Modeling Uncertainties in Mining Pillar Stability Analysis

Risk Analysis, Vol. 29, No. 10, 2009 DOI: 10.1111/j.1539-6924.2009.01237.x

Modeling Uncertainties in Mining Pillar Stability Analysis

Maxime Cauvin,1,2 Thierry Verdel,1∗ and Romuald Salmon2

Many countries are now facing problems related to their past mining activities. One of thegreatest problems concerns the potential surface instability. In areas where a room-and-pillarextraction method was used, deterministic methodologies are generally used to assess thehazard of surface collapses. However, those methodologies suffer from not being able to takeinto account all the uncertainties inherent in any hazard analysis. Through the practical ex-ample of the assessment of a single pillar stability in a very simple mining layout, this articleintroduces a logical framework that can be used to incorporate the different kinds of uncer-tainties related to data and models, as well as to specific expert’s choices in the hazard orrisk analysis process. Practical recommendations and efficient tools are also provided to helpengineers and experts in their daily work.

KEY WORDS: Hazard and risk analysis; mining pillar stability; Monte Carlo simulations; uncertainties

1. INTRODUCTION

Many countries are facing problems related toabandoned underground mines. In France, iron orehas been industrially mined all around the Lorraineregion (Northeastern France) from the middle of19th to the end of the 20th century. The most com-mon mining method was pillar extraction, but aroom-and-pillar partial extraction method was some-times used under sensitive zones to protect housesand surface infrastructures from subsidence duringthe exploitation. In those areas, some collapses havealready occurred, either during the mining operationor a long time after it. Even though numerous stud-ies have already been carried out, researchers still goon developing and improving their works in order tounderstand the reasons for such collapses.

Today, one of the most important issues of stateorganizations having to manage the consequences of

1 LAEGO—Ecole des Mines de Nancy, Parc de Saurupt, CS14234, 54042 Nancy Cedex, France.

2 INERIS, Parc Alata, BP 2, 60550 Verneuil-en-Halatte, France.∗ Address correspondence to Thierry Verdel, LAEGO—Ecole

des Mines de Nancy, Parc de Saurupt, CS 14234, 54042 NancyCedex, France; tel: +33-38-358-4281; fax: +33-38-353-3849;[email protected].

those past mining activities is to deal with the prob-lem of potential surface instability, that is, evaluat-ing the hazard and then the risk of potential insta-bilities. In the Lorraine region, LAEGO, INERIS,and GEODERIS developed a multicriteria analysisin order to develop a risk map, covering all the ironore mining areas.(1) In the Netherlands, a determin-istic method was applied successfully in the recentpast to investigate the necessity of underground sup-port measures to protect the surface from collaps-ing above shallow abandoned room-and-pillar lime-stone workings.(2) Even though those methods areperfectly efficient, they suffer from not being ableto take into account all the uncertainties inherent togeotechnical problems.

Coming back to Knight in 1921(3) and as saidby Haimes,(4) risk “refers to a situation in which thepotential outcomes can be described in objectivelyknown probabilities” while uncertainty “refers to asituation in which no reasonable probabilities can beassigned to the potential outcomes.” Haimes(4) dis-cussed the different meanings of uncertainty froma literature review covering different areas. Hethen proposed a taxonomy helping in understand-ing and dealing with uncertainty. He first suggested

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1372 Cauvin, Verdel, and Salmon

to distinguish between uncertainties caused by vari-ability (also called aleatory uncertainty) from uncer-tainty due to incomplete knowledge (epistemic un-certainty). The first category refers to temporal orspatial variability and to individual heterogeneities.The second category comprises model uncertainty,parameter uncertainty in used models, and uncer-tainty in decision making from modeling. Similarly,for the purposes of a risk analysis, we also sug-gested(5) to distinguish between four classes of uncer-tainty ranging from a very general to a very specific—uncertainty attributed to (1) the scientific, economi-cal, and politic context of the study; (2) the expertiseapplying on deterministic human choices; (3) the useof models; and (4) the randomness and/or the lackof knowledge on data. In the past decades, litera-ture about the use of probabilistic methods as a wayto take into consideration uncertainties in geotech-nical engineering became more and more compre-hensive.(6,7) Most of the probabilistic studies thathave been undertaken concern slope stability anal-ysis. This geotechnical field indeed offers an efficientframework for the incorporation of uncertainty intoslope design(8) because physical phenomena are rela-tively well known and modeled. Nevertheless, prob-abilistic studies are relatively rare in the mining field.Efforts have been started for the last couple of yearsin New Zealand(9) and in Great Britain(10) after sur-face collapse events in order to determine where fur-ther such collapses are likely to occur in the future.Even though those studies give really interesting re-sults and make it possible to quantify the risk of fu-

Fig. 1. Tributary area theory.

ture surface collapses, they do not take into accountall the different kinds of uncertainties.

