or model with geostatistical tools.
The rule is simply that the property must behave consistently within the volume chosen for analysis, study, or modeling. If it does not, then the geostatistical tools which we use will not work properly. Stationarity assumes that a property behaves the same way in all locations of the chosen volume; i.e., that the samples have no inherent trend. If a trend exists, it must be removed before using certain algorithms.
“He used so many five and ten-dollar words in his lecture that now
I’m completely broke...”
- Mathematics student
“
I must say that words simply cannot describe what happened
.
”
- Hairless caveman upon discovering fire
Some simple concepts used in geostatistics
Questions for review:
1. Which concept is used to quantify whether an event has any chance of happening? [probability]
2. Which concept is concerned with simple differences between samples? [variance]
3. Which concept gives a way of comparing two different attributes? [correlation]
4. Which concept is concerned with direction? [anisotropy]
Page Title: Some Simple Examples Consider these simple examples:
Variance
“Samples of porosity in a simple sandstone unit will show much less variance than samples measured in a unit containing several sands and a shale”.
Correlation
“Values of some seismic attributes can show a strong correlation with values of certain petrophysical properties”.
Anisotropic
“My porosity data set is anisotropic because measurements towards the Northeast vary much more than in other directions”.
Probability
“I would like to see those locations in my saturation model where there is a 70 percent probability that values will be greater than 0.6”.
Stationarity
“Porosities in the same geological unit, but in different fluvial depositional
components may not exhibit stationarity as a group because the different particle sorting mechanisms at work may cause the variance of the property in one facies to behave in an entirely different fashion than in another facies” It is for this very reason that facies should be modeled first, then petrophysical properties modeled within facies.
Subject: Part 1 – Statistical Concepts
Section: Basic Concepts and Terminology
Chapter: Descriptive and Analytical Tools Page Title: Preface
Geostatistical data analysis tools rely on the traditional definitions of statistics, and give us a way to measure, describe, and compare certain characteristics of our raw data and the resulting models. Below you can see a display of the most common of these tools and the way in which each presents its measurements to the user. We will discuss each of these tools in more detail.
H i s t o g r a m s U n i v a r i a t e S t a t i s t i c s C r o s s p l o t s
Page Title: Histograms
Below, you see an example of a histogram. This histogram depicts the distribution of all Porosity values of all well logs which fall in the Ness geological unit. What does this histogram tell us?
[Q 1]
1. Each red-colored column represents a “class” (a range of values).
Example: Class 1 has a range of 0.0 to 0.04, Class 2 has a range of 0.04 to 0.08, etc. [X –axis] [q 2]
2. The height of each column shows the number of points whose values fall in the range of the class [Y-axis] [q 3]
3. The overall shape of the histogram shows how the data may be “grouped” Example: The group of points in the first class are smaller and appear to be independent of the rest of the data, suggesting that they could be excluded from the higher, more useful values of this attribute.
Questions for review:
1. A histogram shows us a __________________ _________________ of the data. [frequency distribution]
2. Each red column in the histogram example represents a _____________ of data. [range] or [class]
3. The height of each red column in the histogram example shows 4. [how many points fall in the class]
Page Title: Typical Uses of Histograms
[q 1]
1. For cleaning up of log data
a. look at the histogram of the original log data
b. if you note any clumping of data at either end of the histogram, as in the very low values in the one above, you might want to remove and treat this portion of the data differently, since it could represent a different facies.
2. For quality control after lumping or up-scaling of well logs.
a. look at the histogram of well logs for the property you will be mapping, such as porosity
b. lump the data or upscale the logs
c. look at the histogram of the lumped or upscaled logs; it should have the same characteristics as the original data. If not, the lumping or upscaling
operation did not preserve the character of the data 3. For quality control after modeling
a. using the histogram of the lumped, or upscaled data as the criteria, make sure that the histogram of the model (3d grid) maintains the same
character
b. if not, a different algorithm might need to be chosen
Questions for review:
1. Name one good use of histograms.
Page Title: Univariate Statistics
This set of measurements is simply a way of describing a particular data set with a series of measurements which is unique to a single set of data which is assumed to represent values for one variable; hence- “univariate”. Measurements include:
• Measures of
Size and Location
such aso Number of points, number of null values
o Minimum value, maximum value, mean value, median value, etc.
• Measures of
Distribution Spread
such aso IQR (inner quartile range)
This number shows the range of the middle 50% of the values. • variance
This number represents a measure of how different the data are; in particular, it measures the probability that a data point will deviate from the mean in this particular data set. We’ll learn more about variance in a spatial context when we get to variograms [q 2]
• standard deviation
This number is actually just the square root of the variance, and also is used when describing how the distribution of data points vary from the mean.
• Measures of
Distribution Shape
such asskew which measures how much data distribution deviates to the right or left of a normal distribution: [q 1]
Questions for Review:
1. Skew is a measurement of Size and Location T/F (F)
Page Title: Typical Display and Uses of Univariate Statistics
A typical display of univariate statistics in report format might look like this:
[q 1]
Typical uses of univariate Statistics
1. It is not uncommon to use the univariate statistics of a data set as its “signature”. When voluminous data sets must be managed and manipulated, naming
conventions are sometimes forgotten, but the data signature inherent in these measurements can be used to identify a particular data set unambiguously. [q 2]
2. In geostatistical modeling, monitoring the univariate statistics of a particular data set as it goes through this transformation or another lets you determine if the transformation went according to plan or not
Questions for review:
1. Univariate statistics are usually presented in what format? (graphic)
2. Taken in combination, univariate statistics can be a unique ____ of a data set. (signature)
Page Title: Crossplot What does a Crossplot show?
