Cognitive Classroom - Variograms

hello welcome back to the cognitive classroom today we're talking about very grams and I thought we would spend the time explaining exactly how a very room is acquired how he generated experimental data and what it really means for you as a geoscientist so let's start off quickly by explaining how we gather our experimental very ground data we really use a very gram in the modern sense we use it in in all three different directions so we're going to use a vertical diagram and we're going to also use a horizontal diagram so you can think of this one is a map against X&Y or latitude and longitude and this one here can be our depth direction when we're looking at a typical routine that we get on modern packages will often look at the vertical via gram is one well at the time so you may see a visualization it gives you something that looks like it like a curve like that for an individual well so that's something that we can use for a very ography and in the horizontal direction we'll have a fairly simple map usually we don't bother necessarily showing you all of the normal beauty of a of a geological map but just the individual well locations and we could imagine these is there's a few well names across through here ABCDE and so on now when we're chewing when we're calculating an experimental Vera Graham what we're going to try to do is understand if there's any mathematical structure in the spatial distribution of your datasets and that's really is primary purpose so what we want to do is have an analysis of this information and see do we see for example a particular way that the data set is varying in terms of its its random distributions so in the horizontal direction now let's say we want to try to gather evidence of is there a trend in the way that the randomness is occurring in a particular direction will of course take the orthogonal to that train later on as well so there will be two directions that we're going to acquire and we want to analyze to see do we see a spatial pattern in them in the vertical direction it's going to come down the wellbore and oftentimes you'll see some packages we'll give you the option of orientating the direction in the vertical usually I don't recommend worrying too much about this keep it vertical because each of the individual wells have got different dips as well so it can be slightly misleading their visualization so what we want to do is calculate the difference between our pairs and we're going to collate those as a function against its distance so a distance in this direction so here's our distance here yes and this is going to be our third major direction at this point in time so along this distance s we want to see do we observe any particular pattern in the way these data are being distributed you'll see that we don't observe any pairs in that particular straight line from this measurement of well a so in the real world what we do is we create a being tool to try to require measurements that we can say or are indicative of that direction but not perfectly in that direction we'll define a tolerance angle theta that comes through here and that should be less than 45 degrees because the other data belongs in the other direction and a lot of authors will actually recommend that you do less than 22.5 degrees so that you've really got confidence that it relates to that particular direction we often though you have a limit on the horizonte' on the on the lateral movement that we're willing to go because we don't want to borrow data that's been too far away even if it's within the tolerance angle this is the bandwidth of the of the data set and then finally we want to try to create a statistical set from all of this so we need to be able to make more than one pair land in a particular bin and so what we use is a thing called a lag and that is a radial representation within that distance through here and any any two pairs that occur within those legs will get recognized as as being equal so let's have a look at how we can go so we're going to represent this as these lags so here's our lag distance which is from here to here from those two radii and then we've got one two three four legs we could continue that onwards and that would make the range that we are investigating raise up or we could create finer degrees of beans as well we'll come back to choosing that beginning in a moment so let's have a look in the first direction in this inner space along this area here within that first leg we don't observe another pair but we do at the second leg through here so I start to create a column through here we have lag one two three four five and so on ie does not have a pair with in the first leg distance but it does have a pair at the second leg distance so I can make that calculation and then what we're going to do is calculate the semi variance semi variance is pretty simple to calculate it's going to be a minus B and we're actually going to take the square of that which is going to remove any negative sign and that's the semi variance between those two points and so we take that measurement for you now you can generate some other pairs for us you can generate ie to C and that's going to occur in the third leg so I can make this value here a minus C squared you'll notice it because we're doing - and then squaring it doesn't matter which direction it's going so the Varia gram only needs to be calculated in one direction the other direction is implicitly calculated here at C minus a squared and keep the same answer in this bin though we also get a 2d so we can take a minus D we can square that back to through there and at the fourth pin we have a minus e square that so there we go we've got from a we've been able to generate a total of four pairs in that particular direction so this this lag tool here that we've got is then reapplied at all of the locations across through here and as move that to say B we would notice that the 2d will turn into the I'm guessing about the second leg distance so we could say B minus the square that goes here and B minus C might be over here squared and so on and then that will probably fall into our into a third letter that falls into our this is B minus E and we can square that I'm not strictly