The study presented herein focuses on the prac-tical assessment of a single pillar stability in a verysimple mining layout. It provides a methodology toincorporate different kinds of uncertainties relatedto data, models, or specific expert’s choices thatmay help defining the hazard over abandoned under-ground mines and consequently the risk that affectspeople and infrastructures.

2. CLASSICAL DETERMINISTIC APPROACH

Room-and-pillar method of mining is commonlyused to extract ore from horizontally bedded sedi-mentary strata. Pillars are often left intact to providepermanent support to the undermined roof. In sucha context of exploitation, economical considerationsand safety assessment are the two most important is-sues at stake. Over the past few decades, several stud-ies about pillars have been carried out by a numberof researchers and engineers either to determine op-timal design pillar sizes(11,12) or to assess long-termpillar stability.(2)

The conventional methods used to assess the sta-bility of mine pillars are based on deterministic (em-pirical or analytical) approaches. They usually adoptthe safety factor (SF) as an indicator of the stabilityof the pillar. This safety factor is defined (see accu-rate definition in Equation (1) and Fig. 1) as the ra-tio of the pillar strength (R) over the mean verticalstress acting on pillar (S). Theoretically, an SF value

Modeling Uncertainties in Mining Pillar Stability Analysis 1373

greater than 1 means that the pillar is stable, while anSF value lower than 1 means it is unstable. In prac-tice, a threshold value higher than 1 is generally usedat the design level to incorporate uncertainties.

Several methods can be used to determine bothS and R. S is generally calculated using the tribu-tary area theory, which considers the total overbur-den load directly over the pillar and its close vicinity(weight of the black column over the central pillar asshown in Fig. 1) divided by the area of the pillar:

S = γ × A× Da

= γ × (l + bl) × (L+ BL) × Dl × L

, (1)

where l and L are the pillar width and length, bl andbL are the width and length of the gallery, γ is the av-erage density of the overburden, and D is the miningdepth.

Pillar strength can be calculated by laboratorymeasurements on samples or by back analysis of pastcollapses. In the Lorraine region, several field inves-tigations have been made after historical surface col-lapse events and a back analysis of the data led ex-perts to evaluate the long-term strength of the pillarsto be 7.5 MPa.

In the present approach, the mining layout beingstudied is very simple. It consists of a grid of three-by-three rectangular pillars that allows one to takeinto account the mining environment of the centralpillar (Fig. 2). Dimensions of pillars and characteris-tics of the mining area have been extracted from a

Fig. 2. Extract of a mining map and selection of the nine-pillar mining layout.

huge database containing information about all theLorraine mines. The exploitation panel selected forthe purpose of the study is between 130 and 150 mdeep and 500 m wide. Pillars have a 12-by-8 m rect-angular shape, roads are 4 m wide, and the seam is4 m high. The overburden is assumed to have a meandensity of 25 kN·m−3. Using the tributary area the-ory and a constant-strength value of 7.5 MPa, the pil-lar has a safety factor equal to 1.07 and can thus beassessed to be “stable.”

3. TAKING INTO ACCOUNT THEUNCERTAINTY IN INPUT PARAMETERS

Uncertainty about input data can be linked ei-ther to the spatial and temporal natural variability ofmechanical and geological parameters, or to a lack ofknowledge that makes it difficult to assess exactly theparameter values used in models. This article dealswith the second type of uncertainty, usually namedepistemic uncertainty.

In our case of interest, the nine-pillar mining lay-out is narrow enough to assume that the “physical”parameters, as γ or R, are not affected by naturalvariability. They are therefore only affected by un-certainty due to a lack of knowledge.

Geometrical parameters, such as pillar dimen-sions or mining depth, are usually determined frommining article maps (it is usually impossible to checkthe pillar dimensions on site because the mine now


lies under the water-table level) archived by miningcompanies or regional authorities, and that are gen-erally at a 1:5,000 scale. At such a scale, a pencil-line thickness on the map is equivalent to more than1 m in the mine. For classical Lorraine pillars (seeFig. 2) whose dimensions are in the order of 10 m inlength or width, we may then consider that such mapinaccuracy leads to a 10% (1/10 = thickness of theline/mean length) error on the values used for pillardimensions. This obviously leads to significant errorsin the risk analysis process. It has been considered inthis study that uncertainty due to the map inaccuracybelongs to “model uncertainties” as maps are modelsaiming to help engineers to represent reality.