• Specifically, it displays the values of two variables measured at the same location
[q 1]
o vertical axis is first variable, horizontal axis is second.
• Reveals the degree of correlation between the two variables by the shape of the data cloud. [q 2]
• Look for the cloud of points to form a shape, or even a line to indicate significant levels of correlation. [q 3]
Questions for Review:
1. How many variables are depicted in a simple 2D crossplot? (2) 2. Specifically, what does the crossplot reveal? (Degree of correlation)
3. What do you look for in the crossplot which indicates a strong relationship is present between two variables? (Points form a shape)
Very strong postitive Strong negative No correlation
When one variable increases, the other increases
When one variable increases, the other decreases
Page Title: Crossplot Examples
In general, we can use crossplots to see if there is a significant relationship between two or three variables, especially when the primary variable is under-sampled in some of the areas where you want to map it.
For example, assume the following:
Saturation logs exist only in the North of the reservoir we wish to map. Through study of the seismic attributes, we have an idea that the seismic attribute reflection strength is related to saturation in a particular zone. If, indeed, we can show that saturation and reflection strength are strongly related, then in those areas of the reservoir where we do not have any data for saturation, we use reflection strength as a surrogate value, completing the map in all locations.
Gather the data
A requirement for making a crossplot is that we have a reasonably large set of pairs of values where both target variables have been measured at the same geographic location. For example, a grid of saturation values and another grid of reflection strength could be used, as long as the two grid geometries are identical. Alternatively a well log of saturation values and another synthetic log of reflection strength from the seismic volume would work as well.
In making the crossplot, one of the variables is called the primary, and the other is called the secondary. Typically, the secondary will be the surrogate attribute.
Make the crossplot
When the two attributes are plotted on the crossplot, in Petrel, for example, look for a significant relationship, as defined by a focusing of points along some narrow shape, either curved or straight, preferably thin and narrow. We go through this exercise with the data making the following assumption:
The primary data value at one location can be predicted (calculated) from a single value of the secondary variable at the same location.
Formulate the relationship
It is the job of the crossplot to show IF a relationship exists, and how strong it is. If a relationship is found, then the next step is to formulate the mathematics of the correlation. Essentially, we must know how to calculate a reasonable value of the primary variable at some location from a value of the secondary variable at the same location – what is the formula? Software typically takes care of this, for example, Petrel or LPM. [q 1]
Use the relationship in modeling
The final step is to make use of both the primary and the secondary data sets and the relationship between them in a modeling operation which will produce the desired result. The desired result is that the primary attribute will be defined over its original area as well as the area covered by the secondary variable. When Petrel is used for the modeling operation, many of the steps involved are automated. Note that you must choose an algorithm which actually does allow the input of a second data set. [q 2]
Questions for review:
1. When a relationship is formulated, what kind of function is created? (mathematical) 2. Modeling with correlated data sets requires you to choose an algorithm which
Page Title: Variance Cloud
• This is actually a first step in the creation of a variogram. [q 1]
• Finds all point pairs for each distance (lag) classification • Computes the variance for each point pair
• Shows the variance (how different the points are) for each pair within each classification to help explain the final variogram shape
• The final variogram is simply the average of the variance for each distance class. [q 2] The distance class ranges are not shown here.
Questions for Review:
1. What step is the variogram cloud in the sequence of variogram creation? [first step}
2. The final variogram shows the ___________________ of the variances in each of the distances classes. [average]
.804 .603 .402 .201 Variance 8.25 16.4 24.6 32.8 Separation V a r i a n c e c l o u d
Page Title: Variance Clouds Usage Example
The variance cloud as depicted above cannot be generated in Schlumberger software at the moment; however, Petrel offers a variation of this which is very useful. In the diagram below from Petrel, we see that a histogram of the variance cloud has been superimposed over the final experimental variogram. [q 1]
The advantage of this display is that it gives a relative “weight” to each experimental variogram point by showing you how many point pairs were used in its calculation. In those cases where only one or two pairs contributed to a final variogram value, then that value might be ignored when fitting the model to the variogram. [q 2]
The histogram is useful, but being able to see the results of the specific pairs of points is even more useful, especially if there is any facility for identifying actual well names, which might even lead to the correction of errant logs.
Questions for Review:
1. Petrel cannot display a variogram cloud, but can display a____________ based on the variogram cloud values. [histogram]
2. Does the information displayed by Petrel allow the user to see the number of point pairs used to compute the final values, or the actual variance value for each pair? [number]
Page Title: Variograms
• The input to a variogram is a set of data points from which a model is to be computed - for example, a set of porosity well logs. Variograms should be made from the original or thresholded logs, not the averaged or upscaled logs.
• A variogram is a plot of variance (Y) versus distance class (X)
• Variance measures how different a set of points are, one from the other.
• Based on how far apart two points are, a variogram will show how different they can be expected to be [q 3]
• A basic assumption in geostatistics is that the closer two points are, the more similar their values will be
• The variogram measures unique characteristics about a data set’s variance, such as its Range, Sill, and Nugget, which are defined below.
