v115n7a7 an economic risk evaluation approach for pit slope … · 2015-07-31 · becomes steeper,...

IntroductionThe open pit mine design process seeks todefine the optimum pit limits and sequence ofmining, in order to derive the maximumbenefit from the exploitation of a mineralresource given its spatial distribution and theparticular geological, economic, and minesettings. Pit slope angles are determined usingthe conventional approach, whereby slopestability indicators such as the factor of safety(FS) or the probability of failure (PF) arecalculated and compared with generic accept-ability criteria to define the values to be usedin the mine design process. The maindrawback of this approach is that in spite ofthe effect that the slope angle has on theeconomics of the mine plan, its definition isbased on criteria not directly related to thisaspect of the design. The pit slope designprocess described in this paper attempts to

avoid this drawback. The methodology isbased on a quantitative risk evaluation of theslopes, which has as a central element theconstruction of a risk map that relates theprobability of the impact to its magnitude. Inthis process the economic impacts of slopefailure are calculated and used as the elementson which to apply the acceptability criteria fordesign.

The proposed methodology is an evolutionof the approach described by Tapia et al.(2007) and Steffen et al. (2008), where eventtree analysis similar to that used for safety riskevaluations was applied to the economicassessment of slope failures. This approachwas superseded by a probabilistic method witha less subjective basis, as described byContreras and Steffen (2012). The method wasstill in a development phase at the time of thelatter publication, and was due to be applied toactual projects. Since then, the methodologyhas been used to evaluate two open pit mineprojects, and as a result of that work someimprovements have been implemented, partic-ularly in terms of the concepts of probabilityused for the construction of the risk map. Thegraphs and data used in this paper to presentthe methodology are derived from these twoprevious studies.

BackgroundThe optimum design of a pit requires thedetermination of the most economic pit limit,which normally results in steep slope angles asin this way the excavation of waste isminimized. In general, as the slope anglebecomes steeper, the stripping ratio (waste toore ratio) is reduced and the mining economicsimprove. However, these benefits arecounteracted by an increased risk to the

An economic risk evaluation approachfor pit slope optimizationby L.F. Contreras*

SynopsisIn open pit mine design, it is customary for geotechnical engineers todefine the appropriate slope design angles within practical limits. Theconventional approach to slope angle design is based on the comparison ofcalculated stability indicators, such as the factor of safety (FS) and theprobability of failure (PF), with generic acceptability criteria not directlyrelated to the impacts of failure. A major drawback of this type ofapproach is related to the difficulty of defining meaningful acceptabilitycriteria. An alternative methodology of pit slope design is proposed, wherethe economic impacts of potential slope failures are calculated and used asthe elements on which to apply the acceptability criteria for design. Themethodology is based on the construction of a graph, referred to as a riskmap, that relates the probability of exceeding the economic impact of slopefailure to the magnitude of the impact measured in monetary terms. Theprocess includes the analysis of a selected number of representative yearsof the mine plan and slope sections of the pit areas to define the requiredinputs for the construction of the risk map. The paper discusses theconcepts used in interpreting the probability of slope failure, and describesthe approach followed for the estimation of the economic impacts of slopefailure and the construction of the risk map. Finally, the two main uses ofthe risk map are discussed, including the comparison with acceptabilitycriteria for the evaluation of a specific open pit design and the comparativeanalysis of open pit design options in terms of value and risk to identifyoptimum pit layouts.

Keywordsrisk evaluation, economic risk map, slope design, slope failure, probabilityof failure.

* SRK Consulting, Johannesburg, South Africa.© The Southern African Institute of Mining and

Metallurgy, 2015. ISSN 2225-6253. Paper receivedJul. 2014 and revised paper received Jan. 2015.

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http://dx.doi.org/10.17159/2411-9717/2015/v115n7a7

An economic risk evaluation approach for pit slope optimization

operation. Thus, the determination of the acceptable slopeangle is a key aspect of the mining business.

The difficulty in determining the acceptable slope anglestems from the uncertainties associated with slope stability.Typical uncertainties encountered in the pit slope designprocess are discussed by Tapia et al. (2007) with reference tothe Chuquicamata open pit. There are three main approachescommonly used to account for the uncertainties in slopedesign: factor of safety, probability of failure, and riskanalysis.

Factor of safety approachThe oldest approach to slope design is based on thecalculation of the factor of safety (FS). The FS can be definedas the ratio between the resisting forces (strength) and thedriving forces (loading) along a potential failure surface. Ifthe FS has a value of unity, the slope is said to be in a limitequilibrium condition, whereas values larger than unitycorrespond to stable slopes. The FS approach is adeterministic design technique as a point estimate of eachvariable is assumed to represent the variable with certainty.The uncertainties implicit in the stability evaluation areaccounted for through the use of a FS for design larger thanunity. This acceptability criterion is intended to ensure thatthe slope will be stable enough to ensure a safe miningoperation. Acceptable FS values in mining applications rangebetween 1.2 and 2.0 according to Priest and Brown (1983),as indicated in Wesseloo and Read (2009). Acceptable valuesare based on observations of the performance of slopes atspecific sites and experience accumulated over time.

There are two main disadvantages in the FS approach forslope design. Firstly, the acceptability criterion is based on alimited number of cases and combines the effect of manyfactors that make it difficult to judge its applicability in aspecific geomechanical environment. Secondly, the FS doesnot provide a linear scale of the likelihood of slope failure.

Probability of failure approachIn recent years, probabilistic methods have been increasinglyused in slope design. These methods are based on thecalculation of the probability of failure (PF) of the slope. Aprobabilistic approach requires that a deterministic modelexists. In this case the input parameters are described asprobability distributions rather than point estimates of thevalues. By combining these distributions within thedeterministic model used to calculate the FS, the probabilityof failure of the slope can be estimated. A techniquecommonly used to combine the distributions is the MonteCarlo simulation. In this case, each input parameter value issampled randomly from its distribution, and for each set ofrandom input values a FS is calculated. By repeating thisprocess many times, a distribution of the FS is obtained. ThePF can be calculated as the ratio between the number of casesthat represent failure (FS<1) and the total number ofsimulations.

The advantage of the PF over the FS as a stabilityindicator is based on the fact that there is a linearrelationship between the PF value and the likelihood offailure1, whereas the same is not true for the FS. A larger FSdoes not necessarily represent a safer slope, as the magnitudeof the implicit uncertainties is not captured by the FS value. Aslope with a FS of 3 is not twice as stable as one with a FS of

1.5, whereas a slope with a PF of 5% is twice as stable as onewith a PF of 10%.

