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
algorithms T/F. (F)
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 data
Examples: 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, Truncated Gaussian 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)
5. Probabilistic algorithms typically involve a ________ generator (random number).
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
Minor Horizonal 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.