DENSITY

The density is defined as the amount of mass per unit volume. When using the density of water to normalize density values, the current ASTM wording preference is for the term “relative density”: a former term was “specific gravity”.  The relative density, defined by the ratio of the true density to that of water, is then dimensionless. For brevity, the term density will be used hereinafter and will always stand for relative density. Since most rocks are mixtures of impure minerals, and the density of the minerals is affected by their state and history, the density within an individual rock will vary from point to point. Samples taken at different times and from different locations in a deposit will also vary. In the past, it was difficult or impossible to study such variations. The validated CT technique now makes this possible, as shown below.

A laboratory investigation on a sample of the granite rocks that were of similar type to the 12 test rocks reported the mineral phase composition using x-ray diffraction as shown in Table 2.

Table 2: Principal mineral phases found in the granite rock

The Handbook of Chemistry and Physics (2004) reports the following density ranges commonly found for these minerals, listed in Table 3.

Table 3: Densities of compounds commonly found in rocks

From the data in Tables 2 and 3, it can be seen that the density of the granite will lie in the range 2.6 to 3.0.  It may also be noted that the values are for “pure” minerals, which are rarely found in quarries.

The densities of the 12 test rocks were calculated by dividing the measured mass in air to the measured volume. The mass was measured to an accuracy of 0.1 mg, and the volume had similar accuracy, since the same balance was used. The value used for the density of water was probably only accurate to three figures, however, so that the measured densities were only accurate to about 0.1 %. The CT volume data was also used along with the measured mass of each rock. The results of these two procedures for the 12 test rocks are shown in Table 4. The percent differences are just due to the percent differences between the measured and calculated CT volumes, as can be seen by comparing to Table 1. The average measured densities for each subset set of six rocks (0.5 or 0.75) differ from each other by well less than one standard deviation, as would be expected since the entire set of 12 test rocks were taken from the same population. A few of the test rock densities lie further than one standard deviation from the mean, as would be expected for a (small) random sample.

On the other hand, it is interesting to note that the densities range from a low value of 2.49 to a high value of 2.97. The three significant figures reported are consistent with the accuracy of the measured densities. The high value is 19 % higher than the low value. It is important to remember that these were multi-phase rocks, with components having this kind of density range, so it is not surprising to see this kind of scatter. The Archimedean measurement treats each rock as a uniform solid. However, knowing the average density of each rock allows us to extract information from the CT data about the distribution of density within each rock.

Table 4. Average density of 12 granite rocks by Archimedes and CT

Recall that the output of a CT scan is a collection of 2-D gray scale images that slice through the sample. These image slices are “stacked” (computationally) to recreate the 3-D particle. The gray scale of each voxel is proportional to the density of the material occupying that point in the rock (Kak and Slaney, 2001). By comparing the measured densities vs. the average gray scale for each 3-D rock image, the distribution of density can be measured. It was found that the average density per rock was linear in average gray scale for each rock, as shown in Figure 3. The fitted lines are as follows, with AGS = average gray scale and AD = average density:

                      (5)

The values of R² were 0.91 and 0.97 for the 0.5 rocks and the 0.75 rocks, respectively.

Figure 3:  Average density of each rock plotted vs. the average gray scale of each rock.

It should be noted that the gray scale of CT scans can change based on instrument settings. This is the reason for the difference in y-intercepts between the 0.5 in and the 0.75 in data. However, the instrument settings were approximately the same for the two samples containing the six 0.75 in rocks. Therefore, the linear variation in Fig. 2 for the two sets of data is a real reflection of density differences. The linear offset between the lines for the larger and smaller rocks is probably due to a difference in instrument settings. These kinds of measurements could be better calibrated in the future by including an object of precisely known density in the rock sample. It is possible that a better match between measured densities and images could be made if one worked with the X-ray attenuation coefficients that come directly from the X-ray CT scan, before the reconstruction that results in the gray-scale images. This effort was beyond the scope of this research.

Equations (5) can be used to approximately convert the gray scale in each voxel to an average density. The results are plotted as histograms in Figs. 4 and 5 for each particle, showing how density is distributed within each rock. It should be mentioned that, in the CT scans, the edge elements of each rock have gray scales that include some of the wax matrix. Therefore, the histogram for each rock was only built out of voxels that were more than a few voxel lengths from the surface of the rock. Summing up the area under each histogram will still give unity, however, since the histograms are normalized to the actual volume sampled, which is somewhat less than the actual particle volume.

The information provided by CT may be illustrated by considering rock 0.5-2 in Fig. 4. The peaks in the histogram may be interpreted to indicate that there are at least two component materials within the rock with average densities of about 2.5 and 3.4. The granite is composed of hornblende (darker material in Fig. 1) and granite (lighter material in Fig. 1), so this result is in accord with Tables 2 and 3.  The estimated average density is about 2.8 and the standard deviation is about 0.3. It is interesting to note that while Fig. 4, for the 0.5 rocks, seems to indicate that each rock has at least two distinct phases, Fig. 5 shows that the 0.75 rocks can be considered significantly more uniform. But one should note that the 0.75 rocks could still have the two major phases, but with closer densities, so that their peaks overlap. In the remainder of this paper, we will treat each rock as having a uniform density equal to its average density.

Figure 4: Histograms of density for the six 0.5 rocks.

Figure 5: Histograms of density for the six 0.75 rocks.

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