Some drawbacks of the FS methodology that persist inthe PF approach are the difficulties in defining an adequateacceptability criterion for design and the limitations inpredicting failure with the underlying deterministic model.

Acceptability criteria for PF have been defined bydifferent authors and organizations, and a summary of thisinformation is presented in Wesseloo and Read (2009).However, the actual criteria to be used in a specific minecannot be determined from general guidelines like these, andshould be subjected to a more thorough analysis of theconsequences of failure (Sjoberg, 1999).

Risk analysis approachThe risk analysis approach tries to solve the main drawbackof the previous methodologies with regard to the selection ofthe appropriate acceptability criteria. Risk can be defined asthe probability of occurrence of an event combined with theconsequence or potential loss associated with that event:

Risk = P(event) × Consequence of the event

In the case of slopes, the P(event) is the PF of the slope andthe consequences can be two-fold: personnel impact andeconomic impact.

The PF calculated as part of the design process isnormally based on a slope stability model calculation andaccounts only for part of the uncertainties of the slope.Because risk analysis sets the acceptability criteria on theconsequences rather than on the likelihood of the event, athorough evaluation of the PF of the slope is required,incorporating other sources of uncertainty not accounted forwith the slope stability model. For this purpose and for theanalysis of consequences of slope failure, non-formal sourcesof information (engineering judgment, expert knowledge) areincorporated into the process with the aid of methods such asdevelopment of logic diagrams and event tree analysis. Thesetechniques are described in detail by Baecher and Christian(2003) with reference to geotechnical engineering problems,more commonly in the disciplines of dam and foundationengineering. However, the use of risk methods in open pitmining focuses on safety applications, based on qualitativeapproaches to assess operational aspects.

In the following sections, a description of the proposedrisk methodology for slope design optimization is presented.

MethodologyThe proposed methodology uses the framework described bythe Australian Geomechanics Society (2000) with referenceto the landslide risk management process, characterized bythe following main steps:

➤ Identify the event generating hazards

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608 JULY 2015 VOLUME 115 The Journal of The Southern African Institute of Mining and Metallurgy

1 In non-technical literature, ‘likelihood’ is usually a synonym for’probability’, but in statistical usage, a clear technical distinction ismade. Here, probability of failure refers to the estimated frequencyof FS<1 cases with the model assuming that this conditionsrepresents failure. PF can take values only between 0 and 1. Thelikelihood of failure is a quantity not constrained and refers to thechances of actual failure, given the results of the stability analysis.

➤ Assess the likelihood or probability of occurrence ofthese events

➤ Assess the impact of the hazard➤ Combine the probability and impact to calculate the risk➤ Compare the calculated risk with benchmark criteria to

produce an assessment of risk➤ Use the assessment of risk as an aid to decision-making.

The methodology described in this paper refers mainly tosteps 2 to 5 as applied to the risk evaluation of pit slopes.

The proposed risk evaluation process for slope design isintended to quantify the impact of potential slope failures onthe economic performance of open pit mines. Figure 1illustrates the risk evaluation process and depicts the mainelements of the methodology, which are described in detail inthis paper. The diagram includes the main components of theconventional geotechnical slope design process as describedin Stacey (2009) and incorporates the additional elementsrequired from the mine design process.

The main objective of the methodology is the definition ofthe pit slope angles for mine design by applying projectspecific criteria to the quantified risk costs. The approachincludes the following main steps:

➤ Definition of the set of slope sections for analysiscovering key and critical pit areas during the mine lifeto provide representative cases of potential risks ofslope failure within the mine plan

➤ Calculation of the probability of failure (PF) of theslopes from the analysis of stability of the selectedslope sections

➤ Quantification of the economic impacts of slope failurewith reference to the loss of annual profit or totalproject value as measured by the NPV

➤ Integration of the results of probability of failure andeconomic impact on an annual basis to define theeconomic risk map per year and for the life of mine

➤ Comparison of the risk map with criteria to assessacceptability of the design and to define risk mitigationoptions as required

➤ If the analysis is intended for the comparison ofalternative slope design options, the process is repeatedfor each alternative pit layout and the results arecollated in a graph of slope angle versus value and riskcost where the optimum slope angles can be defined.

A complete risk evaluation process should also includethe evaluation of the safety impact of slope failures. Safetyrisk evaluation is discussed by Contreras et al. (2006),Terbrugge et al. (2006), Tapia et al. (2007), and Steffen etal. (2008), and is not covered in this paper.

Slope sections for analysisThe risk evaluation process requires a programme of slopestability analyses, including the critical pit areas and years interms of potential economic impacts of eventual slopefailures. This means that besides adequate information ongeotechnical conditions defining the likelihood of failures, agood understanding of the mine plan is required to identifythose areas and years in which the impacts of failure arelikely to be greater.

The selection of the sections for stability analysis startswith the selection of the years of the mine life that representdevelopment periods in the mine plan with similar character-istics in terms of pit geometry, production profile, andeconomic scenario. Figure 2 shows an example of thecumulative discounted profit of a mine plan, which is arepresentation of the realization of value with time. Thisgraph facilitates the definition of the appropriate periods andrepresentative years of mine development for the risk modelanalysis, which in this example corresponds to the six yearsdefining the stepped curve.


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Figure 1 – Risk-based slope design approach


In general, probabilities of failure increase through themine life, whereas impacts tend to maintain their levels oreven decrease as mining progresses. The assumption thatrisk conditions of a later year (2027) represent those of earlyyears (2025/2026) is therefore reasonable, with a minor

effect on the results or (more commonly) on the conservativeside. The graph in Figure 2 implies that there is a trade-offbetween rigour and practicality when selecting the years foranalysis. Ideally, every year would have to be analysed,although this would not be practical and is probablyunnecessary in the majority of cases.

The appropriate slope sections for analysis can be selectedby examination of the mine plan in the identified key years.The criterion used for this selection is based on covering theanticipated higher risk areas of the mine, which includelocations where the likelihood of slope failure or theassociated impact is expected to be high. Examples of thepreferred locations for analysis include areas with higher orsteeper slopes, sites with unfavourable geological conditions,areas with distinct characteristics such as those defined bythe geotechnical domains, critical access points to miningfaces, areas close to key infrastructure, and so forth. The pitdevelopment plan sketched in Figure 3 shows an examplewith the selection of 42 sections used in this paper toillustrate the risk process.