.
How is a variogram computed?
• First, find all point pairs in all distance classes (lags). For example, find all those pairs of points which are 10 meters apart, put them in a bin, then find all pairs of points which are 50 meters apart, put them in another bin, etc…
• For each bin, compute the variance for each pair of points in the bin (class). • Now, average all the variance values into one number for each bin.
• Plot the average variance for each class as depicted by the black points below. These points make up what is called the “experimental variogram”.
• Let the user fit a curve thru the experimental variogram to create variogram shape, as depicted by the simple curve in the diagram below.
Range
Sill
[q 1] [q 2]
The range, sill, and nugget are the three most revealing characteristics of the variogram. The Range shows the distance where spatial relationships between data points cease. In other words, two data points which are further apart than the Range have only a random relationship.
The Sill is the variance at the Range.
A non-zero Nugget indicates that there are very close data points which are not very similar.
Questions for Review:
1. The value on the X-axis of a variogram represent [distance class or separation distance]
2. The values on the Y-axis of a variogram represent [variance or semivariance]
3. Variance measures how _____________ points are. [different]
Page Title: Why do we need variograms? Reason 1 – Variograms are required for many algorithms [q 1]
If you choose a Kriging algorithm, or certain other geostatistical algorithms for modeling, you must have a variogram. In most cases, a default variogram will be available, making it unnecessary to actually create and shape the variogram, but still, a variogram is
required for many of the algorithms because the variogram becomes the primary weight function during modeling.
Reason 2 – To determine the natural heterogeneity or inherent granularity of the data in the vertical direction [q 1]
It turns out that the Range of the Vertical variogram is a good candidate for the layering increment within a particular zone. We want a layer thickness which will allow
differences in facies to be seen.
Reason 3 – To determine if there is anisotropy in the horizontal direction [q 1]
Tools are available during the creation of the horizontal variogram which allow you to establish if anisotropy exists and the measure it.
Reason 4 – To have another quality control measurement for comparison before and after modeling operations [q 1]
As you analyze, edit, threshold, and model your data, the variogram just like the histogram is an excellent tool for making sure that the characteristics of the data are preserved after each modeling operation. For example, the variogram of the upscaled well logs and the variogram of the 3D model should be similar, if you are to assume that the proper algorithm was used.
Layering is Vertical cell size in the 3D model Top
Questions for Review:
1. Name two reasons to make a variogram. [It’s required
to determine vertical heterogeneity to determine horizontal anisotropy to verify QC measurements]
Page Title: Variogram Maps
• Uses the same information as computed in a variogram
• Reorganizes the point pairs in a geographic 2D sense by E/W and N/S separation distance
• Produces a contour of the 2D variance surface for unambiguous detection of the direction and extent of anisotropy. If anisotropy exists, then you will see oval-shaped contours whose high or low will be in the center of the map. Think of the oval contours as sets of small to large-sized ellipses whose major axis shows the major direction of anisotropy.
• The map is symmetrical and reversed on either side of the major axis of anisotropy as in the example below: [q 1]
Questions for review:
1. What strong geometric characteristic does a variogram map have? [Symmetry]
Page Title: Using the Variogram Map
The variogram map provides a highly automated way to determine whether or not a data set has anisotropy or not. [q 1]
It performs several single variogram functions at one time. The anisotropic direction, if present, can be seen quickly and clearly measured. If available, it should be the first thing done when investigating horizonal variograms, since so much information can be
revealed at once. [q 2]
Review Questions
1. Is the variogram map a more automated or less automated way to discover anisotropy? [more]
2. If available, a variogram map should be the Initial or Final operation in making the horizontal variogram?
Subject: Part 2- Modeling Concepts Section: Modeling Geometry
Chapter: 2D Grids
Page Title: What are the components of a 2D Faulted Model? The components of a 2D Faulted model are:
1. Grid [q 1]
2. Fault Traces [q 1]
3. Interpolator
The interpolator is a grid-based algorithm which can compute the value of the grid at any arbitrary location. It is this software interpolator which allows you to think of the grid as continuous and existing between the grid nodes, even in the interpolator zone.
• 2D grids can be refined
• Z-values are located at the nodes (intersection of row/columns lines) [q 2]
• Grid geometry is everywhere perpendicular [q 4]
• Number of grid nodes across any fault face depends on the width of the fault zone. [q 3]
Upthrown fault trace
Downthrown fault trace
Fault Face
Extrapolation zone
Single-valued grid
nodes
Downthrown
horizon block
Review Questions
1. What are the 2 real components of a 2D model? (Grid and Faults) 2. Where are the grid values located in a 2D model? (Grid nodes)
3. The number of grid nodes across any fault face in the 2D grid is determined by? (Width of fault zone)
Page Title: Grid Nomenclature
Once the grid node values are computed, the data points are redundant. The 2D model consists of the grid and the faults.
The 2D grid is a collection of X, Y, and Z (elevation) points, but only one Z can be stored for one X, Y location. [q 2]
Row/column numbering is shown with respect to CPS-3 conventions.
