Grain size distribution in-class exercise.

What is the purpose of this exercise? First this exercise is meant to make you think about grain size, a fundamental geologic variable, more complexly. A second purpose of this exercise is to teach you how to construct and analyze histograms, a fundamental first step in the analysis of a single variable (univariate analysis), which we will focus on next week. You will also learn about some basic univariate statistical descriptors (that is a mouthful). Finally, this exercise will give you more experience with the idea of approximation and the importance of assumptions in analysis.

Why is the grain size distribution of geologic materials important? Grain size distribution data is widely used in geology and engineering. Think of sand, silt and mud. The grain size of a particle is crucial in determining whether it is transported in a current or when it comes to rest as a current diminishes. This in turn is important for engineering purposes, such as considering what size rip-rap to put in place to armor a channel bank. Too small and the current takes it away.. Grain size distribution is also a fundamental determinant of intergrain permeability. Grain size is important in an array of chemical reactions because smaller grain sizes increase surface area and aggregate reactivity. Grain size distribution also carries information as to origin - consider the difference between plutonic and volcanic rocks. Example abound where grain size is crucial variable in geoscience and other endeavors. Can you think of some additional examples? Wentworth grain size scale from United States Geological Survey Open-File Report 2006-1195 . Web source: https://en.wikipedia.org/wiki/File:Wentworth-Grain-Size-Chart.pdf .

Background for exercise: A sphere is an elegant shape, one that has a radius, diameter and a distinct middle that is easily found (do you remember how from your high school geometry class?). A perfect sphere has no variation in its radius or diameter. Operationally, one typically measures the radius as the distance from the center to the periphery, and the diameter as the length of a line that passes through the center and goes from one side to the other side of the circle. However, an individual sand or cobble grain shows significant departures from a spherical geometry (see the image of a sand grain below), and this creates an object with variance in the diameter one would measure. The next better geometric approximation for a real grain shape might be an ellipsoid. An ellipsoid has 3 primary axes, each of which would be a diameter by our definition. They intersect at the center. There are also 'other' diameters that exist for other lines that pass through the middle. Real grains of course show departures from a perfect ellipsoid form. Lets define our variable of interest as the diameter of a single grain; i.e. the lengths of lines that pass from the center to the periphery.
Crucial to the concept of diameter is the identification of the grain center. It turns out that finding the center is not a trivial thing. One can take two perpendicular lines and move the intersection about the grain interior until the center point is equidistant from the margins as measured along a straight line. Then this operation can be repeated with two other perpendicular lines multiple times to hone in on the best center approximation. Another approach would simply be to compute the point where the average distance from that point to points on the grain boundary is a minimum. This point would define the center and hence the radii in a different way. This can be solved by brute computer force. By the end of this course you would be able to set this exercise up in Excel if you wanted to, but one might argue about if it is worth the time. However, many grains are close enough to spherical that the concept of grain size as measured by the grain diameter is still meaningful and useful. Experience indicates the approximation is useful. Diagram showing how in cross section a sphere has equal radii, an ellipse has unequal radii, but predictably so, and a real grain has unequal radii, but in a less predictable fashion. Thin section seen under microscope of medium grained sandstone. The sand grains are clear and the brown material in between the grains is carbonate cement.

Assumptions: In order to simplify we will consider a cross section of a quartz grain in thin section. We will assume that this cross section is representative of the grain and that it passes through the middle of the grain and that we can identify the center. Can these assumptions be challenged? It turns out that they can. Imagine an egg shaped grain. When it comes to rest statistically it will likely have its long axis horizontal, just like an egg on a counter top. So the direction of the cross section cut is important. Additionally, whether this cut went through the side or center of the grain is indeterminate since it was basically a random cut. By picking one of the larger grains in the thin section it is more likely that this section passes near the middle. It is not uncommon that assumptions are not perfectly met. It is also important to recognize and analyze one's assumptions. The level of precision needed is an important consideration here. Since we simply want to get some idea of how the diameter of a single grain tends to vary these assumptions are probably closely enough met. If it becomes important to be very precise about the grain diameter measure then you might want to proceed differently. Can you think of how? There is an interesting trade off. To get an increase in precision of 10% you may have to expend 100% more effort. Exercise:

• Select which of the three grains above you would like to do.
• Select your grain center by eye.
• Measure carefully the two diameters associated with two perpendicular lines you draw through the center.
• Bisect each quadrant (by eye) with a diameter, and measure the length of these 2 additional diameters. Bisect each angle left with a diameter and measure these (should be 4 more). Bisect once again to get 8 more measurements of diameters. Keep going until you have >50 diameter readings (enough to get a decent histogram shape).
• For each diameter measurement mark a box with an X above the bin interval the value falls within, on the provided graph paper or within the provided Excel sheet template. Note the bins are each .01 mm wide. The scale of the copies of the grain cross sections handed out in class will be so that 10 cm = 1 mm, or 100:1. Most bins will have more than one reading, and so you stack your x boxes as you have more than one radius that falls within a bin.
• Which bin interval has the most boxes? This is an estimate of the mode.
• What is the difference between your highest and lowest measurement? This is known as your range.
• What will be the difference in the shape or form of the histogram for a very spherical grain versus a very elliptical grain?
• Can you think the histogram expression other gain shapes might have?
• Remember, as always, to ask questions.

@Harmon D. Maher Jr., 2010.