First Section

- In this lecture, we will be talking about a very fundamental concept in Geostatistics, which is called the Variogram. The Variogram is really the working-horse of Geostatistics, and of most common classical techniques. Consider here in this second slide: four probability fields. Obviously, you notice a lot of differences between these probability fields. The question that we, in fact, will be asking in this short lecture, is, How would you quantitatively describe these differences? And, secondly, of course, Why is this at all useful? Before considering the Variogram, we will consider a simple case, namely Autocorrelation in One Dimension. Consider here, for example, on the left, a signal, a time series. So, function Y, that is function of time. And, in that function, we would like to describe whether there's any temporal correlation, which there obviously exists in this particular case. Signal is not purely noise. The way this can proceed is very simple. What we will do is, we will record pairs of observations at times t and at times t plus Delta t. These pairs of observations are then recorded as points in the right-hand scatter plot, and we collect these points by sliding the window over the entire time series, thereby creating this scatter plot. In the scatter plot, we then calculate a correlation coefficient, which is known to be between minus one and plus one, where one means "strong correlation," "exact correlation," and zero means "no correlation." Therefore, the correlation coefficient at a particular interval, Delta t, describes the correlation in this time series between time interval, Delta t. The question now is, How can we extend this particular notion of Autocorrelation to two-dimensional space and three-dimensional space? Really, the largest difference between one-dimensional and two-dimensional space, is the issue of direction. In 1D, there's only one direction. In two-dimensional, there are an infinite set of directions. Consider that we have some data on some regular grids, like, for example, one of the images that I showed in the beginning. In this particular case, therefore, we have to choose not only a distance, a like distance, or, as in the previous case, a time interval. But we also have to create a direction in space. For example, if I consider the blue vector, I have a direction of about 30 degrees from the east-west direction. But, otherwise, I will proceed in the exact same fashion. I will try to collect pairs of observations by moving this vector in this three-dimensional or two-dimensional space. With each pair of vector, comes a point in a scatter plot. Then, I collect these points, all the possible pairs of observations I can collect in this scatter plot, and I can calculate the correlation coefficient. Next, what I will do is increase the distance of this vector, but keep the same direction. In other words, I will try to calculate many correlation coefficients for the same direction, but for longer vectors. What we, of course, expect is that as the vector becomes longer, that the correlation will decrease, and hence, the correlation coefficient becomes smaller. This is observed in this particular case. We observe, for example, that when, for a distance of seven, we observe a much smaller correlation say, than, for a distance of three. Collecting all this information into a single graph, which is shown here, in the middle, is called the Experimental Correlogram. It essentially records the correlation as it decreases, for a specific distance, h, in a particular direction. Unlike one-dimensional phenomenon, in two- or three-dimensional phenomenon we can have spatial anisotropy. This means that the correlation looks different in different directions. Notably, in one-dimension, there's only one direction. Consider these two simple cases, here. In case one, in the picture on the top left, you can see clearly that there's no specific or preferred direction of continuity, while in case two, there's a clear preferred direction of continuity along the 45 degree direction. Therefore, when calculating correlograms along specific direction, as was shown in the previous case, we will see some differentiation in the correlogram. For example, on the right hand case, we see clearly that the correlogram in the north-west direction and the north-east direction, are very different, while in the left case, this is not the case. Therefore, we call case one an "isotropic" case, and case two, we observe spatial anisotropy. In Geostatistics, we often do not calculate just correlograms, but we calculate variograms, although their relationships aren't very clear. As a matter of fact, there are three functions, or characteristics, of spatial continuity that are used in spatial modelling. One is the Autocorrelation Function, which we have just discussed. The second one is the Covariance Function, which is often used in Statistics. And in Geostatistics, we often calculate the Variogram. As you observe, and there are some theoretical reasons for calculating the Variogram, and there are also equations that relate one to the other, which I will not show, we can go from one to the other. Essentially the Autocorrelation function, when multiplied with the variance of your property, become the Covariance function, it's an un-standardized function, and when we flip this function, we get the Variogram function. We notice, therefore, that the Variogram function increases as the distance increases, while the Autocorrelation function decreases when the distance increases. We could, therefore, say that the Autocorrelation function is a measure of similarity between two points in space, while the Variogram is a measure of dissimilarity between two points in space, and that dissimilarity tends to increase with distance. Therefore, when calculating a Variogram from, say, well data, porosity Variogram, permeability Variogram, or other Variograms of other reservoir properties, we typically, along one specific direction, get this particular behavior. First of all, we have the Nugget Effect. This Nugget Effect is this jump that occurs at the origin, and is often due either to measurement error, or to the fact that sampling is insufficient, and that some of the spatial variability is missed. We then see a gradual increase from the origin, either as a straight line, exponential function, or other types of function, and we notice that the variogram reaches a certain plateau at a certain level. In Statistics, this is often called the Correlation Length. In Geostatistics, this is called the Range. The Range is essentially that value beyond which we no longer see any spatial correlation. The plateau where this is reached is called the Sill, and is often equal to the variance of the phenomenon. When dealing with real data and real reservoir situations, obviously we cannot calculate an exhaustive variogram from an exhaustive data set. We have to calculate a variogram from well data. For example, here, on the left, we see porosity data recorded along wellbores. There are many good sofware out there to calculate Variograms from this more irregularly-created data, and so I'm not going to go into details how this is done. What I will, however, point out, is that when you calculate Variograms in reality, what we find is that, along the vertical direction, which is the Variogram shown at the top, here, we get a nice Variogram with a nice structure, we see a Range, we see a Nugget, we see a Sill, but due to the sparsity of wells, often when we have few wells, say, only ten or twenty or thirty wells, then the Variogram in the horizontal direction, becomes very difficult to calculate simply because we do not have enough observations. Therefore, in summary, this first really short overview lecture on Variograms, I would like to summarize that Variograms are a means of expressing spatial continuity, variability, of a reservoir property. That, in reality, we expect that most reservoir properties exhibit spatial anisotropy. This is certainly true if comparing the vertical direction with the horizontal direction, and that often the reality of variogram calculation from well data, is that horizontal variograms turn out to be very noisy. Which does not necessarily mean that there is no spatial correlation, it is just that we do not have enough, or sufficient, data to calculate meaningful variograms.