Thinking in circles: Visualization of depth registered data sets for intuitive multiscale reservoir description

From numerical simulations to the interpretation of large data sets, we know how much relies on visualization. Whether we try to communicate information or discover hidden patterns in data, there is nothing like a good visualization to capture the essence of a subject.

Geomaterials are complex, they are 'multiscale'. In fact, they are just like any physical object. And despite a relatively modest R&D spending, the Geoscience industry is confronted daily with one of the toughest problems there is: navigating just about 12 orders of magnitude in length scale.

In his 1902 essay entitled ‘Science and Hypothesis’, Henri Poincaré wrote this: “No doubt, if our means of investigation became more and more penetrating, we should discover the simple beneath the complex, and then the complex from the simple, and then again the simple beneath the complex, and so on, without ever being able to predict what the last term will be. We must stop somewhere, and for science to be possible we must stop where we have found simplicity. That is the only ground on which we can erect the edifice of our generalizations.”

So which scales are appropriate for practicing Geoscience? Is there such a thing as a representative elementary volume (REV)? Think of how strain can be simultaneously expressed at the kilometer and at the micron scale. Are our brains even wired for comprehending multiscale objects?

A few months ago, I needed to gather some ideas on the topic of multiscale petrophysics. Though the concept itself might mean different things to different people, one of the key issues, I thought, was visualization. How does one even look at several scales, let alone attempting to make sense of the whole thing? Let’s try to offer an answer with a little ‘circular thinking’.

Color wheel and mean of a circular quantity

In the following, I will use the result of a rock typing exercise based on publicly available slabbed core photographs of a portion of the Green River Formation (more info here). This result is used for illustration purposes only. Rock types can be derived from any combination of variables (through e.g. data clustering) and greatly simplify the picture of changes in rock characteristics with depth. Rock types are often identified by numbers, which can be turned into colors when visualized spatially. However, even though they result from the integration of a lot of information, they still suffer from being ‘trapped’ into the spatial resolution they are obtained at, and after all these efforts, one might still not be able to see the big picture. Hence this question: How does one upscale rock types?

The simplest way of predicting what data might look like if acquired at larger scales is to compute a running average (i.e. smoothing). This is what would be done for instance to compare high resolution continuous core data with well logs. With rock types, however, the task is a bit less straightforward. As they are not actual measurements, one needs to question the meaning of averaging in this case.

Let’s assume that we have identified 10 rock types in a data set that has a spatial resolution of 1 inch, and we wish to look at them at a one-foot spacing. We then would want to compute a running average with a window size of 12 along the profile. But what to make of the fact that rock types 1 and 9, 2 and 8, 3 and 7, 4 and 6 will all give the same average of 5? Chances are that our ability to distinguish various combinations of rock types at larger scales will be somewhat compromised. The use of colors offers a path to addressing this issue.

The figure below on the left shows a spectrum color wheel which can be used to attribute colors to rock types by equally distributing them around the circle. Suppose we want to propose a color for the rock type resulting from the combination of rock types A and B. The first way of achieving this is to ignore the circular space and compute the simple average of the values associated with the rock types, which will lead to the black line (linear mean). From a visualization standpoint, however, it makes more sense to average them according to their closest distance around the color circle, which leads to the grey line (circular mean). By using the circular mean, we avoid converging to a unique value too quickly while allowing more intermediate color combinations among rock types.

Now, that doesn’t take care of everything, and one might want to allow for more unique combinations and hence open another dimension. This can be done, for instance, by using the surface of the RGB cube (right hand side of the figure), which roughly equates to freeing the saturation component in an HSV color space.

Let’s see what happens to our column of rock types (5500 values at 1-inch spacing) when they are averaged with a very large sliding window of 24ft using the linear, circular, and ‘RGB cube’ means.

The figure on the right shows that the use of the linear mean clearly favors about a half of the color wheel and fails to extract a number of intervals that the naked eye identifies when studying the initial column. The circular mean provides a very different picture which better retains the dominant tones and is more consistent with ‘natural’ color mixing. The ‘RGB cube’ mean brings out the saturation component with moderate (yet present) differences compared to the circular mean.

The wrap-up

We can now pick a color mixing rule and run our rock type upscaling on the initial column with a varying averaging window size reflecting all the possible scales of observation. This typically results in a triangle-shaped image ending with a single value computed in the middle of the column and comprising all the data. From experience, those triangles are a little hard to interpret. So what if we could use a compact representation that can accommodate any range of depth and sliding window length? The figure below represents a section of the upscaling triangle (using the circular mean) wrapped around itself. The depth increases along the contour of the disk, and the radius carries the length of the averaging window (i.e. the degree of upscaling). Some major markers from that section of the Green River Formation are added for reference. At the largest radius, we find our initial rock types. As we travel towards the center, we can see how they all combine into new ‘units’. We can see for instance how much of the inch-scale variability might be missed by the well logs, or what units might emerge beyond them. In this case, I picked a rather large range of averaging window length, but the diagram can be easily set to better study the plug to well log scale transition for example.

That's all for now, hope you'll agree that there is something to be said about visualization when trying to think multiscale. Thank you for reading, and don't hesitate to comment below or get in touch.

Have a great week,

Laurent Louis, 30 September 2019.