very much sugar there but just mental arithmetic Euler's I'm trying to accumulate it so what are we doing we're starting to accumulate some semi variances at a whole bunch of locations we're going to apply that all the way across the field to this this object here this blue object that I've drawn does not actually have a spatial position it just has an orientation and a definition of its band of its lag areas and it's going to acquire it's going to operate all the way across the map and it's also going to be able to go through the well system so what we're going to do so through the vertical sections so just going to generate a whole bunch of values along through you we tend to observe when we do this in the horizontal directions that we might get next to nothing occurring the first leg we might get something in the second leg you might find that you start getting a lot of different calculations occurring that then tend to pop up at fairly regular leg distances so you'll see spikes like this that's your well spacing even on the offshore but certainly on shore we tend to find that we've drilled our wells at roughly the same spacing because we obviously got similar drainage points between the wells and so we often have these spikes once this has been inquired entirely we're going to do an equation we're going to take the average of all of these results here so we're going to calculate the average variance or semi variance here and we can do that because you can average variances and this is the reason why we use variants not standard deviation in this calculation we can do that averaging we're also going to record the number of pairs that count and the reason why we record the number of pairs is that we want to recognize what students statistically significant because you'll see in particular orientation I really don't have a statistical sample here the average is at one number if I even achieve that I'm not really going to be very useful so commonly in the horizontal direction you'll all be recognizing me end up with horrible numbers appears very small numbers be gaps and really large areas that we are beating together within the typical oil field so in the horizontal space we very rarely get a good quality very go the verticals quite different you will see some programs will provide you with the same being tool you'll even be able to rotate its orientation against the wellbore and if you select the well name you might notice that it will rotate if you select it different well it might suddenly rotated in another direction so you can get quite confusing you might start trying to orientate this thing in order to match one of the worlds but be aware that it's just a often unmarked direction actually is the way you're looking so I recommend just keeping it vertically and in essence for greeting because we're working within I J columns you really don't really need to worry about this too much couple of notes on the being sizes I don't worry about the log sampling people often tell me that my logs have a resolution or two samples per foot the reality of course is you have often wealth log sampling or two samples per foot and but the resolution of the measurement can be very different so resistivity can be much coarser if it's on the deep resistivity same with the sonic you might have a much finer quality spectral gamma-ray resolution but the sampling rate is not its resolution however what we're trying to achieve through here is we're trying to pick up the major changes that are occurring within the system so we want the bins to be representing the frequency of those cyclicity that are occurring within that and we're trying to avoid any aliasing that can develop by having in Korea being exampling the other thing is that we do the transformation is very grand transformation is actually usually done on the normalized data set so we're viewing it like this what we would normally have done we would have come through here and would have known this is fine costing upwards in this case if this was a ferocity and so a an experienced interpreter might remove but patent and say that well that that's not random so we can take that out of the dataset and then instead of viewing the raw data we will be viewing that their transformed data but even after we removed that pattern we're going to do a another transformation and we're going to do that for the purposes of making our mathematical trickery simply what we're going to do is what we call a z-score transformation or a z-score for the Americans we're going to take to calculate the z-score value for any observation we are going to take its value X we'll subtract the mean of the data set so and then we're going to divide it by its standard deviation so why do we do this if I run it through that transformation or if I use a more elegant say quantile quantile transformation what's going to happen in the end because I'm basically dividing itself in standard deviation by itself my standard deviation for the z-score in the design spacing is going to always equal 1 why is that useful well we refer to standard deviation so we refer to variance as that squared because variance is simply the square of the standard deviation so this value that we've calculated through here the average over here that variance is actually also equal to 1 after we're transformed but through you now that's the reason why you're very round goes to a still of one simply because you've transformed it into this space right the interesting point to note of course I've taken out this trend through here so if I take away this trend and I remove that from the data set I now no longer have the original observation it's going to be a fire data if this was porosity given my death trend so it's actually a transport it's also going to have a residual that's plus or minus around zero because these data would be positive these data will be negative relative to that line and so we'll find that minus its mean is going to be minus zero we're going to end up with a distribution that has a mean of zero and a standard deviation and a variance of 1 now we like variants for doing this computation because we are allowed to collate them together and average them we can't do that with standard deviations but we can