Introduction

Calculating the volumes of material and aggregate in mines is critical for the general operations of the mine. These volumes can be ore, overburden, tailings, waste, etc. Depending on the composition, these volumes are generally stored in a pile, pit, or pond. There are many ways to determine how much aggregate is being placed, but one of the easiest ways is through the use of passive geophysical methods. These range from the aerial methods of lidar and photogrammetry, to the water-based method of bathymetry. Many times, a mine might require a survey completed many times in the same location to determine the change of material over time. This article will go over ways in which this data is used or interpreted. We will also look at a delta matrix, which is a useful method of calculating volumes deposited using survey data.

Simulating the Depth Matrix

I won't explain Lidar, Photogrammetry or Bathymetry, but each of these methods can be used to determine volumes of material in a mining site. For this article, we will focus on bathymetry, but the same principles can be applied to both Lidar, or Photogrammetry. Below you can see some comparisons of a tailings pond in a northern Canadian mine from 1985 to the present day. This shows change in lake size as the mine matures, it gets substantially bigger, and is filled up with fine tailings and bitumen. I wrote some Python code to generate a simulated depth matrix for the pond at various times throughout the years. The resolution is very coarse, but it serves the purpose to explain the principles of a depth matrix. As I'm sure you've guessed, the colour gradient indicates the depth of the pond, red is shallow, and blue is deep.

1985

This is the earliest photo before the mine is occupying this area, the pond you can see here is very small.

2010

This picture was taken almost thirty years later, the mine has had substantial operations in the southwest, and the lake has physically shifted locations, although, there could be a datum error in the original photograph.

2017

This was taken seven years from the previous photograph. The mine has turned the small lake into a tailings pond. Also, note the beginnings of a small pond to the west.

2018

A year later and the tailings pond has grown considerably.

2021

This is the most recent image of the pond in summer, a side note is the immense difference in the pond to the west from the previous year.

2021

Finally, this is a theoretical change in pond bottom topography in the 2021 model. If you look closely, you will see that there are two locations where the material has been deposited as indicated by the shallower readings.

Comparing Moments in Time

A delta matrix is what happens when we take measurements of a pile or pond before and after a change happens to it, and negate the matching data within the models. This gives us a precise model of exactly how the topography has changed, and is hugely important for ore piles as it can help us estimate processing rates. It is equally as important for tailings ponds as it can show settling, or dispersion patterns of the tailings post-deposition.

Delta Matrix

For this article, I created two models of the same pond, with a notable difference in the topography. I have simulated how tailings may have settled after a deposition program. When we examine these matrixes on the left, there is a clear difference of shape, but it may be hard to see. We can very easily show this with some simple linear algebra. Fortunately, I saved you the trouble and did it in Python, we subtracted the matrices from each other to leave a precise volume representation of the new deposit. This is seen in the delta matrix on the right, which we used to calculate an exact volume, as well as a visual representation.

Conclusion

Volumes are very important to miners, but they tend to look at it from a layman's perspective. The best we can help is to show them precisely how much material is being delivered or piped to a location. They know this, and they are starting to take advantage of modern surveying techniques. The delta matrix is a very powerful tool for this purpose, and should always be considered when determining volumes in mining operations.