Mine waste dump stability analysis

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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 [2]. 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. This page is a brief overview of the items to be considered for design based on a 4 month stability analysis the author performed. It is by no means an exhaustive list, but rather a place to start.

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:

  1. Design the structure such that it will retain the material behind it, and still be stable.
  2. 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.

Design Considerations

As Dr. Steve Vick stated in his 2011 Cross Canada Lecture (paraphrased):

"Do we know what the problem is?"
"No."
"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.

Foundation

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 Bob 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.

Groundwater

Since effective stress is a key part of geotechnical engineering, understanding both the groundwater regime as well as the groundwater response to load changes is incredibly important. The pore pressure response can be measured by installing piezometers in several foundation layers and constructing a test-fill above it. The pore pressures are then measured as follows (Fredlund, 1986):

Pore Pressure Relationships.png

Where B (bar) is the relation between the change in pore water pressure (u) as a result of the change in vertical stress. On the other hand ru is just the present ratio of pore water pressure to total overburden pressure. Both of these relations are used in practice and both have strengths and weaknesses. ru is a parameter that is much better defined (directly measured), however as Bob Cameron points out, since there a slope applies different earth pressures from crest to toe, there may be several different ru values needed along a single material layer. On the other hand B (bar) is likely to be an value generated from several tests, but may be applied throughout a soil layer. it is the opinion of this page's original author that ru is more appropriate for analyzing a single slope (during or post-construction) whereas B (bar) is more appropriate for pre construction design.

Geomorphological Processes

The surrounding geology of the area is another key factor. If construction is required near erosional areas such as river banks, the long term change in the surrounding topography must be taken into account. In the river bank example, it is reasonable to assume that, unlike pit operations, the mine waste facility will be around for 1000+ years. If the river adjacent to facility errodes sufficiently, a large part of it could be lost into the river. This effect can be increased if any waterways are diverted to make room for the mine facilities (ie. the pit or the waste dump). For this reason, detailed geomorphological studies should be carried out on anything that could affect the facility.

Soil Strength Model

Additionally, the soil strength model must be determined. In almost all effective stress cases, this will be the Mohr-Coulomb model. The Hoek-Brown model is typically for a rock mass, whereas waste facilities are constructed from blast rock, soil, or mine tailings, which have a more soil-like behaviour. Mohr-Coulomb based models available for the modelling of waste rock or soil are (examples are based on personal experience):

  • Undrained: Φ=0° and all strength comes from the c' value.
    • Example: Rapid loading.
  • Anisotropic:The soil has different strengths in the vertical and horizontal directions.
    • Example: Pre-sheared Shale.
  • Depth dependant: Soil gets stronger as a function of depth.
    • Example: Normally consolidated fines (usually have data to indicate this).
  • Overburden dependant: Soil gets stronger based on vertical effective stress.
    • Example: Liquefied sands and other very soft materials.

The final case is assessing the seismicity of the waste facility. There are many tests that can be done in-situ to determine the likelihood of either liquefaction or strain-softening behaviour which are described by Rauch. It should be noted that it is possible, and good practice, to sufficiently compact construction materials to prevent liquefaction. If seismicity is possible, there are ways to determine if it is a potential at your site depending on the type of model being run. How each method models the seismic force is discussed in Section 3.

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, where there is enough data to support a meaningful statistical analysis.

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 difficult 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.

Stress Conditions

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. The soil behaviour while undrained and differ quite significantly, however with good sampling practises, they can be determined thru Unconsolidated-Undrained tri-axial testing. Long term stability is modelled using effective stress conditions the parameters for drained strengths are best determined by a Consolidated Drained tri-axial test (but if water pressure during the test is measured it is much faster to perform a Consolidated-Drained test). 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.

Stability Models

Computer models are like a captured spy: they will tell you anything if you torture them for long enough. This is paraphrased from an undergraduate mine waste design class taught by Dr. Ward Wilson, now of the University of Alberta. It shows how a computer model should only be used as a tool and like any tool, the operator should know how it works. 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 and will not be discussed in this page.

Additionally, there are

Limit Equilibrium

(thumbnail)
A Slope divided into slices. The forces on the slices are added together to determine the FS of the slope
(thumbnail)
The Ordinary Method does not account for horizontal forces on the slice or shear forces
(thumbnail)
Similar to the ordinary method but it does account for horizontal forces. Shear forces are still not taken into account
(thumbnail)
Morgenstern-Price Force Diagram - note how the shear force is accounted for (small vertical line)

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 Morgenstern-Price. They all work by taking a slope and defining 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. this method is repeated for many slip surfaces and the lowest calculated FS is the 'critical' surface. Obviously it is the role of the designer to determine whether the calculated critical slip surface is reasonable.