transform it back once we've got close you take the square root of your variance through here and you're having the standard deviation the other thing is if we don't do it on the on the z-score transform data set if you actually do it on its natural coordinates then the the SIL is no longer a variance of 1 it's a standard deviation of your data set which is a nice little trick for me personally anyway helps me understand what my very gram is doing so it's nor a very grand there is our typical Vera Graham display we have an access through here with a distance yes that was this direction through here on this axis we usually show gamma which is the semi variance it could also be denoted as Sigma squared or if you do it in the way I like to do it you could actually take the square root of that and just show it as n deviation if we were working in the traditional format you probably know and both that will go to a SIL of one if you've done it in the way I like to show it you know it's natural for this will be the standard deviation of your data set so let's take our data set let's just say we've got this input data set here porosity and problems I think and we've got a data set that looks like this we have a mean here that's what the meeting is in this particular case we've got to standard deviation through here and we kind of recognize this particular data set because we look at that day is it all the time so the standard deviation that I see through here let's say that's about for porosity units that's familiar to me and that would be what my still is going to trend towards in the end we'll come back to how we get to get there in a moment right so we've got our plot up through here there's one access that's missing now on our plot and it's usually on the right side over through here and this will be the count of the tears and we show that because some day that's more significant than others so out of the first bin will often see a histogram looks like this second one might be a little bit better the third one might be huh denied white always seems to be in a third one seems to have some information I might go back into a low again as they get to my world spacing and then of distance I might see another spike up through here and then perhaps as I go beyond that distance the the the subtle differences in between South average in the heart and I might get a slightly better up here so we show that histogram why why have you showed me this do well when we plot these average values through here we are going to see how much of these we need to care about so let's say we've got something that looks like this right what does that mean so as an experienced interpreter I will come to this and I will first off throw a degree of skepticism across across this model I'm going to look at this and recognize I've got very little data here and I've also got less data here than I do on either side of it I expect my variance to get up to my sill so if you're in the standard space you're expecting it to go to one if it gets to a stationary state or in this case in standard deviation I'm expecting it to achieve its standard deviation that just means that at a great distance an observation point tells me nothing more than I'm inside my own dataset that's it so I've got no structural information of a nice spatial correlation but close by we're inferring hopefully that the observation is going to tell me something about its adjacent point and if it does that that's that's a powerful piece of information for us to use now we can't just use the experimental very bad day in a real bottle application because when we do a simulation we're going to need a value for this variation at every point for us so we're going to have to infer this with a model and we're going to use any continuous function actually to describe this it is perfectly legitimate for me to use any continuous function there is no reason for it to be exponential Gaussian and spherical those just happen to be three of the common ones people might often look at this and say well I can see some cyclicity could I honor that well actually we can you can honor some cyclicity in very Aquafina you can do that using a hole very ground so a hole very ground has some some cycles to it not saying that's a particularly good match and we often refer to it as we often use a dampened hole effect very round if we're going to use that so it's a sinusoidal function and the degree of sinusoidal behavior is is depreciating over time that's the dampening of the pinhole effects bury ground what does that mean to me well it means that at some distance as I get further away I start to see a similar behavior to this observation again that's actually quite a reasonably common situation in geology that's when we've got some reached regional scale or some some some larger scale features like some large June system perhaps or or some Paris events behaviors that we're occurring in the vertical direction so we get progressively more dissimilar and then it starts to become more similar to that observational point of view the dampening is because over distance those the thicknesses of those cycles isn't perfect and so the sampling rate starts to add positive and negative relationships together and interventions sorts itself out the whole effective area gram though isn't commonly used partly because there's not available a lot of the packages some people find it quite difficult to understand you can put yourself back into understanding talking about what it's referring to say that this becomes this tells me something about so the the frequency of the wavelength of my events in the geology Assoc not too hot to be honest but we often end up using something a little bit simpler now how would I come without waiting it well I'm going to look at this observation here and say there's not a lot of statistical information so I can actually do all that value if I want to or at least not honor it it's in reality it's got a lot of potential error associated with it likewise with this one isn't there still some error with that this piece here starts to have less error because I'm saying I've got calling out more statistically significant so realistically we could think about our very rounds to something