By Jack Caldwell - Mining Engineer - Robertson GeoConsultants

Let me discuss modelling, calculating, and thinking as part of the decision making process. In particular, I write of those essential modern aids to judgment, namely conceptual models and the running of computer codes.

Formulation of a conceptual model and the use of the model as the basis of numerical calculations is a process that is equally applicable in most analyses leading to a judgment in the engineering of mine waste disposal facilities. The processes I describe here applies as well to slope stability analyses, as cover performance modelling, as groundwater modelling. As this latter type of modelling is the most prevalent and the most difficult of the modelling challenges of mine waste disposal management, I talk only of groundwater modelling. But you can apply the principle to any analysis leading to a decision.

Groundwater Modelling in Waste Disposal Engineering

First a word on groundwater modelling as applicable to mine waste disposal facility management. Water seeps into, through, and from tailings, waste rock, and heap leach facilities. Often the seepage incorporates contaminants. The issue is most often: will the contaminate seepage impact local and region groundwater resources or bodies of clean surface water?

Site Characterization

As in all engineering and decision making, the first step is to characterize the site and the proposed or existing waste disposal facilities. I have written extensively about site and waste facility characterization elsewhere, so no more need be said here than that at the least you should have some information about:

  • Geology
  • Soil and rock hydraulic properties
  • Surface water
  • Groundwater
  • The layout and properties of waste disposal facilities.

Model Types

Two types of models applicable to mine waste engineering are:

  • Baseline Modelling or Existing Conditions Simulation. The primary purpose of such models is to see if you can get a numerical model that, once calibrated, simulates to an acceptable level of accuracy, the existing conditions.
  • Predictive Modelling or Proposed Engineering Changes Simulation. Once the numerical model is calibrated and you are sure it is a reasonable representation of reality, then you proceed to change boundary conditions. For example you place a seeping tailings impoundment on top of the upper soil layers. Or you put in an open pit next to which is a waste rock dump. With these proposed changes in place in the model, you proceed to run the model to quantify the changes that the new facilities will cause. This is sometimes referred to as stressing the model.

Conceptual Models

Now proceed to formulate a conceptual model of baseline conditions and the conditions that need to be predictivelymodelled. Conceptual models are key aspects of mine modelling, analysis, and decision making. With our limited thinking ability, we need concepts, or models, of reality before we can fully comprehend, understand, and mentally manipulate reality.

Thus arise the key questions:

  • What is a conceptual model?
  • How do we formulate a conceptual model?

The answers depend on whom you ask. There are different answers if the responder is a mining engineer, a civil engineer, or a groundwater expert. The mining engineer needs a conceptual model of the ore body so that he may formulate ways to mine the ore. The civil engineer, charged with designing, operating, and closing the mine’s waste disposal facilities, needs conceptual models to evaluate slope stability, erosion, seepage, and impact on the environment. The groundwater specialist needs a conceptual model so that they may do those groundwater analyses that predict impact or absence of impact on precious water resources at and around the mine.

Like any model, a conceptual model is a pale reflection of reality. A model is an attempt to capture the essential features of a real situation in words, figures, and numbers. Being a model, it may cannot and need not replicate the infinite variety and complexity of reality. It is a model after all: a simplification of essential aspects of reality. A toy that helps us think, analyze, judge, and decide. For if a conceptual model has any use, it is to be the basis of numerical calculations and analyses.

If you need to calculate the factor of safety of the perimeter slope of a tailings facility, you need a conceptual model of the real situation so that you can put pen to paper, or turn on and run the computer code, and do the numerical calculations that result in a number that is the ratio of resisting to disturbing forces.

If you need to quantify the change in the concentration of a contaminant emanating from the waste rock dump, seeping through attenuativeclays, and emerging into a creek where the salmon spawn, you need a conceptual model before you undertake simplistic calculations using high-school math or sophisticated computer codes.

So what is a conceptual model? Simply stated, it is a description in words, figures, and numbers of reality. Perforce, it is a simplified representation of reality. It is a replication of reality that our limited, human mind can conceive and work with. It is the basis of a further step in science and engineering: namely the process of using equations and numerical analyses to quantify the probable performance of reality. And maybe study the range of possible and statistically probable performances of the real thing.

In the mining context, a conceptual model must incorporate at least the following:

The domain of the mining facility under consideration. Examples include: the distribution of rich ore, the open pit slope; the tailings impoundment and seepage therefrom; and the groundwater hydraulic regime and the passage of contaminants from the waste facility to receptors.

The Boundary Conditions. Examples include: the location of strata that contain valuable ore; pools on the tailings facility that generally are kept at an established elevation; and the watershed divides away from which all groundwater seeps.

The Geology and Its Strata. Examples include: the soil or rock layers in which the valuable gold, platinum, oil, or iron ore resides; the aquifers and aquitards that allow and limit flow of pollution from waste facilities; and the faults that may limit groundwater seepage and force water to the surface to springs, creeks, and rivers.

The Properties of the Principal Strata. Examples include: the ounces of gold per ton of rock; the pollution concentrations in waste facility waters; and the hydraulic conductivity of the soils & rocks through which seepage will occur.

To formulate your conceptual model, first gather all the data you can about the field situation you need to model and make decisions about. Document these data in words, figures, and tables. That is your conceptual model.

Get colleagues and peer reviewers to examine your conceptual model. For your concepts may be faulty; your understanding of reality limited; you intellect too rusty to jump to elegant models; and your prejudices too ingrained to admit your preferences and biases.

Once you have an agreed conceptual model, you may proceed to the next step: definition and selection of numerical models and analytical methods to put numbers to the performance of the model and hence a pale numerical replication of reality.