Xmin Ymin C o l u m n 1 2 3 4... [q 3,4] R o w 1 2 3 . . . Ymax Xmax
One grid cell [q 5]
Small cells mean high resolution Large cells mean low resolution
One grid node
Nodes area at the intersections of the lines . Values are computed for each node, if possible, from surrounding data points Fault Trace [q 1]
Used to segregate data points
Questions for review:
1. Fault traces are used to ___________________ data points. (segregate)
2. How many Z-values can be stored for one X, Y location in a 2D grid? (only 1) 3. Columns are numbered from ______ to ______ in CPS-3 convention. (left, right) 4. Rows are numbered from ______ to ______ in CPS-3 convention (top, bottom) 5. Smaller grid cells cause a/an ____________in model resolution. (increase)
Page Title: Grid Refinement
Grid refinement is typically a 2D operation, with 3D modeling systems tending towards cellular models, where only one value per cell is allowed, and cells are unable to be refined. It is grid refinement which allows a 2D grid-based model to be thought of as existing everywhere, even though grid values exist only at row/column intersections. As evidenced by contours, which are traced across all locations of the model, a 2D grid is always thought of as a continuous surface, rather than a collection of cellular blocks. Refinement typically occurs in discrete multiples, i.e., refinement by two, by three, etc.. Below is a depiction of an original 5 column by 6 row grid which was refined “by three”, i.e. each cell was subdivided into three sub-cells. The new grid is composed of both the dark lines and the dotted lines and has 13 columns and 16 rows. The extra cells are computed only from the original cells. Original data is not used. The extra cells give the model a smoother, and in many cases, a more accurate quality up to a point. [q 1, 2, 3].
R O W COLUMN 1 1 2 3 4 5 2 3 4 5 6
Questions for Review
1. Refinement allows the 2D model to be thought of as existing ___________. (everywhere)
2. A refined grid is _______________ than its original version (smoother) 3. 3D grids typically cannot be refined (T/F). (T)
Page Title: 2D Grid Examples
In this view of a 2D grid displayed in a 3D volume, the continuity and refinement are still apparent.
Likewise, in this traditional manifestation of the 2D grid through contouring, we see that the lateral variation of 2D grids can be broken into many levels by the use of refinement, which, in many 3D systems is not available.
Below is another figure which shows how the light blue lattice of the 2D grid cells are typically refined for purposes of contouring, in contrast to the geo-cellular 3D grid below it, whose cells are typically not refined during display. They are not refined since the concept is that only a single geo-cellular value exists at the center of each cell, whereas the values in the 2D grid are assumed to exist at the corners.
Subject: Part 2- Modeling Concepts Section: Modeling Geometry
Chapter: 3D Grids – Property Models Page Title: Components of a 3D Model What are the components of a 3D geo-cellular Model?
1. Grid Cells and Values
2. Fault Blocks (Fault “Segments” in Petrel) 3. Units (“Zones” in Petrel)
4. Layers
Typically, the cells cannot be refined and so the grid-based interpolator used by the 2D grid geometries is not needed. Also the fault blocks are pre-defined in 3D models so that fault traces and grid-based interpolation is not needed for those reasons, too. [q 5]
Grids are geometrically partitioned by Fault Block, as well as by Zones (Units) [q 2]
Zones are subdivided by Layers [q 3]
Grid has X, Y, Z, A components, where A is the attribute being modeled. Originating 2D structure grids provide the initial geometry [q 1]
Cells are still basically rectilinear in X,Y, although truncated by the faults Sides of cells are typically orthogonal except at faults
3D grids do not allow refinement 3D grids do not require fault traces
A single value is positioned at the center of each cell [q 4]
Unit or
Zone
Block 1
.Block 2
Layers
Cell value
in center
Review Questions
1. The initial geometry for 3D grids is typically provided by __________. (2D structure grid)
2. 3D grids are partitioned by __________ and _________. (fault block and zone) 3. 3D grid zones are subdivided into _______. (layers)
4. Values for 3D grids are typically located at __________________. (the center of the cells).
Page Title: Example of 3D model
An example of a 3D facies grid showing the inherent granularity of 3D grids compared to 2D grids.
Subject: Part 2- Modeling Concepts Section: Modeling Geometry
Chapter: 3D Grids – Reservoir Models Page Title: Reservoir Models
Components are basically the same as for 3D Property grids. The biggest difference is the orientation and shape of the cells whose purpose now is to not only honor the structure, but the flow within it. [q 1,2]
• Faces of cells oriented along faults, perpendicular to FLOW [q 3]
• Faces of cells also oriented radially around wells, when needed. • 2D structure grids provide basic geometry
• Model is organized by fault blocks • Attribute value in center of cell
• Zones or Units are subdivided into layers • Sides of cells may or may not be vertical • No fault traces required
• No refinement
Today, a 200,000-cell grid is an average grid for simulation.
A 500,000-cell grid is considered large.
[q 4]
A 200K-cell grid is a cube 60 cells on a side.
3D
Flow
Vector
Review Questions
1. The purpose of the simulation grid is to model ________. (flow) 2. One major difference between simulation grids and property grids is the
______________________ and ____________________ of cells. (orientation and shape)
3. Faces of cells in a simulation grid are ideally oriented _____________ to the fluid flow. (perpendicular or normal)
4. Today, an average sized simulation grid has about _____________cells. (200,000).
Page Title: 3D Simulation Grids
There are two basic types of simulation grids, structured and unstructured. [q 1]
Structured grids are the most commonly used; unstructured grids are used only occasionally.
