Mine waste dump stability analysis
Mine waste facilities are an essential part of any mining process, and a unique engineering challenge. Mining economics make their construction and maintenance very different to those of conventional water retaining dams. While they are among the largest structures humans have built, they are purely a cost to the mine, and have historically not had much design thought put into them. In several cases such as Los Frailes(1) the design failure of the tailings facility caused massive financial damage to the mining company. More recently, the breach of the Mount Polley Dam in British Columbia caused a 45% drop in the value of the company, and at the time of writing is expected to cost the company around $100 million in clean up costs . Every waste facility is unique, since the geological factors differ at each mine, however most waste facilities can fall into the category of either waste dumps, or tailings ponds. Understanding of the geological conditions is paramount as they will dictate the location, size, and constructability of the waste facility.
The most important part of the stability analysis is determining what you are trying to model (or: What problem are you trying to solve?). In the case of mine waste facilities, the objectives are simple:
- Design the structure such that it will retain the material behind it, and still be stable.
- Try to make the structure take up as little space as possible.
In the case of point 1, the rock slope or tailings dam slope should be as flat as possible to return maximum stability. On the other hand, the second point states that the slopes should be as steep as possible in order to minimize the amount of land taken up by the waste facilities. For this reason, judgment must be applied to several items of the design for it to be stable, but not prohibitively expensive.
As Dr. Steve Vick stated in his 2011 Cross Canada Lecture (paraphrased):
"Do we know what the problem is?"
"Good, Run FLAC."
The purpose of this anecdote, as well as his Fall lecture, is that there are many things that must be taken into consideration at the conceptual level before any modelling should take place. These items include The geological model of the site, the failure criteria, the target Factor of safety (FS) and the model that is to be used.
Geological Model of the Site
Understanding the geology at the Mine site is of the utmost importance. Through various site investigations, The geological model of the site is developed. This information is vital to the design as it allows the engineer to bound the engineering properties of the mine waste and foundation. As a quick example, knowing that the foundation materials at the mine site have an internal angle of friction of between 20° and 40° will lead to a very vague and likely over-conservative design, whereas knowing that in the areas of the dumps, the values are between 25° and 30° confines the problem far better, and saves several steps in a sensitivity analysis. In addition to soil and rock parameters, the groundwater conditions must be taken into account.
The foundation conditions are critical to the stability analysis, as there is usually much less control over the foundation conditions (unsuitable fill can usually be treated prior to placement, whereas any major treatment to the foundation of the mine waste area gets very expensive). As shown in Cameron's 2013 Lecture, the foundation can heavily influence the critical slip surface as weak layers within the soil create preferential slip planes. In this example, if a circular slip surface was used, it would return a much larger FS.
Strength parameters - Hoek-Brown vs Mohr Coulomb - depends on what information is available and what the material is. in the case of waste facilities it is usually classified as a soil, and Mohr Coulomb failure criterion are used.
Standard of Design
ASD vs LRFD
One important part of stability analysis is determining what standard of design to use between Allowable Stress Design (ASD) and Load Resistance Factor Design (LRFD). The current state of geotechnical engineering is such that ASD is typically used except in cases where there is enough data to obtain a statistical analysis of the materials. Since LRFD is probability based, it is typically used only in civil engineering applications, such as pile foundations.
ASD is simpler in that the FS must encompass the total uncertainty of the problem such as material strengths, groundwater conditions and seismic considerations. Generally, a higher FS is required for situations where there is more uncertainty, but a higher FS typically means an increase in construction cost for the mine. It is for this reason that selecting the FS is very important.
The FS is most heavily impacted by uncertainty, which is why the geological model is so important. Beyond the model however, several things can be done to lower the uncertainty of the design such as monitoring with geotechnical instrumentation, and careful construction monitoring. Current practice (Fall 2014) allows for bounding the problem by modeling two separate strength conditions: the conditions that have the greatest chance of being present in the field, and a second with the probable worst case conditions (as stated by Denis Russell, the conditions are probable because if you were to select a value (say φ=12°) as your worst case, it is hard to claim that 11.9° is not possible). The probable worst case can have a lower target FS, as long as there is adequate monitoring of key strength parameters and the ability for contingency measures. Without these options the FS for the probable worst case must not be reduced.
The model will require determining what stress conditions the waste rock is experiencing. Any rapid change in stress conditions is usually modelled using total stress conditions, whereas long term stability is modelled using effective stress conditions. This is an important side of judgement that is required before any modelling can take place, as picking the wrong one can lead to overestimating or underestimating the strength of your structure.