In geotechnical engineering, a very efficient wayof taking into account uncertainty on input parame-ters is to define statistical distribution functions foreach of the parameters.(13) Unfortunately, availableinformation is frequently poor and test data are ofteninsufficient to characterize the statistical moments re-quired in the distribution functions. When test dataare unavailable, those parameters may be estimatedfrom the literature or judgment and experience of ex-perts.(14)

Once distribution functions are defined, MonteCarlo simulations may be used to propagate uncer-tainties through the deterministic model. The princi-ple of such simulations is to take one random value

Fig. 3. Three mining layouts computed from random simulations (SF = safety factor; τ = extraction ratio).

from the distribution of each input parameter and tocalculate the output value of the function. Each cal-culation gives one outcome. A great number of sim-ulations allow one to obtain a frequency distributionof the output values. Mean value, standard deviation,or other statistical moments of the studied parame-ters may thus be estimated. The greater is the numberof simulations, the more precise are the computedstatistical moments.

For this study, a simulation method has beenbuilt, allowing one to define all the parameters by amean value and a standard deviation. This methodhas been developed using the Mathematica software.For the construction of the mining layout, each posi-tion of the corners of the nine pillars is randomly dis-tributed around a mean value. In order to considerthe real mine design, the assumption was made herethat the roads are cut in a straight line. Map inaccu-racy is supposed to lead to a 1-m error on pillar di-mensions. Fig. 3 presents different layouts that canbe obtained from random simulations. For illustrat-ing the difference between them, safety factors (SF)and extraction ratios (τ ) were calculated for a 140-mthick overburden.

In order to assign probabilistic distribution func-tions to the other input parameters, several assump-tions have been made. All parameters have beencharacterized by a mean and a maximum error.


Normal distributions were chosen to fit the param-eters. Such an assumption may be valid for parame-ters that have been determined from back analysis ofseveral past events (as R) or that are considered to betraditional mean values of sedimentary rock proper-ties (as γ ). It is more open to criticism for parameters,as h or D, that are described in the available databaseby a minimal and maximal value. However, we willshow later that, in this study, the choice of a specificdistribution function has only a weak influence on theresult (see Table II). The maximum errors chosen forthe parameters are considered to be the boundariesof the 95% confidence interval of those normal dis-tributions. When no information was available, max-imum errors were roughly assumed to be 10% of themean value. Table I summarizes the statistical valueschosen for each of the input parameters.

Using the previously described method, the sta-tistical analysis of a large number of computed sim-ulations made it possible to evaluate the mean valueand the standard deviation of the safety factor. Fig. 4presents the obtained distribution function. From apractical viewpoint (to evaluate the hazard), it wouldbe of interest to estimate pf, the probability of “de-sign failure,” defined as the probability that the com-puted safety factor is less than the target value of sta-bility, that is, 1:

pf = Prob(SF < 1). (2)

pf in this case is equal to 35%.While using the Monte Carlo method, an impor-

tant parameter to take into account is the number ofcomputed simulations. The greater is the number ofsimulations, the more precise is the computed statis-

Table I. Statistical Values Used forInput Parameters

Mean Maximal DistributionParameter Symbol Value Error Law

Pillar length (m) L 12 1 UniformPillar width (m) l 8 1 UniformLength and width of the gallery (m) b 4 1 UniformPillar height (m) h 4 0.5 UniformMining depth (m) D 140 10 UniformAverage density (MN · m−3) γ 0.025 3.43e-3 NormalLong-term strength (MPa) R 7.5 1.47 Normalk-strength constant (Equation (5)) (MPa) k 5.5 1.09 NormalMining span (m) A 500 (–) ConstantNumber of pillars (–) N 31 (–) ConstantCoefficient of geostatic stress (–) K0 1 0.5 UniformYoung’s modulus of pillar material (GPa) Ep 13 4 UniformYoung’s modulus of wall material (GPa) Ew 7 4 UniformPoisson’s ratio of pillar material (–) νp 0.3 0.1 UniformPoisson’s ratio of wall material (–) νw 0.3 0.1 Uniform

Note: (–), when occurring in the first column, means that the parameter is dimensionless;when occurring in the fourth column, it indicates that such a parameter is a constant.

Fig. 4. Distribution of the safety factor when uncertainties exist ininput parameters.

tical moments but the longer the computation. In thisstudy, a parametric study has been carried out in or-der to investigate this computation time. Fig. 5 showsdifferent numbers of simulations, the correspondingcomputation time, and the corresponding range ofvariation (difference between maximal and minimalvalues) of computed means of safety factors. To getthe latter range of variation, we repeat the simulationprocess several times and we compare the mean val-ues of the safety factors obtained in each repetition.It can be shown that for 30,000 simulations the com-putation time is still reasonable (around a minute)while the range of variation of the computed safetyfactor (mean value) is less than 1% (which is littlecompared to the usual inaccuracy for this parameterin geotechnical engineering, which is often estimatedto largely more than 10%). All this means that anyset of 30,000 simulations will give almost the sameestimate of the mean value of the safety factor.