Slope stability analysisThe results of the slope stability analyses are reported interms of PF values, which are calculated with the appropriate

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Figure 3 – Example of selection of slope sections for risk analysis

Figure 2 – Realization of value with time as a criterion for defining yearsof risk model analysis

slope stability models in accordance with the relevant failuremechanisms in each domain. The methodology used for thecalculation of the PF is in part determined by the type ofdeterministic model used for the calculation of the FS of theslope. A compilation of the methods commonly used in slopedesign can be found in Lorig et al. (2009). In a probabilisticstability analysis, the input parameters that represent theuncertainties are described by probability distributions. Thesedistributions are combined within the deterministic model todefine the distribution of the FS, which is used to estimatethe PF of the slope. The PF is calculated as the ratio betweenthe number of cases representing failure (FS<1) and the totalnumber of cases of FS described by the distribution. Simplemodels, such as those based on the limit equilibrium method,can incorporate built-in routines to perform Monte Carlosimulations that enable the PF to be calculated relativelyquickly. However, the use of more elaborate models based onstress-deformation analysis, with higher computationaldemands, restricts the calculation of the PF to those methodsrequiring a reduced number of FS entries to define itsvariability. Examples of such methods include those based onTaylor series expansions, the point estimate method, and theresponse surface methodology. Descriptions of these methodsin terms of their conceptual basis are given by Baecher andChristian (2003) and Morgan and Henrion (1990). Theresponse surface method has been used in risk-based slopedesign applications as described by Steffen et al. (2008). Thisapproach has the advantage of combining the rigour of aMonte Carlo simulation with the practicality of requiringfewer FS calculations with the geotechnical model toconstruct the response surface used as a surrogate model inthe process.

Due to practical limitations, the PF values calculated withslope models are typically the result of considering theuncertainty of the strength properties of rock masses andstructures, without consideration of any other potentialfactors contributing to slope instability. Therefore, these PFvalues are incomplete representations of the likelihood offailure, and need to be adjusted as discussed later for thepurpose of a risk consequence analysis.

Interpretation of probability of failure (PF) of the slopeA slope failure event could be regarded as a Bernoulli trial(also called binomial trial), which is defined as a randomexperiment with only two possible outcomes, success orfailure, and in which the probability of success (or failure) isthe same every time the experiment is conducted. Accordingto this definition, and considering failure as the target eventof analysis, if p is defined as the probability of failure, then q= (1-p) corresponds to the probability of no failure. Examplesof Bernoulli trials include a ‘head’ after tossing a coin(p=50%, q=50%), a ‘one’ after rolling a dice (p=16.7%,q=83.3%) and, under certain assumptions as explainedbelow, a failure after excavating a slope (p=PF, q=1-PF). Thesuccessive repetition of Bernoulli trials constitutes a Bernoulliprocess. The probability of success (or failure) is revealed ina Bernoulli process with a large number of trials. It ispossible to verify that after rolling the dice a hundred times,the number of ‘one’ cases will be close to 17 and as moretrials are considered, the better the approximation will be tothe ‘one in six’ probability of getting a ‘one’.

Strictly speaking, a Bernoulli trial refers to a discreteindependent event, which is not exactly the case of thecontinuous process in time or space that characterizes theexcavation of a pit slope. However, the consideration of theslope excavation process as a series of discrete situations, forexample, excavation of consecutive slope lengths along a pitwall or annual exposure of slopes through the mine life, is avalid assumption within the framework of the risk model forslope failure, as failure events are associated with specificslope sections that are selected precisely to represent distinctconditions in terms of time of exposure and location withinthe pit.

The association of open pit slope failure events with aBernoulli process enables the following interpretations basedon the number of trials of the process.

Bench slope failure in a homogeneous domainA bench slope failure in an open pit situation could be seenas a Bernoulli process involving many trials. The probabilityof bench failure in a benched slope within a homogeneousstructural domain corresponds approximately to the ratiobetween the cumulative length of failed benches and the totallength of constructed benches in that domain. In this case,the entire slope could be considered as a series of consecutiverealizations of a unitary slope with a length given by thetypical failure width. This case is illustrated in the sketch inFigure 4 and is comparable with the situation of rolling a dicemany times to verify the probability of getting a ‘one’. In fact,the bench slope case can be seen as if a bench of length ‘b’ isconstructed many times, with a percentage of thosecorresponding with failure situations.

Inter-ramp slope failure in a homogeneous domainThe case of a hangingwall in an open pit mine located withina homogeneous geotechnical domain could be looselyassociated with a Bernoulli process with several trials. In thiscase the probability of failure of the inter-ramp slopes for thelife of mine could be approximated by the ratio between thecumulative volume of inter-ramp slope failures having


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Figure 4 – Interpretation of the bench slope failure case as a Bernoulliprocess with many trials


occurred and the total volume of rock excavated during thelife of mine in the hangingwall, as illustrated in Figure 5.

Overall slope failure in a heterogeneous domainThe case of overall slopes in open pits in heterogeneousgeotechnical domains could be associated with a Bernoulliprocess with few trials or even with a single Bernoulli trial.The probability of failure of these slopes is not revealed in aphysical manner and the estimation can be based only onsimulation trials with geomechanical models representing theslopes. In this case the slope could be seen as a uniquerealization or trial that is not repeated in time or space,similar to the situation of a dice rolled once with two possibleoutcomes in terms of getting a ‘one’, success or failure. Theoverall slope failure case as a Bernoulli trial is illustrated inFigure 6.

Estimation of PF values for risk analysisThe PF values to be used in a risk evaluation process need torepresent all the exposed areas of the pit in the year ofanalysis, and to account for all possible uncertain factors thatmay lead to slope failures. The PF values calculated withslope stability models refer to specific sections of the slopesand typically account only for the uncertainties associatedwith variability of geotechnical properties. Therefore, theselimitations need to be accounted for in the set of PF valuesresulting from the geotechnical analysis, such that they aretruly representative of the likelihood of failures in the pitareas and mine plan years of analysis. For this purpose, twotypes of adjustments are required to the PF values calculatedwith the geotechnical models: one related to the estimation ofthe PF of the pit wall as opposed to that of the section ofanalysis; and the other to the estimation of the total PF asopposed to the model PF.

Section and slope wall PFFigure 7 shows the difference between the PF resulting froma stability analysis with a representative section of the slopeand the PF value reflecting the likelihood of slope failure in apit wall with a length greater than the expected width of thefailure.