Structured grid geometry (FloGrid):
• Cell edges are vertical, but faults can be sloped • Grid is distorted in X,Y to honor faults
• Tops and bases follow geological model • Cells are typically 6-sided
• Viewed in plan view, grid is relatively homogeneous wrt cell size
• Structured grids have a fixed number of rows, columns, and layers. [q 2]
Questions for Review:
1. What are the two basic types of simulator grids? (structured and unstructured) 2. Structured grids have a _________number of rows, columns, and layers. (fixed)
Page Title: Structured Grids
Two types of Structured Grids
1. Block Center
• Flow connections are center to center
[q 1]
• Cells are always rectilinear and orthogonal
• No information about the geometry of neighbor cells
• FloGrid does not use this.
2. Corner Point
• Flow connections are face to face
[q2]
• Geometry of neighbor cells are known
Questions for Review:
1. In Block Center grids, flow is assumed to be from ________ to ________. (center to center)
2. In Corner Point grids, flow is assumed to be from ________ to _______ . (face to face)
Page Title: Unstructured Grid Geometry (FloGrid)
Unstructured grids can model flow more accurately, especially relative to a particular object or feature, but are time-consuming to define and use.
Unstructured Grid Geometry (FloGrid):
• Cell edges are generally vertical, but faults can be sloped
• In plan view, cells may not be homogeneous with respect to shape. (dense in some places, sparse in others) [q 1]
• Honors flow orthogonality not only at faults, but at wells.
• Use is not as common as structured grids.
Review Questions
1. When viewed in plan view, an unstructured grid cells may not be ________________ in shape. (homogeneous, consistent)
Subject: Part 2- Modeling Concepts
Section: Data Transformations During Modeling Operations Chapter: Data Transformations
Page Title: Thresholding Transform Name – THRESHOLDING
Description: Remove/ignore borehole values below specified Min/Max Cutoffs
[q 1]
Workflow:
• Load boreholes
• Use previously decided cutoffs or verify cutoffs with histogram • Use thresholding tool to reset the Min and Max
Question for review:
1. Thresholding will _________ values along a borehole along some specified limit. (remove or ignore)
Page Title: Upscaling
Transform Name – UPSCALING or AVERAGING by CELL
Description: Average the borehole values falling in the same cell [q 1]
Workflow
• Define number of layers in unit (correlation scheme) • Load boreholes
• Pick best averaging technique for the original or thresholded data • Upscale the data, specifying the averaging method.
• Each cell contains one averaged value, while there may be hundreds of values along the borehole.
Red circle represents upscaled data, one point per cell (horizontal subdivision of a layer)
Upscaled well logs from Petrel
Question for review:
Page Title: Segregation Transform Name – SEGREGATION
Description: For model quality, data is segregated by fault block so that grid values computed in one block will use data only from that block. [q 1]
Another example of segregation is during Sequential Gaussian Simulation in Petrel, were the input data such as porosity logs are optionally segregated by lithology.
Workflow
Questions for Review:
1. Data is segregated by fault block to control the ___________ of the model. (quality)
2. Porosity data can be segregated by lithology because porosity may behave differently in different facies T/F. (T)
Page Title: Masking Transform Name – MASKING
Description: Discrete grid values from one grid can define a location template to determine where to populate an output grid. For example, during Sequential Gaussian Simulation in Petrel, an existing facies grid can be used as a look-up table, so to speak, during petrophysical modeling to tell the algorithm when it is gridding at a “channel” location, or a “fine sandstone” location, etc.
Workflow
• Create a masking grid with discrete values at those locations where you want an attribute computed, for example, at particular facies locations. [q 1]
• Pick the masking option when specifying output grid during population, or when performing the grid to grid operation.
Questions for Review:
1. Masking can be used to control the __________ where certain operations are performed. (location).
Page Title: Removing a Trend Transform Name – REMOVING A TREND
Description: As you will learn, Kriging algorithms make certain assumptions about their input data sets. In particular, the assumption of “stationarity” (another expensive word which, for our purposes can just mean “behaving everywhere in the same manner”) [q 1]
insists that there be no inherent trend exhibited by a data set if it is to be Kriged properly. Thus, one of the most common data transforms to perform in 3D modeling is to remove the trend from data sets before making the final variograms and modeling with a Kriging algorithm. The workflow is shown below.
Workflow
• Before Kriging any data set, determine if it has a “trend” by either inspecting its histogram or variogram for horizontal convergence to a sill, or by some other means. [q 2]
• If a trend is identified, then remove the trend using the appropriate tool within the software, krig the data, and add the trend back into the result. Note that this operation is not to be confused with attemping to make use of an external trend during modeling
Trend removal is also discussed more in detail in the section which describes Kriging algorithms.
Determine if
a trend exists in the dataRemove the
trendMake the final
variogram
Krig the data
Add the trend
back into the resultDo not confuse the removal of a trend from a data set with various Kriging algorithms such as “Kriging with an External Trend”. They are separate concepts. [q 3]
Questions for Review:
1. Removing a trend from a data set ensures that ______________ is maintained (stationarity).
2. You can use either a _____________or a ______________ to determine if a data set has an inherent trend.
3. Kriging with an External Trend is the best way to remove a trend from a data set. (T/F) (F)
Page Title: Normal Score Transform
Transform Name – NORMAL SCORE TRANSFORM
Description: Some Kriging algorithms require your data to be in a normal distribution before the algorithm will work properly. An example is Sequential Gaussian Simulation. Petrel provides a simple way of doing this for you and will automatically perform the inverse transform on the output. A data set which has been transformed to a normal
distribution will have a mean of 0.0 and a standard deviation of 1.0. It is easy to identify data such as this with a simple histogram. The first histogram below shows data in its raw state, the second shows the same data after being transformed to normal score format.
