- As the experimental variogram represented as a set of points on a graph are computed based on the limited, measured data resources for a number of lags, they should be geologically understood and interpreted, be mathematically modeled, then can be used in geostatistical estimation and simulation. This is a typical variogram graph, or variography. The variogram value is on the y-axis and the lag h, or distance, on the x-axis. A variogram value at a given h is the average squared difference between the values of the paired locations. If two locations, u and u + h, are close to each other in terms of the distance measurement, two values are similar, so the difference in their values, Z- Z, will be small. As u and u + h get farther apart, they become less similar, so the difference in their values, Z- Z, will become larger. As the variance of the difference increases with distance, so the variogram can be thought of as a dissimilarity function. At a certain distance, the model levels out. The distance where the model first flattens out is known as the range. This is because sample locations separated by distance closer than the range are spatially autocorrelated, whereas locations farther apart than the range are not autocorrelated. The range depends on direction for anisotropic reservoir. Usually, horizontal ranges are larger than the vertical ranges. Theoretically, at zero separation distance, that is, lag is equal to zero, the variogram value is zero. However, at an infinitesimally small separation distance, the variogram often exhibits a nugget effect, which is some value greater than zero, the interception with the y-axis. The nugget effect can be further divided into Measurement Error and Microscale Variation. The measurement errors result from location measurement error, property value measurement error, or sparse data. The microscale variation errors result from the smallest lags are greater than this scale variability. Since either component can be zero, the nugget effect can be composed of one or the other. The value on the y-axis that the variogram value attains at the range is called the sill. It is the variance of data used for variogram computation. It is equal to 1.0 if the data have been normal score transformed. It is often composed of two parts: a discontinuity at the origin, that is the nugget effect; and the contribution, or partial sill, added together. In some cases of non-stationarities, the variogram does not level off at lag h; or even it levels off, but it's not equal to the variance of data. In this case, using the variance of the data is more practical. Contribution concept is often used in nested variogram data, which is the partial sill for its corresponding variogram curve. It will be addressed in variogram modeling. Let's investigate horizontal variogram. It is an areal measurement. It's not reliable due to the sparse data. Usually, the horizontal wells, seismic data, conceptual geological models, and analog data are incorporated. When directional continuities are not evidently detected from geological information, horizontal variograms are calculated for a series of directions, starting at zero degrees, that is, looking North, and increasing by defined increment to 180°. In the above case variograms are produced in directions zero, 30, 60, 90, 120, 150 degrees. The Tolerance is a angle which defines a conical search area in which data will be considered. For example, data will be used which falls between directions of given angles, plus 10 degrees and at minus 10 degrees. Normally, the horizontal variogram is calculated first to give an indication of the strike orientation of the data. This direction can then be used to define the direction for the vertical variogram. There are computed variograms in six angles: zero, 30, 60, 90, 120, and 150 degrees. Six gray lines under the variogram graphs are used to illustrate six ranges. You can see they have different ranges. Let's put these six ranges into their corresponding directions in the lower graph, that is zero, 30, 60, 90, 120, and 150 degrees. Extending six ranges into their opposite directions to construct a rose diagram. The directional continuity of anisotropic effect in geostatistics is referred as Geometric Anisotropy. Next, let's investigate the Vertical Variogram. Vertical Variogram is computed using all lags in vertical direction. The vertical range in red is shown on the right side variogram diagram. This is an example of experimental vertical variogram. You see, there is no sill defined as the variogram value, that is, the difference of data values keeps increasing as the distance between data pairs increases. How to handle this issue? The trend curve can be derived, for example, by moving window average method. The moving window average is a smoothing method. Simply define a window size, proceed along the profile, taking the average value of all points contained in the window. The definition of a window size is arbitrary; a value is used that will provide reasonable smoothness, but not so large that the profile is excessively distorted. Trend could be the function of z or the function of x, y and z. Then, subtract the trend curve from the observed curve in order to obtain residuals. They are shown in the right side. Now, let's compute variogram from the residual data, that is, try and remove the data. There is a sill which is the variance of the residuals. The geostatisical estimations and simulations are performed on the residual data, using the variogram created from the residual. Then, the trend is added back to the results over the entire reservoir model. The trend can also be modeled as "power" or "fractal", but they cannot be used in sequential Gaussian simulation method, as there is no sill. This is also referred to as hole effect. As geological phenomena often occurs repetitively over geological time, the repetitive variations in reservoir properties are discovered. This implies a cyclic behavior to the variogram, that is, the variogram will show positive correlation going to negative correlation, then going to positive correlation and so on, at the length scale of the geologic cycles. These cyclic variations often dampen out over large distances as the size or length scale of the geological cycles is not perfectly regular. This is an example of Geological Periodicity for variogram cyclicity, the series of events in which a rock of one type is converted to one or more other types and then back to the original type are discovered. It is a continuous process by which rocks are created, changed from one to another, destroyed, and then formed again. Zonal Anisotropy is an extreme case of geometric anisotropic where the range of correlation in one direction exceeds the field size, which leads to a direction variogram that appears not to reach the sill or variance. In this diagram, you can see that the Vertical variogram sill is lower than the Horizontal variogram sill. Suppose the horizontal variogram sill is equal to the variance of the data. The vertical variogram should be handled by setting vertical range as a very large number. The zonal anisotropic variogram are interpreted as the geological phenomena. Here is a map view with two wells. Well A is in the higher-valued area. Well B is in the lower-valued areas. Both of them don't go through the entire variability as the areal variogram goes through. Let's review what variogram interpretation covers. It covers Range, Nugget, Sill; Horizontal Variogram; Geometry Anisotropy; Vertical Trend; Variogram Cyclicity; Zonal Anisotropy. This is the end of the presentation of Variogram Interpretation. Next presentation is Variogram Modeling.