Numerical Models

Conceptual models are well and good in their own right. If the physical situation is simple and the details sparse, the conceptual model may be all you need to think and decide.

In most situations, however, you will have to use the conceptual model as the basis of numerical models. The two basic categories of analytical models are:

  • Equations, usually closed-form equations that you can solve by pumping in parameters and solving by means of simple mathematical techniques.
  • Computer Codes which are usually numerical models involving the use of finite difference or finite element methods.

Ido not delay here to list the many equations or computer codes you may use to analyse the impact of a mine waste disposal facility on groundwater resources. They are many, complex, and mostly readily available in the literature or from software sellers.

A word of caution: it is not necessary to use only one equation or computer code to numerically analyse your conceptual model. Depending on the situation, the performance of some parts of the conceptual model may be studied with simple equations, such as Darcy’s law which tells you how much water will flow through soil or rock as a result of energy differences in the system. The performance of some parts of the conceptual model may be studied by sketching simple two dimensional flow nets. Only when unavoidable, should you fall back on a numerical computer code in order to study the performance of the most complex parts of the conceptual model.

The point is: be parsimonious and selective in selecting the analytical methods you will use to numerically quantify the performance of your conceptual model. The analyses take a lot of effort. Before you begin make sure the results are likely to be worth the effort.

Calibration

Once you have selected one or a number of suitable equations and/or computer codes, you may proceed to calibrate the model. Here are some guidelines for calibration. Regardless of whatI say here, recall that calibration takes skill and experience. Professional judgment is essential. There are no absolutes. Calibration is an art.

Calibration may be defined as the iterative process undertaken to get good correlation between field measurements and the values calculated with your equations or computer codes. To calibrate an equation or code, you run the equation or code with a range of different parameters for key system characteristics until the fit between measured and calculated is considered good enough for practical engineering decision making.

The usual things compared in a standard calibration include:

  • Water levels, as measured in the field with piezometers
  • Flow rates at discharge points such as springs
  • Contaminant concentrations as measured in water samples.

You should know the field value of thesefactors before undertaking model calibration. Watch out for errors in field measurements. You will seldom be able to calibrate a model to faulty field data.

Then select input parameters to the equation or code and vary them intelligently. Common parameters that are varied in calibration include:

  • Soil and rock permeability or storativity if you are studying a transient condition.
  • Input fluxes resulting from infiltration to the surface from rain, snow-melt, or seeping waste disposal facilities.
  • Strata attenuation or dispersivity if you are studying contaminant transport.
  • The number and geometric properties of key soil and rock strata.

Use judgment in deciding which of these parameters to vary first. Do not vary them all at once, for the result will be amuddle. Vary the parameters only within reason—consider just how much the value of the parameter may vary in practice. Do not wonder beyond the bounds of reason and physically reality.

A word of caution: be on the look-out for the non-uniqueness plague. This is an ever presentdanger in all modelling and particularly in calibration. The problem arises from the fact that in theory there is no single unique set of parameters that will yield a give answer. Keep in mind that groundwater flow is ultimately governed by a Laplace differential equation. And that equation allows an infinite number of possible coefficients to yield the same answer. The only way to guard against calibrating your model to the wrong set of parameters, is to be on your guard and use common sense. If the set of parameters that yields answers that correspond to what you measure in the field is nevertheless crazy, unconceivable, or outside the bounds of common sense, they probably are, and you should start again with more reasonable parameters.

Predictive Modelling

Once you have a reasonably calibrated baseline model, you may proceed to predictive modelling. As we have noted this involves adjusting the baseline model to include the mine waste disposal facilities you will operate at the site. Then you run the model that incorporates these new facilities in order to quantify the response of the system to the new facilities.

A common example is to incorporate the proposed tailings impoundment in the model. Then run the model allowing the tailings to seep from the base into the groundwater. This may cause the water table to rise, flow rates to get larger, or constituent concentrations to increase.

If you plan to place a liner beneath the tailings impoundment, the influx of water at the area of the proposed impoundment may decrease significantly. This may result in a drop (lowering) of the water table, less flow, and drying up of springs that animals rely on.

If you are planning on excavating an open pitand dumping the waste at the perimeter of the pit, you may increase the infiltration flux at the area of the rock dump. This may occur because the waste is more permeable that the soil that you strip to form the foundation of the dump. Hence there is greater infiltration at the upper surface of the rock dump than through currently in situ soils. The result is an increase in flux to the groundwater. But recall you are digging the pit, so now groundwater flow may be towards the holeyou dig rather than away from the area to the local valley. Increasing input flux and altering flow directions, may cause porepressures in the slopes of the open pit that are detrimental to pit slope stability.

Sensitivity Analysis

Sensitivity analysis is the act of varying key parameters within a reasonable range in order to establish the sensitivity of the equation or computer code outcomes (calculated results) to parameter variability.

As part of calibration or as a separate exercise, you may undertake sensitivity analyses of your conceptual model. You may undertake sensitivity analyses of the baseline model. But sensitivity analysis is more often done as part of predictive modelling.

Thus you may be curious as to how sensitive the predicted elevation of the water tale in the bedrock is to the estimated seepage from the waste rock dump. To quantify this, you would run and rerun the code with a feasible range of seepage rates. Hence you may decide to install liners or drains, or compact the lower layer of the waste rock to reduce its permeability.

Other Readings

There are many books, papers, computer code user manuals, and guidelines on all the topics we deal with above. The best are the ASTM standards for groundwater modelling. Consult them if you need more detail.