[q 1, q 2, q 3]
Questions for Review:
1. The mean of data which is in Normal Score format is equal to ____. (0.0) 2. An example of a kriging algorithm which requires the data to be in a normal
Page Title: Weighting Transform Name – WEIGHTING
In modeling, individual data points are typically weighted (given more or less importance) during many calculations.
Description: Input data collected during gridding or population operations is typically weighted by distance before it is used to calculate a value for the grid node. The weighting scheme ensures that data points which are closer to the grid node being
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computed have more weight than far points. Close points are assumed to be more
representative of the attribute being mapped than points which are further away. Attribute values of none of the points are changed, they are simply given more or less importance, based on their distance from the node. [q 1, q 2, q 3]
As you can tell by the shape of the weight curve, most of the weight is given to very close points, and little weight is given to further points.
HIGH WEIGHT
LOW WEIGHT
CLOSE POINTS FAR POINTS
.
.
Close Points Have Higher WeightsFurther points have Lower weights Grid node to be
computed
Data points
Distance to data point determines weight.
Questions for Review:
1. Points used in the calculation of grid values are typically weighted as a function of ____________ (distance).
2. Far points have _______weights, near points are assigned ________weights. (low, high).
3. When weighting data points, the algorithms raise or lower the original value, based on the weight assigned. T/F (F).
Subject: Part 3- Modeling
Section: Gridding Methods
Chapter: Algorithm Classifications Page Title: Introduction
In this section, we’ll talk about the geostatistical algorithms which compute 3D facies and property grids. Gridding is sometimes referred to as “population”.
“Gridding”, or “population” is the process by which randomly-spaced data samples of some property are transformed into a complete model in the form of an organized 3d grid lattice.
Well Logs
2d grids
3d grids
Constants
algorithm
model
Input
Data
Page Title: How Are Gridding Algorithms Classified?
Geostatistical algorithms vs. Traditional algorithms:
Most of the algorithms we’ll be discussing here fall in the category of geostatistical algorithms, including Kriging and Simulation. These algorithms include standard statistical techniques in their operation. There are also non-geostatistical
algorithms available such as Nearest Neighbor, Distance To Nearest Neighbor, Inverse Distance, and Inverse-Distance Cogridding.
Kriging algorithms vs. Non-kriging algorithms
Kriging is one type of geostatistical algorithm which has many variants. The traditional algorithms listed above are typically non-kriging algorithms. We will see what this means later. [q 1]
Continuous vs. discrete properties
Some algorithms are designed specifically for discrete, tabular data values, such a rock type, facies or lithology codes. Other algorithms are designed for continuous data such as porosity. [q 2]
Estimation (Deterministic) algorithms vs. Simulation (Probabilistic) algorithms The mechanics of Estimation algorithms are totally different than Simulation algorithms. Deterministic algorithms try to create a model which follows the data literally, while the Probabilistic algorithms create a model which if faithful to the statistical characteristics of your data. One of the biggest decisions to make is to choose which one to use for a particular property or facies. [q 3]
Single or multiple data set
Many algorithms will allow the specification of a secondary, correlated data set. This is particularly useful when the primary data set is sparse and correlated data exists which covers a larger area. An example of this would be the situation where you have very few wells for your primary data, but you have one or more seismic attributes which can be correlated with the property you wish to model. [q 4]
Review Questions:
1. Traditional algorithms are typically non-kriging algorithms. T/F (T). 2. Facies logs would be considered to be __________data. (discrete)
3. Probabilistic algorithms are synonymous with _______ algorithms (simulation). 4. Some algorithms allow you to use more than one _________ . (data set)
Page Title: More About Kriging and Non-Kriging Algorithms What are the Differences between Kriging and Non-Kriging Algorithms?
The comparison below applies only to the modeling of petrophysical properties, not structure.
• Kriging algorithms use variograms to guide the weighting of the data points; non-kriging algorithms do not.
• Kriging algorithms allow valid statements to me made relative to the probability of the results of certain calculations; non-kriging algorithms typically do not. [q 3]
• Kriging algorithms produce grids whose variance is minimized; typically non-kriging algorithms do not. As a result of this characteristic, kriging algorithms tend to produce grids whose values remain within the range of the data whereas non-kriging algorithms sometimes tend to project slopes
inherent in the data. [q 1]
• Kriging algorithms tend to produce grids which preserve the percentages of data ranges inherent in the original data.
• Kriging algorithms will decluster two close data points, providing better weighting for both; non-kriging algorithms do not. [q 2]
• Kriging always accommodates anisotropic weighting; only a few non-kriging algorithms will do this.
• Kriging allows for a variety of alternative “extrapolation” algorithms to cover grid areas having only very sparse data.; in general, non-kriging algorithms have only one alternative.
• Non-kriging algorithms tend to be simpler to use, not requiring the creation of a variogram.
• Kriged maps can appear noisy compared to maps made by non-Kriging algorithms. This is because many non-Kriging algorithms have built-in smoothing algorithms which are designed for more aesthetically pleasing results, and may not take established probabilities into consideration.