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:

Ordinary Eqn.png

Where:)

  • 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.

and

  • 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:

Bishops.png

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 to give FS values within a few percent of the "true" FS (Wright, 1985). Krahn notes that this is because a circular shape can slide and maintain its shape - so the internal forces are less important. this reasoning breaks down when it comes to non-circular slip surfaces.

All models discussed thus far were developed for circular slip surfaces. The downside is that if the critical surface is not circular, their assumption that vertical shear forces are zero starts to affect their accuracy. in these cases the most rigorous and complete models are called general limit equilibrium models. The two that have most commonly in use are the Spencer method and the Morgenstern-Price Method, which solve The moment FS, the Horizontal Force FS, and then iterate values to equate them.


The exact details of how these methods work are too long to include in this article, but can be found in Krahn's 2004 paper. Briefly, the models assume a function for an interslice shear forces: in the case of Spencer, the shear forces are assumed to be a constant ratio to the normal forces of each slice. It then iterates each slice to determine the minimum factor of safety. Morgenstern-Price on the other hand, uses a user-defined function (typically half-sine) to determine how the shear forces change from slice to slice along the slip surface. The result is that the Morgenstern-Price method is more robust, but more complicated than the Spencer method. Neither method is practical for hand computations, but computers can easily handle between 500 and 10,000 slip surface calculations per minute. These models also fall under the assumption of all limit equilibrium analyses, where they do not incorporate any field stresses into the model and they assume that the whole slip surface is mobilized at the same time. While the two assumptions usually can be ignored, there are tools available fro when they cannot be. Further, no limit equilibrium model incorporates constitutive stress-strain relationships, and therefore do not factor in effects of the sliding mass deforming. The main consequence of this, is that the stress distributions that are calculated by the model are not always accurate, and therefore should not be relied upon if they must be understood.

Finite Difference and Finite Element

Finite Difference/Element models are known as numerical methods. This section will explain the general principles used in the two models types, as well as the differences between the two. Both of these methods run iterative processes that calculate equilibrium to a point defined by the user. For example, all computer models have an option of the level of convergence that is required (such as 0.1 or 0.0000001). The finer the convergence required, the longer the calculations take, so care must be taken in selecting a reasonable value.

Finite Element Model (FEM)

Finite Element Models break the project area up into small elements that are either rectangular or triangular in shape. The material areas are defined, and then a mesh is generated based on the requirements of the user. Choosing the correct mesh is important because too many elements take a long time to calculate, whereas too few may not return a high enough parameter resolution. When stresses are applied to these elements they deform and affect the surrounding elements according to their constitutive relationships. The deformed elements then affect the elements surrounding them and again throughout the model. The model uses iterations to calculate the stresses based on boundary conditions and material properties until the model is in equilibrium.

Choosing the boundary conditions are important for accurate representation of the model. Finite element programs have two main boundary conditions, smooth-rigid and rough-rigid. Smooth-rigid boundaries prohibit movement across the boundary but the material or stress flows are free to move along the boundary. Rough-rigid also prohibits movement across it, but also prohibits movement along the boundary itself. These boundaries must be selected with care, as the wrong boundary conditions can return unrealistic results. For example: If modelling a slope the outer edges of the model will both be at geostatic stress conditions (which can be checked using any finite element program). Using a "smooth-rigid" boundary on the bottom of the model will require the material to hold the change in stresses from one side to another. Depending on the material parameters used, this difference can cause strange planes of weakness that would not necessarily develop in-situ. Changing the bottom boundary to a "rough-rigid" boundary forces the boundary to accommodate the change in stresses rather than the foundation. In most cases this will be more representative. As an aside, seismic forces are modelled as a pseudo-static force like the limit equilibrium model.

FEMs calculate the FS of a slope by using the Shear Strength Reduction (SSR) method. The material must be defined as elastic plastic in order for the deformations to be taken into account.The FS is calculated as follows, with the SSR parameter defined as F:

SSR eqn.png

where

SSR Def.png

The designer first computes several values of f* and c* for various FS. These values are then inserted into the model as the c' and φ' parameters for an FS and the model is run. The FS is gradually increased until the model no longer converges (indicating no equilibrium). The value of F just before the model becomes divergent is the factor of safety. The parameters such as Young's modulus Poisson's ratio don't have a significant effect of the FS but do affect the calculated deformations (Hammah et al, 2005).