like this we need to make a model that fits through those lines through those error bars and so I could come up and do something that ties through those one but let's say that's a roughly very cool example so we've got a model through here maybe exponential and then we're going to plot through them so one of the elements of the model of matter we need to know the model name so the model that we're fitting here and let's say this is a solid spherical say this is a spherical model so that tells the computer immediately how that function is going to get defined I need to know it's nugget I hope it need to know its range and actually don't need to know it's still the SIL is actually gonna get used in most situations it just comes from the data set that's not actually taken from the very ground execution right so couple points nuggets if you don't recognize it it comes from from the golden history where all this technology was developed in South Africa back in the day and it reflects that instantaneous change that can occur between two data's two data points and that's at the of the immediate change point now a lot of companies and including the one that I first learned this methodology with how they have a rule of thumb that say that we must always have a low nugget in fact one of the companies I work for had a mandate that nugget was always zero and why a long time to find out why we did that and this is clunk obvious the reality is not to do with geology there are plenty of publications out there that make that claim that it's to do with geology because the two points between between observations are often similar but that's not the real truth the reality is is because we do a simulation simplification when we take two cells in into a dynamic flow simulation model we are going to make a calculation to determine how many how much fluid flows from one side to another given a certain amount of time and we make that time step quite large because we're trying to save on computational cost beauty computational cost we want to have a large time stepping between the between these wells in order to minimizing our time it's going to take to compute and the problem is if this cell here wants to try to send 100 barrels across to that cell in that one time step but this one can only receive 50 barrels we're going to have to find somewhere else to send those other 50 barrels or I'm going to have to increment on my time step so this becomes a convergence problem in closing elation that is entirely because of a computational simplification there's nothing to do with geology and so anyone tells you that the Nuggets is always lower in geology because the things are similar they need to recognize that there is also a component in the way we have that rule of thumb that relates back to this flow simulation part of the problem with this assumption of Loyd nugget is because most people doing is in score transformation we're dividing it by the standard deviation now if I take out a trend out the data set my original standard deviation is for ferocity units take that trend out I might be able to reduce that cross that standard deviation down to 0.02 you know that I'm really not doing nothing to change the instantaneous variation between or very little the main change is actually over a greater distance so the Nugget is referring to the changes are carried over a very short distance so it shouldn't have changed but because I'm now dividing it instead of dividing it by 0.04 I'm dividing it by a smaller number my nugget will seem to climb and my sill is already being forced back to a value of 1 so if I take the trend out it would mean no it would appear to climb and so it looks like we're breaking this rule of thumb but the actual reality is not gonna break it so it's okay if that nugget climbs up if you're working in Zen school space if you take in your trend zone that's that's just what's going to happen because your distribution said it's gonna get progressively narrower so there's just a rough summary of what we do in very ography when we come to applying the very rare in the property simulation space what we're going to be using is a representation to say that if I have a grid cell my model through here or a quick model up and we're saying okay I've got some observation points I've got some some unknown points if I'm trying to make a guess if I've got an observation in this cells or a well sample through here if I come to guess what's in the cell next to it okay I can come along to this and say that well that observation is within distance so I need to be within this much variation of that observation so that's how we get to preserve some of the spatial structure once we get beyond the range we come to make a question mark over through here that's not going to give us any information so that's how we're going to use the very ground later on there are lots of different methodologies both for continuous and Static so continuous and discrete properties that we can use very plausible but I hope you can see that this is just a potential way of describing the mathematical structure that might exist inside your data sets it's one that we use in the industry because it's nice and easy to acquire particularly in systems with pretty poor amounts of information but it's not the best it's not the only and it's just one form of a branch of spatial mathematics that is used has become popular in our industry in part to do with some reasons for flow simulation calculations is useful in that and it also helps us try to describe the structures in our data but in my personal experience I do prefer to focus on trying to understand the geological structures and link that back to geology rather than just focusing on the very ground but nonetheless this is how you calculate a very rare I hope you found this useful if you've got any questions and you'd like to to bring up some other points please make some comments below but otherwise I'll see you back here at the cognitive classroom next time

Channel: Cognitive Geology

Views: 1,947

Keywords: cognitive geology, geology, variograms, geomodelling, geomodeling, geomodels, geostats, geostatistics, petrel, sgs

Id: cBNnhsvRAuM

Length: 26min 25sec (1585 seconds)

Published: Fri Jul 05 2019