Questions for review:
1. Kriging algorithms minimize the _____________ of the error. (variance). 2. Kriging algorithms will __________ close data points. (decluster).
3. When using kriging algorithms, accurate statements can be made about the ___________of certain results. (probability)
4. Non-Kriging algorithms do not use ________________ . (variograms) 5. Kriging algorithms are typically easier to use than non-Kriging
Page Title: More About discrete and continuous data
The most common properties to model with geostatistics are facies and petrophysical properties. Discrete data is data whose values are based on a classification scheme (0=floodplain, 1=levee, 2=channel sand), and continuous data is data whose values represents real numbers, such as porosity, whose value can be in the range of zero to 1.0. Most geostatistical operations distinguish between these two types of data, providing separate algorithms for each type. For example, Sequential Indicator Simulation is a type of simulation algorithm which should be used with discrete data, whereas
Sequential Gaussian Simulation is used for continuous data such as permeability or porosity.
Review Questions:
1. Discrete data values are based on a _________. (classification scheme). 2. Sequential Gaussian Simulation is used for _______ data sets. (continuous).
Page Title: More about deterministic and probabilistic algorithms Deterministic (creates a single grid) [q 1]
Deterministic algorithms are also called “Estimation” algorithms Use this method when you have plenty of dataExamples: Nearest neighbor, inverse distance, kriging, Probabilistic (creates single or multiple grids) [q 1]
Probabilistic algorithms are also called “Simulation” algorithms Use this method when you have sparse data, or very complex facies [q 3] Examples: Sequential Gaussian Simulation, Fluvial Simulation, TruncatedGaussian Simulation Deterministic
These algorithms take the data values literally and assume that computed grid values between data points have almost a geometric relationship with the points, based only on z-values, slopes between points, and closeness to the node. No attempt is made to preserve the distribution characteristics of the input data. This means that the histogram of the computed grid values may or may not resemble the histogram of the input data.
Probabilistic
These algorithms take the data values literally as well, but what happens in between the points is a function of more statistical measurement such as frequency and distribution of z-values, both horizontally and vertically. In addition, the algorithm uses a random technique in the selection of data for each computed grid node. A fundamental characteristic of probabilistic grids is that, for any given data set and its parameters, one output grid is equally probable (“stochastic” ) as the next. In fact, it is common to generate “multiple realizations” (versions) of a property and then study their differences and distributions to determine global probabilities as the basis of the final model. [q 2], [q 5]
Advantages/ Disadvantages Deterministic
Advantages: Many to choose from/ Simple to use/ Intuitive [q 4]
Disadvantages: Tend to smooth out highs and lows/ Inappropriate for flow simulation/ No probability information/ Tend to require plentiful data for good models
Probabilistic
Advantages: Works well even with sparse data/ Gives a more realistic model in the case of sparse or difficult data/ Retains highs and lows/ Provides assessment of global probability /Retains distribution character of the original data.
Disadvantages: More difficult to use/ Time consuming
Review Questions:
1. Deterministic algorithms typically create only _____grid, while probabilistic algorithms create ____________grids. (one, multiple)
2. Multiple realizations simple means multiple __________. (versions) 3. Probabilistic algorithms are used when you have _________data (little).
4. One advantage of deterministic algorithms is that they are _________ (plentiful, simple, intuitive)
Subject: Part 3- Modeling
Section: Gridding Methods
Chapter: Variogram Basics
Page Title: Variogram Roles in Gridding & Geostatistics
At this point in this primer, we will introduce only the most basic facts about variograms. A separate sections of variograms will be presented later.
Above, we see a typical variogarm, a measure of variance with respect to distance classes. We will not worry for the moment how to create a variogram, only what it is used for by the gridding algorithms. We will be learning a lot more about variograms, but for right now, all we need to know is
1. the variogram is created directly from the data to be used to create a model 2. the most important measurements shown by the variogram are
• Range – which shows that distance between data points where they cease to have any statistical relationship
• Sill – which shows that value of the variance associated with the Range where the variogram curve begins to flatten out.
• Nugget – which shows how well the data set honors the assumption that close points will be very similar in value.
This portion of the variogram curve becomes the weight
function for gridding
What part do variograms play in geostatistics?
It is not possible to discuss geostatistics without variograms. They are the basis of the geostatistical algorithms and must be present for the algorithms to be useful. We can summarize the need for variograms as follows:
They are required for geostatistical algorithms [q 1]
They are very useful as a data analysis tool for the following: [q 2]
Determine layer thickness (using vertical variogram)
Determine directions/degree of anisotropy (using horizontal variograms) They are used as quality control tools to judge the quality of your model [q 2]
How are variograms used during gridding?
The mechanism of how to create variograms and how to use their analytical capabilities will be covered as a separate topic. Here, we simply want to understand how they are used by the algorithms.themselves. Think of a variogram as a packet of information which helps assign weights to the data points used in the calculation of individual grid values. In particular, variograms provide the following information:
1. a weight function for all data to be used in the calculation of grid values. This weight function is defined in the three primary directions, X, Y, and Z. It can be thought of as an ellipsoid.
2. the range in each direction (X, Y, and Z) for which the variogram weight function is valid. These numbers represent the Major, Minor, and Vertical Ranges. If most data collected to compute a grid values is further away than these ranges, then the grid value is computed using an alternative method which does not rely on the variogram’s weight function. [q 3]
Following will be a few more important facts about variograms which you should know before we can continue with the comparison of 3D modeling algorithms. You will learn the mechanics of making variograms in a later section.