Finite Difference

(thumbnail)
Calculation cycle for each time step in the finite difference model (McKinnon, 2014)

Finite Difference models are similar to FEM in the sense that the materials are separated into grids. Unlike FEM however, the area between the grids do not represent elements, but the corners, or grid points, are used to determine the movement or stress of the model. The cycle the model uses is shown on the right side of the page.

Similar to FEM, no slip surface needs to be determined; the model will find the critical slip surface itself through stress relationships. Another similarity is that defining the mesh is a key part of finite difference modelling, but with a key difference in that finite difference models also allow for defining more boundary conditions. The boundaries can be both stress conditions as well as dynamic conditions for seismic analyses.

The cycle of a finite difference model calculates material displacements through an iterative process. The first step is to define the stress conditions, which act as forces on the grid points. The forces cause accelerations, which are calculated by the equations of motion (F=ma) the accelerations are integrated to determine the grid point velocity and then integrated again to get the displacements. These displacements are then converted into strains, which are then converted into stresses using constitutive models. This cycle usually repeats itself until an equilibrium is reached. If equilibrium is not reached, the finite difference model can still return a solution. Since it calculates grid point displacements, movement of the soil mass can be modelled over time, and complex landslide models can be generated without convergence.

The time step is defined such that the movement will only affect one grid at a time in order to keep the calculations possible. The Δt does not represent any real time time. The exception to this rule is when the boundary conditions are dynamic, such as seismic conditions. In this case Δt must correspond to real time, as the boundary conditions are defined as real time.

List of Other Important Considerations

As discussed above, the most important part of a mine waste design is understanding of the site. The topics highlighted above are brief and act as a compass to guide the design. The discussion on stability models shows the wide range of complexity and capability of the available models, and as such choosing the right model for the job can save considerable amount of time, as long as the additional complexity is not required. An analogy would be digging a hole: if a hole is needed for a tree sapling, a large excavator can be used, but is very excessive. Whereas digging a basement for a house with a garden trowel can be done, it would just take an excessive amount of effort. There are several key parameters which indirectly impact a stability analysis that are outside the scope of this page, which include:

  • Construction material availability
  • Construction Sequencing
  • Filter design on tailings ponds
  • Hydrological models
  • Using the tailings facility as water storage
  • Distance of waste from the pit or mill


References

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

Canadian Geotechnical Society, (2006), "Canadian Foundation Engineering Manual, 4th Edition," BiTech Publisher Ltd, ISBN 0-920505-28-7, January 2007.

Coduto, D. (1999), "Geotechnical engineering: Principles and Practices," Upper Saddle River, NJ: Prentice Hall.

Fredlund, D., & Barbour, S. (1986). "The Prediction of Pore-Pressures for Slope Stability Analaysis." University of Saskatchewan Slope Stability Seminar April 28, 1986, pp. 3-6.

Hammah, R., Yacoub, T., Corkum, B., & Curran, J. (2004), "Stability analysis of Rock Slopes Using the Finite Element Method." Proceedings of the 53rd Geomechanics Colloquium.

Hammah, R., Yacoub, T., Corkum, B., & Curran, J. (2005), " A Comparison of Finite Element Slope Stability Analysis with Conventional Limit-Equilibrium Investigation." Proceedings of the 58th Canadian Geotechnical Conference.

Krahn, John (2004), "Stability Modelling with SLOPE/W, an Engineering Methodology", GEO-SLOPE International Ltd.

McKinnon, Steve (2014), "Numerical Modelling," Mine 469/823 Lecture Slides, Downloaded from Queens University D2L, November 6, 2014.

Russell, Denis (1998), "Comment on 'The PMF Does Have a Frequency,'" Candian Water Resources Journal, Vol. 23, No. 4, 1998

Rauch, A. F. (1997), "EPOLLS: An empirical method for predicting surface displacements due to liquefaction-induced lateral spreading in earthquakes" (Doctoral dissertation, Virginia Polytechnic Institute and State University).

Wright, Stephen G. (1985), "Limit Equilibrium Slope Analysis Procedures," Design of Non-Impounding Waste Dumps, pp. 63-77, American Institute of Mining Engineers.

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