It is clear that the PF of the longer slope wall in Figure 7is greater than that of the shorter slope shown. Consideringthe shorter slope as a unitary slope with a length comparableto the expected width (d) of the failure, then the longer wallwith length (L) could be seen as a series of consecutiverealizations of the unitary slope (Bernoulli trials). If the PF ofthe shorter slope is given by the probability of failure (ps)resulting from the analysis of a typical section of the slope,then the PF of the longer wall (PFw) can be estimated withthe following expression:

PFw ≈ 1 – (1 – ps)L/d [1]

This consideration is useful to ensure that the possibilityof failure of every exposed slope in the pit is included in the

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Figure 6 – Interpretation of the overall slope failure case as a Bernoullitrial

Figure 5 – Interpretation of the inter-ramp slope failure case as aBernoulli process with several trials

Figure 7 – Interpretation of the probability of failure of a slope wall in ahomogeneous rock mass as a function of its length

risk analysis. However, the applicability of this adjustment isrestricted to those situations where the assumption ofhomogeneity of the wall represented by the section analysedis reasonable; otherwise, the analysis of additional sectionsneeds to be implemented.

This consideration is also important in conventionalgeotechnical design procedures where PF values fromgeotechnical analysis are compared with acceptability criteriaof reference, since complying with the specified criteria on asection basis does not necessarily guarantee that the criteriaare met for the slope wall.

Model and total PFThe analysis of consequences of slope failure requires thatthe PF of the slopes be a true reflection of the likelihood ofoccurrence; therefore, all possible situations leading to slopefailures need to be included in the analysis. Due to practicallimitations, the PF calculated with the geotechnical slopemodel typically accounts only for the uncertainty of thematerial properties, and is referred to as the model probabilityof failure (PFmodel) in the following discussion.

The estimation of the PF incorporating other sources ofuncertainty not accounted for with slope models wasdiscussed by Contreras et al. (2006) and by Steffen et al.(2008) using a methodology based on the analysis of sourceresponse diagrams (SRDs). The methodology is based onconcepts presented by Chapman and Ward (2003) withreference to project risk management processes used in awide range of industries. The method enabled the quantifi-cation of the contributions to the PF caused by departuresfrom the normal conditions assumed for the design of theslopes. These variations were evaluated within variouscategories such as groundwater conditions, geologicalfeatures, operational factors, or occurrence of seismic events.The estimated contributions were added to the PF valueresulting from assuming normal conditions of design tocalculate the total probability of failure (PFtotal) to be used ina risk analysis.

The methodology presented in this paper is analogous tothe SRD approach described by Steffen et al. (2008), butadds some considerations regarding time in order to reflectthe gradual increase, with time, of exposure to the atypicalconditions evaluated. The method is appropriate for theassessment of types of uncertainties characterized by analeatory nature. Other uncertainties not associated withfrequency of events would be better treated with an expertopinion approach, with a greater reliance on experience andintuition.

There are two main types of uncertainty in geotechnicalengineering – aleatory and epistemic. The former is due tothe random variation of the aspect under analysis, and thelatter to the lack of knowledge of the aspect. Uncertainties arequantified with probabilities, which in turn can be interpretedas frequencies in series of similar trials or as degrees ofbelief. Baecher and Christian (2003) provide a detaileddiscussion on the topic of this duality in the interpretation ofuncertainty and probability in geotechnical engineering,indicating that both types of probabilities are present in riskand reliability analysis and pointing out that the separationbetween them is a modelling artifact rather than animmutable property of nature. Some aspects of geotechnicalengineering can be treated as random entities represented by

relative frequencies, and others may correspond to uniqueunknown events better treated as a degree of beliefrepresented by expert opinion.

Subjectivity associated with probability estimates is a wayof capturing and integrating expert judgment, only some ofwhich may be based on hard data, and is what formalmodeling of uncertainty and risk is about. Analysis, whichmust be based on hard data, is inherently partial and weak.The topic of subjectivity and expert opinion as a key elementof risk and reliability analysis in geotechnical engineering isdiscussed in detail by Vick (2002) and by Baecher andChristian (2003).

The atypical conditions treated with this methodology areanalysed on an annual basis, therefore each year they eitheroccur or do not, and their annual occurrence is determined bythe same underlying probability derived from a common setof conditions judged for the life of mine, either from harddata or from expert opinion or from a combination of both.These conditions suit those of a Bernoulli process andsupport the gradual increase of likelihood of occurrence withtime estimated with the approach.

Given the probability of occurrence of a particularuncertain atypical situation leading to slope failure (Patypical)associated with a defined mine life duration in years (n), theannual probability of occurrence of this situation (patypical)can be calculated with the following expression:

patypical = 1 – (1 – Patypical)1/n [2]

The probability of failure of the slope, given that theatypical conditions occur (PFmodel│atypical), could be evaluatedwith the slope stability model. The results of such analysiscould be expressed as a factor (fatypical) of the modelprobability of failure evaluated under normal conditions. Thisfactor could be the result of sensitivity analysis wheredifferent scenarios of the atypical condition are evaluated.Therefore:

PFmodel│atypical = PFmodel × fatypical [3]

Finally, the probability of failure of the slope due toatypical conditions (PFatypical) can be calculated for aparticular year (i) of the mine plan as follows:

(PFatypical)i = PFmodel│atypical × (1 – (1 – patypical)i ) [4]

The probability of failure of the slope due to atypicalconditions (PFatypical) is added to the model probability offailure (PFmodel) from the geotechnical analysis under normalconditions of design to define the total probability of failure(PFtotal) appropriate for the risk evaluation process. Theaddition of the probability values is carried out with thefollowing generic expression, which is based on the conceptof system reliability:

PFtotal = 1 - (1 – PFmodel) × (1 – PFatypical) [5]

The method of calculation of PFtotal from PFmodel isillustrated with an example where the contributions fromuncertainties related to groundwater, geology, and mining areadded to the PF calculated with the geotechnical model for theslope represented by Section 3 of the mine case shown inFigure 3. Equation [5] can be extended to account for thesethree aspects as follows:

PFtotal = 1 - (1 – PFmodel) × (1 – PFgroundwater) � (1 –PFgeology) × (1 – PFmining) [6]




The results of this analysis are indicated in Table I andFigure 8, and the input probabilities and factors are indicatedin the footnotes to the table. Equations [2], [3], and [4] areused to calculate the terms in Equation [6]. The calculatedcontributions of each uncertain aspect to the PFtotal are shownby the curves in Figure 8.