Question for Review:
1. We need variograms because they are _________ for geostatistical algorithms (required).
2. The variogram serves as both _________ and_________ tools during modeling. (data analysis and quality control)
3. The variogram has a range in each _____________ for which a weight function is required. (direction)
Page Title: Facts to Remember about Variograms
In order to continue our discussion of geostatistical algorithms before going into a major discussion of variogramming, it is only necessary to make sure that we remember the following simple facts about variograms.
Variogram Fact #1
Variogram Fact #2
The RANGE of the variogram is that distance where data points in your data set begin to LOSE AUTO-CORRELATION. Stated another way, points in your data set which are closer together than the RANGE have a spatial
significance in the correlation of their values, but points which are further apart do not. [q 1]
That portion of the variogram curve from zero Distance out to the RANGE, when inverted, becomes the weight function used internally by geostatistical algorithms when computing grid node values. Data which is further than the RANGE to the grid node being computed uses a different weighting scheme.
[q 2]
This portion of the variogram curve becomes the weight
function for gridding
Variogram Fact #3
Variogram Fact #4
Conceptual ellipsoid formed by the 3 variogram ranges
In 3D geostatistical modeling, three variograms are defined for each data set to be modeled – one in the vertical direction and two in the horizontal direction. If the data is isotropic (having no natural directional bias), then the two horizontal variograms are the same. Together, the ranges of these three variograms define a three-dimensional ellipsoidal weight function which is used by the geostatistical component of the chosen algorithm.
The variogram RANGE and the SEARCH RANGE which is specified for a particular gridding operation are two separate concepts. The variogram RANGE has already been described. The SEARCH RANGE specifies how far away from a grid node data will be collected for use in the gridding. The user sets this value. Data further than the SEARCH RANGE will not be used. As suggested above, data closer than the RANGE SEARCH, but further than the variogram RANGE is handled differently. Clearly, the SEARCH RANGE
should be LARGER than or equal to the variogram RANGE. [q 3]
M ajor Horizontal Range V ertical Range
Review Questions – Variogram Basics
1. The variogram range is that distance within your data set where the individual points begin to lose ___________________ . (autocorrelation)
2. That portion of the variogram curve from zero out to the Range is inverted and then becomes the ____________________ during gridding. (weight function).
3. The _________range determines how much data is collected to use in gridding a particular note, but the _____________range determines when the weighting scheme changes. (search, variogram)
Subject: Part 3- Modeling
Section: Gridding Methods
Chapter: Terminology and Diagrams Used During Description of Algorithms
Page Title: Disclaimer
Disclaimer
Those of you who have attained a good understanding of geostatistics, and maybe even GSLIB, will recognize that some of the depictions of gridding mechanics appear oversimplified, especially from a programming or data management point of view. Our intent here is to present large building-block concepts that can be quickly understood and allow the student to become effective with the tools. We do not presume to trace the actual structure and organization of grid-building computer code, nor the fine details of internal data manipulations or programming techniques.
Subject: Part 3- Modeling
Section: Gridding Methods
Chapter: Traditional Estimation Algorithms
Page Title: Mechanics of 3D Traditional Estimation Algorithms A simple view of the mechanics of 3D Traditional estimation algorithms is given below:
Position at a location where a value requires computation (grid node within some
selected zone.
Collect points in the search zone (defined by the user in various ways) [q 1] In this example, the Search Distance, D, is a Horizontal Search Range. The
search thickness, T, is measured vertically.
Weight the collected points by distance from the grid node. [q 2]
Compute the value of the grid node using the selected algorithm and parameters. Move to the next node and repeat the process
If a minimum number of points are not found inside the Search Zone, the node
value becomes null.
0.23
W
W
W
W
Grid node to
be calculated
Search zone
Data points
D
T
Questions for Review:
1. Points used in the calculation of the node are collected in the _______zone. (search).
2. Collected points are weighted by their_________from the node to be computed (distance)..
Page Title: A Survey of Some Traditional Estimation Algorithms
Nearest Neighbor
Each grid node takes on the value of the closest collected point. [q 3]
Used for Lithology, Rock Type Distance to Nearest Neighbor
Each grid node takes the value of the distance to the closest collected point. Not really a population algorithm, but used for calculations and analysis Inverse Distance
Each grid node takes the value of a distance-weighted average of all points collected within the search limit. A variable power parameter determines the weighting – at maximum, the algorithm gives a distance-irrelevant simple average, at minimum, it is equivalent to the nearest neighbor. Anywhere in between, points closer to the node to be computed are weighted higher and further points are weighted lower.
Inverse Distance Cogridding
If the primary data to be gridded is sparse, then the grid of a secondary, correlated data set can be used to help guide the gridding of the primary data. If the secondary data, for example has been shown to be correlated to the primary data, then the system will automatically compute the correlation function and transform the secondary data to the domain of the primary.
When both primary and secondary data are available within the set of collected points during gridding, then the secondary data is weighted lower than the primary. Other than simply providing more data with secondary weighting, this algorithm works the same as the inverse distance algorithm.
Review Questions – Non-Kriging Estimation Algorithms
1. In the Nearest Neighbor algorithm, each node takes the value of the _________ collected point. (closest)