In order to decide how to model stability, the likely failure mode and stress conditions must be understood. Some models include (in increasing complexity):
• Limit equilibrium (Rocscience Slide, Geo-Studio SLOPE/W)
• Finite element (Rocscience Phase2, GeoStudio SIGMA/W, Plaxis)
• Finite Difference (Itasca FLAC)
• Distinct element (Itasca UDEC)
In the case of mine waste facilities, there are usually relatively low stresses, and the structures are usually constructed from soil like materials. Therefore limit equilibrium analysis is a commonly used model as it is simple and relevant. Mine waste facilities are typically processed material such as blast rock or milled material, which is placed above ground. Thus, the Distinct Element Model is rarely used except for specific cases.
A limit equilibrium analysis, although the simplest of the models, is applicable to many slope design problems. The limit equilibrium method is a great tool as it can calculate many different shear surfaces, and determine the critical slip surface. There are several methods that are used to calculate the FS of the landslide. There are non-general equilibrium methods such as Ordinary, Bishop, and Janbu, as well as two general equilibrium methods: Spencer and Morganstern-Price. They all work by taking a slope and selecting a slip surface, which is the modelled landslide. The landslide is then cut up into vertical slices, and calculating the moments and or forces that affect the landslide. In almost all cases, the method of slices is run assuming that the slope is a 1 m (or 1 foot) section, perpendicular to the page. It should also be noted that the equations presented below are based on Mohr-Coulomb Failure Criterion, as it is most commonly used when assessing soil strength.
Non-general equilibrium methods have the advantage that they are simpler to use and can be calculated by hand. However this comes at the expense of not satisfying all aspects of equilibrium. The Ordinary Method was the first method of slices to be developed, and calculates only the moment equilibrium for the slope using the following equation for the FS:
- W is the weight of each slice.
- c' is the cohesive strength of the soil.
- φ' is the internal angle of friction in the soil.
- l is the length of the slice along the slip surface.
- α is the angle of the slip surface from horizontal.
This method makes the assumption that the shear forces as well as the result of the horizontal forces are are equal, and therefore not calculated. This assumption makes the problem statically determinate however, as can be seen in the Ordinary slice, the force vector diagram does not close, and is therefore not in equilibrium. The advantage of this is that the equilibrium equation can be computed by hand, however it typically gives low factors of safety, meaning an overly conservative design. The assumptions made by the Ordinary Method typically give a factor of safety that is up to 20% less than the "true" FS.
Bishop developed an improvement over the Ordinary Method in 1955. In the Bishop Method, the shear forces are assumed to be equal and opposite (and are therefore not calculated), and the horizontal forces are assumed to be colinear but not equal, as shown in the Bishop Model. The equation that Bishop uses is:
Where the values are all the same, except m represents the horizontal width of the slice, as opposed to l which was the length along the shear zone.
As can be seen from the Bishop Equation, the model is now statically indeterminate, and must be solved using iterations. The equation can still be solved by hand to gain a general idea of the slope FS, but if possible a computer can compute the answer far more quickly. An additional benefit to the Bishop method is that for circular analyses, it has been shown (Wright, 1985) to give FS values within a few percent of the "true" FS
Finite Difference and Finite Element
Finite Difference/Element models are vastly more complex than Limit Equilibrium models. This section will explain the general principles used in the two models types, as well as the differences between the two.
List of Other Important Considerations
Key design parameters that are outside the scope of this page:
- Construction material availability
- Construction Sequencing
- Filter design on tailings ponds
Budhu, M. (2008), "Foundations and earth retaining structures." Hoboken, NJ: John Wiley & Sons.
Cameron, Robert (2013), "Case Studies in Soil Parameter Selections for Clay Foundations," Canadian Geotechnical Society Cross Canada Lecture Slides, pp. 25
Coduto, D. (1999), "Geotechnical engineering: Principles and Practices," Upper Saddle River, NJ: Prentice Hall.
Krahn, John (2004), "Stability Modelling with SLOPE/W, an Engineering Methodology", GEO-SLOPE International Ltd.
Russell, Denis (1998), "Comment on 'The PMF Does Have a Frequency,'" Candian water Resources Journal, Vol. 23, No. 4, 1998
Wright, Stephen G. (1985), "Limit Equilibrium Slope Analysis Procedures," Design of Non-Impounding Waste Dumps, pp. 63-77, American Institute of Mining Engineers.
"Canadian Foundation Engineering Manual, 4th Edition, Canadian Geotechnical Society