The uncertainties considered in the example of Table Iand Figure 8 are intended to present the concept of addinguncertainties of random character not included in thegeotechnical models for slope analysis. However, the relevantuncertainties not included in the models need to be identifiedand assessed on a project-specific basis. It may be thatfactors such as unknown stress conditions, actual pitgeometry variations, or other specific situations are the morerelevant aspects that would contribute to the overall PF in agiven project. Also, the best way to treat a particularuncertainty needs to be defined based on its prevalent nature(i.e. aleatory or epistemic).

In the slope stability evaluation process, the considerationof the potential effect of atypical situations leading to failuremeans that no matter how stable a slope might appear interms of the calculated stability indicators, the probability offailure for the risk analysis is never zero and therefore therisk of failure is always present.

Model uncertaintyModel uncertainty in the slope stability analysis can beevaluated through the critical FS value (FScritical) used todefine failure with the model. This type of uncertainty arisesthrough systematic biases in input parameter determinationsand idealizations in the calculation process, leading to theresult that failure occurs for some FScritical value that may notbe unity, as commonly assumed. Bias in parameter determi-nation is inevitable, and is handled by calibration to slopeperformance. Model idealizations arise from simplificationsrequired to represent the geometry, material behaviour, etc.Some aspects of model idealizations will tend to reduceFScritical, while others might raise FScritical. The effect of theparameter bias and model uncertainty is to produce anuncertainty band that is centred on the underlying bias. Anevaluation of FScritical based on the comparison of actuarialfailure rates versus nominal factor of safety was carried out

for the risk study of the Chuquicamata pit as described byTapia et al. (2009). Unfortunately, this approach requireslocal historic records, which are not always available;therefore, judgement as well as reference to similar projects isthe only practical option left to account for this uncertainty.

Estimation of economic impact of slope failure eventsThe economic impact of a slope failure can be measuredthrough the quantification of the effect of this event on thevalue of the mine plan as measured by the NPV. The NPVcorresponds to the cumulative discounted annual profitsduring the life of the mine and is normally defined as theresult of a mining scheduling and optimization processcarried out with specialized software. In general, theeconomic impact of a slope failure is a result of the disruptionof the planned ore feed during the time required to restore thesite, and the additional costs caused by these activities.Figure 9 illustrates the conceptual basis for the estimation ofimpacts of slope failures. The economic impact of a slopefailure is defined as the difference between the NPV ofreference (mine plan without failures) and the re-calculatedNPV incorporating the effects of the failure on production andcost components.

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Table I

Example of estimation of PFtotal

Year Mine plan 2 4 6 8 11 14

Year 2015 2017 2019 2021 2024 2027

Section 3 PFmodel 1.0% 1.3% 1.8% 2.3% 2.8% 3.0%PFgroundwater 0.0% 0.1% 0.2% 0.4% 0.6% 0.8%PFgeology 0.1% 0.3% 0.6% 1.0% 1.6% 2.1%PF mining 0.0% 0.0% 0.1% 0.1% 0.2% 0.3%PFtotal 1.2% 1.7% 2.6% 3.7% 5.1% 6.1%

Notes:Input data on uncertainties:Groundwater: P 10% in 15 years (p annual = 0.70%)

fgroundwater = PFmodel⏐groundwater/PFmodel = 3Geology: P 15% in 15 years (p annual = 1.08%)

fgeology = PFmodel⏐geology/PFmodel = 5Mining: P 5% in 15 years (p annual = 0.34%)

fmining= PFmodel⏐mining/PFMODEL = 2

Figure 8 – Calculated contributions to the PFtotal due to uncertainatypical conditions not included in PFmodel

Production may be disrupted by different factors such asinterrupted access to the mining faces, covered ore, variationsof grade when alternative sources of ore are used to mitigatethe effects of the failure, and so forth. The additional costsare caused by the additional material handling and re-scheduling of equipment required to restore the site affectedby the failure.

A simplified approach to quantifying the impact of afailure consists of calculating the differential NPV due to thefailure, using a cash flow model that includes the estimatedeffects of the failure on production and costs. The impact onproduction is simulated by means of a reduction factor of themined tons, which is estimated by considering aspects suchas the magnitude, location, and time of occurrence of thefailure and the flexibility of the mine plan to providealternative ore feed sources. Engineering judgment andsupporting reference data are normally used to estimate theimpact factors from each failure event.

The simplified cash flow model should include productiondata per mining phase, revenue calculations, as well asoperating and capital costs, and needs to be calibratedagainst the reference NPV in the mine plan. An example ofthe structure of the simplified cash flow model used for thecalculation of economic impact of slope failures is shown inFigure 10. The example illustrated shows that the impact onproduction affects the plant product tons and the revenue,which, together with the additional costs of restoring the site,ultimately reduces the net benefit and consequently the NPV.

One drawback of the simplified approach is that thecomplex effect of variations of the planned grade feed whendrawing from stockpiles cannot be simulated accurately. Forthis reason, the calculated impacts need to be validated withresults derived from a thorough evaluation of selected key



Figure 9 – Conceptual basis for estimation of the economic impact ofslope failure

Figure 10 – Structure of simplified cash flow model for slope failure impact assessment


events in a similar manner as they would be evaluated in areal-life situation, where specific re-designs of the plan wouldbe carried out to minimize the impact of the slope failure.

Risk map for economic impact analysis of slopefailureThe results of probability of failure and economic impactcalculations for individual failure events are used to constructthe economic risk map per year and for the mine life. The riskmap defines the relationship between the probability of aparticular economic impact and the magnitude of that impact;and accounts for different situations of occurrence of eventsin a year, including isolated occurrences, concurrentoccurrences of the different possible combinations of theevents, and no occurrence of any event.

The risk map construction process is based on the conceptof event tree analysis. The event tree is a diagram thatconnects a starting event with the ultimate consequenceunder evaluation through a series of intermediate eventsbased on a cause-effect relation. The events are quantified interms of their likelihood of occurrence, thus enabling theassessment of the final outcomes in terms of theirprobabilities of occurrence. The event tree methodology foreconomic impact, originally described by Tapia et al. (2007)with reference to the case of the Chuquicamata mine and laterdiscussed by Steffen et al. (2008) and in Wesseloo and Read(2009), relies on subjective inputs of probability for theevents in the tree to produce an assessment of the expectedlikelihood of three categories of economic impact (forcemajeure, loss of profit, and minor impact). The maindrawbacks of this methodology are that there is no consid-eration of the possible occurrence of various events in a yearand that the impacts are assessed only in terms of likelihood,without a clear definition of the magnitude of these impacts.

