Additive and Non-Additive Variables II: Examples of non-additive geotechnical and metallurgical variables

Following our series of posts on additive and non-additive variables, today we will discuss the case of geotechnical and metallurgical ones. Remember that you can find the basic definitions and properties in our previous post.

Example of geomechanical variables

In the following examples, the intention is not to address the estimation of non-additive variables, but rather to review basic elements that allow for the establishment of a coherent conceptual model and an estimation methodology that adequately addresses the problem.

Prior to defining geomechanical variables, it is appropriate to review the concepts of intact rock, discontinuity, and rock mass. Rocks are characterized based on the properties of the material that constitutes them, such as their rheological behaviour (deformation of materials), their density, porosity, etc.

Intact rock corresponds exclusively to the material located between discontinuities, whereas the rock mass is the material that contains the discontinuities.

To describe a zone, it is necessary to use indicators that account for both the properties of the intact rock and the rock mass. Figure 1 presents the variables of a rock mass that are contained in the variable called Rock Mass Rating (RMR) developed by Z.T. Bieniawski between 1972 and 1973, subsequently updated.

Additive variables in geotechnics:

Among the variables used in geotechnics, such as moisture content (%), soil or rock density (g/cm³), contaminant grades (ppm, %), and stratum thicknesses are variables that can be averaged linearly between samples. According to what has been reviewed, in the density example shown in the previous blog, whose mixture estimated from the relative weights of its volumes adequately represents the expected density (performed in the laboratory).

Non-additive variables in geotechnics:

Among the non-additive variables in geotechnics are shear strength (which depends on cohesion and friction angle), deformation modulus (non-linear with applied stresses), and permeability coefficient (logarithmically related to pore size). These are variables that require transformation or special modelling, such as using logarithmic scales, or that are not estimated directly, but rather are estimated from their additive components. Other examples of this type of variable are: FF (Fracture Frequency), RQD (Rock Quality Designation), UCS (Uniaxial Compressive Strength), RMR (Rock Mass Rating), MRMR (Mining Rock Mass Rating), Q (Rock Tunnelling Quality Index), and GSI (Geological Strength Index).

Table 1 shows the composition of the RMR variable, which corresponds to the sum of partial variables that allow for the calculation of the RMR variable; to do this, the valuation of each coloured box is added. For example, if the RQD of the sample is 65%, the assigned valuation is 13; this calculation is performed for each of the variables that compose the RMR. If we observe the valuation assigned to each section of the RQD variable, it is not related in a linear form with respect to the measured values of this variable; this has the consequence that the RMR variable is "non-additive" and non-linear, so it cannot be estimated directly. The recommendation is to estimate each variable separately and subsequently calculate its valuation. It should be considered that variables such as RQD, FF, and intact rock strength are directional variables, so they are not additive, but they can be estimated under certain conditions.

Directional variables:

When we speak of directional variables, we must understand that the spatial orientation in which the variable is measured will condition the value of the sample, that is, it depends on the direction in which it is measured, such as FF, RQD, and compressive strength.

In Figure 2, the red and blue lines represent one-metre-long supports; it is easy to verify that, in the vertical direction (blue colour), FF = 9 [fractures/m]; whereas for the inclined borehole (red colour), FF = 2 [fractures/m]. This simple example shows the enormous directional dependence that the variable can have.

The same occurs with other geotechnical variables such as RQD. In the case of RMR, it depends, among other things, on the spacing between discontinuities and on RQD, both directional variables. To estimate these variables, we can establish that the value used will be that of the lowest quality so that the values to be estimated represent the worst condition, and that this condition is reflected in the RMR value; it also helps to establish domains with low data variance. We must consider that rocks with greater plasticity, such as tuffs, or those that present high clay content due to alteration, will present fewer structures than rocks subjected to the same stress, but more rigid, such as unaltered basalts. In summary, establishing geological domains helps to address the estimation of non-additive variables.

Another case is the intact rock strength variable; when measuring, for example, the UCS (Uniaxial Compressive Strength) variable in a metamorphic rock (slate) as shown in Figure 3, due to being measured in different directions, its values will be different, because measurements made parallel to the maximum deformation axis are lower than those made perpendicular to them. This variable presents directional variations that are related to the deformation ellipsoids in metamorphic rocks, but if the measurements are made in intrusive rocks, we can assume isotropic behaviour, in addition to assuming additivity in the different lithological domains; therefore, we can perform estimations of these variables in each domain. The geological context allows for simplification of the problem with this type of variable.

Another important property of geotechnical variables is the correlation that exists between them. On one hand, these correlations present in the data must also be observed in the estimated model and can be used as one of the validations of the estimations performed (see Figure 4 and Figure 5). On the other hand, these same variable relationships can be used in the estimation, for example, the residuals method, which allows for both the preservation of the correlation between variables and the conservation of their variability.

Example of Metallurgical Variables

Now we will analyse the response variable Work Index (Wi), defined as the work index that requires the necessary power to grind a material from a theoretically infinite size to one such that 80% passes 100 microns, thus allowing for a good estimation of the energy necessary for grinding (kWh/tc). To calculate the Wi index, experimental data must be obtained under strict operating conditions.

Calculation to obtain Wi for ball mill where:

P80 [µm] is the sieve size that allows 80% of the final product to pass.

F80 [µm] is the sieve size that allows 80% of the feed material to pass.

Gbp [g/rev] refers to grams per revolution; it corresponds to the grindability index of the ball mill and is a response variable from a test.

p1 [µm] corresponds to the opening of the cut mesh used to close the circuit (100% passing size of the product).

Yan & Eaton (2003) studied the behaviour of the Bond index in physical mixtures of high-competence (Wi = 14.0 kWh/t) and low-competence (Wi = 6.0 kWh/t) ore. They considered that the mixture should correspond to the arithmetic average of the input values (Wi = 10.0 kWh/t). However, the physical mixture tested in the laboratory yielded a value of 12.23 kWh/t (Figure 6).

This difference is due to the definition of the variable; being a response function (test) to a treatment (kWh/t), it does not consider internal variables of the samples or of the system, so it does not respond to all mixture conditions, such as the one indicated above where the soft material acts as a cushion for the more resistant material, remaining longer in the mill to reach the desired size.

One way to address this type of variable is to group the data into domains with low dispersion, so that mixing of materials of very different competencies is avoided. Since the material will be mixed in the plant anyway, the different mixture proportions can be tested and values assigned from a regression or other method to blocks that present mixture ranges.

From this example, the need to know the behaviour of response variables when mixing samples with different characteristics (tests) becomes apparent. In this example, the formula does not indicate clay content as a variable (which could modify the result), and it is precisely this mineral that under certain conditions generates an unexpected test response. Therefore, it is important to know the test and the deposit to review which conditions are modifiable and test them.

In the next blog, the variable Cu concentration in % will be presented. Its properties and characteristics will be reviewed, so that we can understand this type of variable and what considerations we should take into account when estimating.