The combined analysis of probability and economicimpacts with event trees is discussed in detail by Baecher andChristian (2003), including examples of consequenceanalysis where the probabilities of events and the respectiveimpacts in monetary terms are multiplied to produce risk costvalues used as a measure of the risks. One drawback of thisapproach is that the outcomes of the analysis do notrepresent actual possible impacts, but rather amountsweighted by the respective probabilities. This characteristic ofthe risk calculation is referred to by Baecher and Chirstian(2003) as ‘risk neutrality’, where high-probability low-consequence outcomes are treated as equivalent to low-probability high-consequence outcomes, as long as theproduct is the same. The reality is that the events either do ordo not occur and consequently the impacts will either becaused or not – intermediate results are not possible.

The proposed risk evaluation approach is carried out witha separate accounting for probabilities and impacts and theend results from the event tree branches are used to constructthe risk map. The method is illustrated in Figure 11 for thesimple case of a pit with two major slopes named East andWest, with PF values of 5% and 10% and impacts of 100 and50, respectively. The sum of the probabilities of the fourpossible outcomes depicted with the tree is 100%, indicatingthat all the possible combinations of events have beenadequately accounted. The risk map constructed with theresults of the event tree is shown in Figure 12. The

cumulative probability curve of particular impacts constitutesthe economic risk envelope of the pit.

The risk map of a more realistic case, such as the mineplan described in Figures 2 and 3, is constructed for theindividual key years selected to represent the various periodsof the mine plan, which are then used to define the overallrisk map for the life of mine, as shown in Figure 13. Thegraph at the top shows the various risk envelopes and thegraph at the bottom shows the details of the failure events ofyear 2019 used to construct the envelope. The risk envelopesare cumulative probability distributions of impacts and areinterpreted as indicated in the graph at the top of Figure 13for the case of impacts with a 10% probability of exceedance.The result for the indicated case would be a 20% probabilityof having an annual impact of at least $160 million over aperiod of 15 years. The display of the individual events in therisk chart is useful to identify critical events causing anincrease of the risk level as measured by the envelope, asdepicted in the example shown on the graph at the bottom ofFigure 13.

Probability concepts for construction of risk mapThe slope failure events considered for the construction ofrisk maps correspond to large-scale failures and are analysed

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Figure 12 – Risk map from results of event tree analysis in Figure 11

Figure 11 – Event tree for economic impact of slope failure of pit withtwo major slopes

on a year-by-year basis. The events are treated as Bernoullitrials and are characterized by a probability of occurrence (p)given by the calculated probability of failure of the slope (PF)and the respective impact (i) estimated in monetary terms.The risk map construction is based on the calculation of theprobability (P) of having an economic impact (I) consideringdifferent possible situations of occurrence of the events asexplained below.

In the following expressions, the terms with sub-indices i,j, and k (in bold) represent the occurring events, and thosewith sub-indices r, s, and t (in italic), refer to the non-occurring events:

➤ Occurrence of single events:Pi = pi × (1 - pr) × … × (1 – pt)Ii = ii

➤ Multiple occurrence of events:Pi…k = pi × … × pk × (1 - pr) × … × (1 – pt)Ii…k = ii + … + ik

➤ A particular case of the multiple occurrence describedin (2) is the occurrence of all the events in a year:

Pi…k = pi × pj × … × pkIi…k = ii + ij + … + ik

➤ No occurrence of any of the events:Pr…t = (1 – pr) × (1 – ps) × … × (1 – pt)Ir…t = 0

The total number of possible cases of occurrence ofevents (T) for (n) independent events in a year effectivelycorresponds to the number of branches of the respective

event tree, and is given by the following expression:

T = 2n [7]

From this number, n cases correspond with theoccurrence of isolated events and one case to the non-occurrence of any of the events. The remaining N casescorrespond with the occurrence of combinations of two ormore events. The generic expression to calculate the numberof combinations (N) of 2 or more events that can be obtainedwith (n) events is:

[8]

or

N = 2n – (n + 1) [9]

The calculation of all possible probability and impactpairs can be done without constructing the respective eventtree, which would be a cumbersome task as the number ofbranches of the tree increases exponentially with the numberof annual events. A summary of the probabilities and impactsof the different possible combinations of 7 events per year ispresented in Table II. In this table, p corresponds with theprobability of occurrence (failure) and q with the probabilityof no occurrence (no failure) of the respective events. Thenumber of cases in Table II is calculated with Equation [8]and the total number of possible occurrences of the 7 eventsis 128. This is the number of data points available toconstruct the risk map as described in the following section.

Construction of the risk mapAn example of the input data required for the construction ofthe risk map is presented in Table III. The data includes theprobability of slope failure and the associated impact of sevensections per year and six years of analysis, on the mine planof 15 years’ duration, as described in Figures 2 and 3. The PFvalues in Table III are based on the results of the geotechnicalanalysis of the respective sections and cater for the atypicalconditions leading to failure discussed previously.

The data in Table III is shown in graphic form in Figure14 to illustrate the variations of the probability of failure andassociated impacts with pit development. The graph at the leftof Figure 14 is consistent with the increasing likelihood offailure of the slopes expected as the pit grows deeper. Thecurves in the graph at the right of Figure 14 do not show aunique trend in the variation of impact with pit growth, asimpacts are dependent on the particular characteristics of oreexposure and ore access during the development of themining phases.

The risk map construction is carried out per year and thedata is used to calculate the pairs of values of probability andimpact associated with all possible combinations of failureevents using the expressions in Table II. The 128 data pairsfor each year of analysis are sorted and used to construct therespective probability distribution graphs of impacts. Thesegraphs include a frequency distribution histogram and thecorresponding cumulative frequency curve as shown in thegraph at the top of Figure 15 for the year 2019 of theexample in Table III.

The risk map result is shown in the graph at the bottomof Figure 15. The graph contains the probability distributionplots with the axes swapped to conform with the typical way



Figure 13 – Example of a risk map for economic impact of slope failure,showing risk envelopes for key years and for life of mine (top), anddetails of events shaping the risk envelope for year 2019 (bottom)


in which risk acceptability criteria is presented, as discussedin the following section. The graph also includes the datapoints representing the various possible occurrences of theevents. The blue data points correspond with isolated events,the green points with the concurrent occurrence of

combinations of events, and the red point on the horizontalaxis represents the particular situation of no occurrence ofany of the events. Not all the data points are visible becausemany of them correspond with low probability values outsidethe range of the logarithmic scale used in the graph.

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Table III

Example of data for construction of the risk map for impact on NPV

Year 2015 2017 2019 2021 2024 2027

LOM year 2 4 6 8 11 14

Section PF % Impact PF % Impact PF % Impact PF % Impact PF % Impact PF % Impact M$ M$ M$ M$ M$ M$

1 0.1 109 0.2 72 0.4 55 0.7 52 1.7 54 3.7 592 0.1 78 0.9 96 5.8 26 11.1 64 17.6 62 23.0 523 1.2 25 1.7 70 2.6 34 3.7 27 5.1 35 6.1 294 3.7 41 5.9 36 8.0 12 10.3 15 13.9 43 16.1 605 0.6 16 5.2 166 9.5 155 12.0 65 15.4 68 19.4 446 0.1 18 0.4 92 1.2 47 2.9 14 7.3 48 10.1 407 0.1 14 0.8 83 2.9 42 6.4 11 9.4 43 12.0 34

Note:– NPV of reference M$ 5.000

Table II

Number of possible cases of occurrence for the situation of 7 events per year

Probabilities and impacts of combination of events

Description No cases P I

Isolated events 7 p1.q1.q2.q3.q4.q5.q6 i12 events 21 p1.p2.q1.q2.q3.q4.q5 i1+i23 events 35 p1.p2.p3.q1.q2.q3.q4 i1+i2+i34 events 35 p1.p2.p3.p4.q1.q2.q3 i1+i2+i3+i45 events 21 p1.p2.p3.p4.p5.q1.q2 i1+i2+i3+i4+i56 events 7 p1.p2.p3.p4.p5.p6.q1 i1+i2+i3+i4+i5+i67 events 1 p1.p2.p3.p4.p5.p6.p7 i1+i2+i3+i4+i5+i6+i7No event 1 q1.q2.q3.q4.q5.q6.q7 0Total 128Notes:Numbers identifying the p, q and i terms in the expressions to calculate P and I should be interpreted as indices that are cycled through the 7 individualevents to generate the number of cases indicated in column 2.p = probability of failureq = probability of no failure = (1 –p)i = economic impact of individual eventP = probability of occurrence of combination of eventsI = cumulative impact of combination of events

Figure 14 – Input data for construction of economic risk map, probability of failure of the slopes (left) and impact on NPV (right)

Nevertheless, these low-probability events have an influenceon the final result, which is captured by the cumulative distri-bution curve. Typically the risk map excludes the frequencydistribution histogram in order to avoid an overcrowdedgraph. A practical way of defining the cumulative distributioncurve of impacts is through a Monte Carlo simulation wherethe seven failure events are modelled with Bernoulli distrib-utions (also called yes-no distributions) and the impactscalculated accordingly.

The probability values given by the risk envelope shouldbe interpreted as probabilities of exceedance of the respectivevalue, as this curve corresponds to a cumulative probabilitydistribution associated with all possible combinations ofevents considered. The risk envelope defines the economicrisk profile for the respective year. The analysis of thepatterns shown by the data points representing theoccurrence of individual events is valuable for identifyingcritical events that push the risk envelope towards the upperright side of the graph. One example of such an event wouldbe the slope failure associated with Section 5 in year 2019 asshown in Figure 15.

The risk envelopes of the six representative yearsincluded in Table III were used to construct the economic riskmap for the life of mine as shown in the graph at the top ofFigure 13. The procedure is based on compounding theprobabilities of the various years for fixed values of impact,considering the periods of the mine life represented by eachyear as shown in Figure 2. The probability values are addedusing the concept of reliability of a system. In this particularexample the probability of an economic impact for the life of

mine (PLOM) for a given impact is calculated from thecorresponding annual probabilities using the followingexpression:

PLOM = 1 - (1-P2015)2 x (1-P2017)2 x (1-P2019)2 x(1-P2021)2 x (1-P2024)3 x (1-P2027)4 [10]

The exponents in this equation correspond to the numberof years represented by the probability value in the respectiveterm. The sum of these exponents is 15 and corresponds withthe total number of years of the mine plan.

A different perspective of the economic risk could beprovided by the analysis of impacts on annual profits,because in this way, future amounts are not discounted topresent values, which in some cases causes a perceiveddistortion of value. Risk maps based on the impacts onannual profits can be calculated following a similar process tothat described for impacts on NPV. Furthermore, the analysiscan be carried out with impacts measured in terms ofcommodity product rather than monetary units, in order toavoid possible distortions caused by the assumptions oncommodity prices.

Uses of the risk mapThere are two main uses of the risk map described in thispaper; one is for the evaluation of a specific open pit designin terms of economic risk by comparing the result withacceptability criteria, and the second refers to the comparativeanalysis of open pit design options, in terms of value andrisk, to identify optimum pit layouts.

Comparison with acceptability criteriaThe risk map can be used to assess a specific pit design bycomparing this result with acceptability criteria specificallydefined for the project. The result of this analysis enables theidentification of the more appropriate risk treatmentstrategies to advance the project. In particular, thecomparison with acceptability criteria is useful for the identi-fication of those years of more relevance in terms of potentialeconomic impacts and the respective critical pit areas causingthose risks. This information is valuable for the definition ofthe areas requiring more investigation in further stages ofstudy and for the evaluation of mitigation strategies to reducethe risks.

Risk acceptability criteria are normally described in theform of a matrix in which risk is categorized in terms oflikelihood of occurrence along the horizontal axis andseverity of the impact up the vertical axis, to define high (H),medium (M), and low (L) risk levels. This type of matrix wasoriginally developed for use in qualitative methods of riskanalysis, with the scales adapted or adjusted to suit differenttypes of application (Joy and Griffiths, 2005). However, amore precise definition of the scales of likelihood and severityresults in acceptability matrices especially suited for the usein quantitative risk evaluation methods such as that based onthe risk map construction described in this paper. Anexample of a risk acceptability matrix is shown in Figure 16,where likelihood and impact categories are defined specif-ically for the project setting at hand. The risk matrix alsoprovides guidelines for risk treatment actions to follow, basedon the risk results.

The use of the risk acceptability matrix in Figure 16 isillustrated in Figure 17, where the risk map results shown in



Figure 15 – Construction of the economic risk envelope for year 2019 inexample from Table III; probability distribution graphs (top) and riskmap result (bottom)


Figure 13 are compared with the acceptability criteria. Thecriteria presented in Figure 16 are intended to adjudicate riskenvelopes of individual years and need to be converted to theappropriate values for the analysis of the LOM envelope. Theconversion is carried out with the same approach used tocalculate the LOM envelope from the annual curves. Thisinvolves adding the annual probabilities using the concept ofsystem reliability, considering a 15-year time span.

In the example presented in Figure 17 the grey curves areincluded for reference but are not intended to be comparedwith the displayed risk zone categories. The evaluation of theindividual years (top graph) indicates a low to moderate riskprofile for all years, with the envelope of year 2019 showinga local elevated risk associated with conditions of Section 5,as depicted in Figure 15. This finding constitutes a pitoptimization opportunity and illustrates the way in which therisk envelopes can be used to identify areas requiringattention in further stages of study. The evaluation of theLOM risk envelope illustrated in the graph at the bottom ofFigure 17 suggests a moderate risk level of the overall mineplan.

Value and risk analysis of design optionsThe risk map can also be used to define risk cost values ofalternative pit slope design options that need to be comparedin terms of economic risk performance. Risk cost values areused to construct the value and risk profile for changing slopegeometries, which provides the elements for screening ofoptions in an early design stage and facilitates the identifi-cation of the main features of pit geometry for an optimumdesign.

Generally, a base case pit slope design is available, whichis the result of conventional slope design methods based onFS or PF criteria, or local experience in terms of slopeperformance in particular geological settings. The base casemine plan typically corresponds with a balanced riskcondition, therefore slope design options on both sides of thebase case are required to define the relationship between theslope angle and the value and risk condition of the pit layout.An example of the construction of alternative pit slopegeometries for the risk analysis from the base case layout isillustrated in Figure 18. In this case the alternative slopedesigns are generated by flattening the base case by 5° andsteepening by 5° and 10°, resulting in nominal slope designangles of 35°, 40°, 45°, and 50° for the east wall and 40°,45°, 50°, and 55° for the west wall.

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Figure 16 – Example of risk acceptability matrix for economic impact(top) and the appropriate risk treatment options (bottom)

Figure 17 – Comparison of risk map in Figure 13 with acceptabilitycriteria in Figure 16, for the evaluation of results of individual years (top)and LOM (bottom) Figure 18 – Example of definition of alternative slope design options

The risk maps for the four alternative pit design optionsare constructed using the respective slope stability resultsand economic impact assessment of slope failures. Anexample of the risk envelopes for the life of mine of the fourslope design options shown in Figure 18 is presented inFigure 19. The risk envelopes are compared with the accept-ability criteria (Figure 16), adjusted for a life of mine of 15years. The graph also includes the risk cost values read fromthe envelopes for probabilities of exceedance of 10%, 50%,and 90%, which are used to assess the options in terms ofvalue and risk.

The risk envelopes in Figure 19 indicate that the basecase -05° (BC-05) is in the low to moderate risk threshold,the base case (BC) and base case +05° (BC +05) are in themoderate risk area, and the base case +10° (BC+10) optionfalls in the high risk area. The comparison with the accept-ability criteria does not provide sufficient elements toestablish a clear contrast between the options in terms oftheir risk performance.

The risk cost values indicated in Figure 19 are used toconstruct the value and risk profiles of the slope design asshown in Table IV and Figure 20. These results show thevariation of value in terms of NPV and risk cost for thevarious slope design angles. The design options have beencategorized in terms of the risk results as conservative,balanced, aggressive, and maximum, for the slope designcases of BC -05, BC, BC +05, and BC +10, respectively. Therisk cost or costs of impact of slope failures have an inverserelationship with the probability of incurring those costs,with higher probabilities of small impacts and lowerprobabilities associated with large impacts.

The graph at the top of Figure 20 shows the typicalincrease of risk cost with increasing slope angle for variouslevels of likelihood of impacts. The risk cost values were usedto construct the NPV with risk curves shown in the graph atthe bottom of Figure 20. This graph shows a steady increasein NPV with increasing slope angle when no risk aspects areconsidered. However, once the risk cost is included in theanalysis, the curve of value shows an inflexion point as theslope steepens, defining the angle that represents theoptimum balance between value and risk. The results inFigure 20 would serve to confirm the adequacy of the basecase design, and would suggest a possible optimizationopportunity by steepening the slopes by up to 3 degrees.

Information such as that included in Figure 20 constitutesa valuable tool to optimize the pit design and to bracket theoverall slope angles for further phases of study.

ConclusionsThe methodology presented provides a rational approach todefining, at an early stage of a mine, the main features of pitgeometry reflecting the appropriate balance between value



Figure 19 – Example of risk envelopes of alternative pit design optionscompared with acceptability criteria and risk cost values indicated forthree levels of likelihood

Table IV

Value and risk cost of the pit design options

Case no. Slope angle Design class NPV (M$) Risk costs (M$) NPV with risk (M$)

option (°) P 10 % P 50 % P 90 % P 10 % P 50 % P 90 %

1 BC –05 conservative 4.935 157 88 53 4.778 4.847 4.8822 BC balanced 5.000 170 115 70 4.830 4.885 4.9303 BC +05 aggressive 5.050 205 160 112 4.845 4.890 4.9384 BC + 10 maximum 5.090 275 230 198 4.815 4.860 4.892

Figure 20 – Risk cost (top) and project value (bottom) variations withslope design angle for risk levels of 10%, 50%, and 90%


and risk, in accordance with the specific conditions of theproject. The process considers both the likelihood ofoccurrence of individual slope failure events and the resultingeconomic impacts from all possible combinations ofoccurrence of these events on an annual basis and for themine life. The economic risk map constructed for a particularpit slope layout can be used in an optimization process bycomparing this result with project-specific acceptabilitycriteria. When the process is used for the evaluation ofalternative design options, the risk maps can be used to makerisk cost estimations to calculate the variation of project valuewith slope angle. These results enable the definition of themain features of pit geometry reflecting the appropriatebalance between value and risk, in accordance with thespecific conditions of the mine, which allows the rational-ization of requirements of geotechnical information atdifferent stages of project development, once risk criteriahave been defined.

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