Fig. 5. Computation time VS log (number of simulations) and range of variation of computed safety factor VS log (number of simulations).

4. TAKING INTO ACCOUNT MODELUNCERTAINTY

Uncertainty in risk analysis can also be at-tributed to some of the expert’s choices. Assessing arisk generally requires using specific methodologiesand mathematical models. For example, in our cur-rent example of pillar stability assessment, the choicehas been made to compare the strength of the pillarto the stress acting on it. Two models have thus beenused to estimate both of those parameters. Actually,models are no more than abstractions of the stateof nature, and no matter how sophisticated they are,they are unable to capture nature entirely. The mosteffective approach to estimate this “model uncer-tainty” would be to rely on field observations and usedatabases to do back analyses. However, in geotech-nical studies, it is usually very difficult to have histor-ical observations to compare with results of modelpredictions, and most of the time, only the uncer-tainty related to the choice of specific models by ex-perts is taken into account.

In a study by Husein Malkawi et al.,(15) un-certainty associated with the use of different slopestability models was addressed by evaluating therelative performance of some simplified methodscompared to the most rigorous and accurate one. Inthis study, it has been decided to consider models asinput parameters. Such a choice allows one to takeinto account both the existence of several mathemat-ical models and the inability of the analyst to identifythe best one.

In the previous part, two mathematical modelshave been used: the tributary area method has beenchosen to calculate the stress acting on the central pil-lar of the mining layout, and a constant value model

has been used to evaluate the strength of a pillar.Nevertheless, even if both of the models are reallyconvenient because simple, they also have disadvan-tages.

The two main drawbacks of the tributary areamethod are that (1) it makes the assumption thateach pillar is carrying the load of a vertical rock col-umn over it, which may only be valid for centralpillars from large horizontally stratified mining envi-ronments, and (2) it does not take into account thenature of the overburden. Even though numericalsimulations such as finite element methods are todaythe most common methods used to compute stressesacting on pillars, they can only be used for specificsites where geology and mining geometry are known.Nevertheless, for the purpose of this study, whoseaim is to be as general as possible, it has been cho-sen to use results from Coates’s work about stressesacting on pillars.(16) His approach makes it possibleto take into account the extent of the mined area,the stress component parallel to the seam, the rela-tive deformation properties of pillar, roof, and floorrocks, and the positions of the pillars in the miningzone. The general solution for the mean pillar stressbecomes:

S2 =

γ D

⎛⎜⎜⎝1 +

2τ − K0hA

(1 − 2νw)(1 − νw)

− νp

(1 − νp)K0

hA

Ew

Ep

hA

Ew

Ep+ 2(1 − τ )

(1 + 1

N

)+ 2

lτA

(1 − 2νw)(1 − νw)

⎞⎟⎟⎠

(3)

with τ = 1 − Ll(L+ b)(l + b)

, (4)


where τ is the extraction ratio, K0 is the coefficientof geostatic stress, A is the extent of the mining area,N is the number of pillars, and Ep, Ew, and νp, νw

are the Young’s modulus and Poisson’s ratios of pil-lar and wall (roof and floor) materials.

Table I presents the values chosen for all the pa-rameters. Some of these had been extracted from lo-cal databases and some are roughly estimated.

In our case study, the use of the tributary areatheory leads to a stress value of 7 MPa while Coates’sformula gives a value of 6.8 MPa. It confirms the gen-eral belief that the tributary area theory is a ratherconservative approach.

Concerning the strength of a pillar, it is now acommon assumption that it depends on the strengthof the material of which it is composed as well asits dimensions, and more specially its width-to-heightratio.(11,12) Using Van der Merwe’s method appliedto a very extensive database of failed and stable Lor-raine iron ore pillars, it was found that the strengthof the pillars can be expressed as follows:(17)

R2 = k

√w

h(5)

with w = 2Ll

L+ l. (6)

Here, w is the equivalent width that has to beused for rectangular pillars and k is a constant valuethat can be related to the strength of the pillar ma-terial. The determination of k assumes that, on aver-age, pillar failure occurs for a safety factor equal to1.

Using Equation (5), the strength of the studiedpillars equals 8.5 MPa. This value is only slightly big-ger than the constant value of 7.5 MPa.

The different physical models have been imple-mented in the simulation. Basically, as for the inputparameters, one random model is chosen among thetwo that exist to evaluate the strength of the pillarand the stress acting on it. The computation is thendone following the same procedure as described inSection 3. A degree of confidence can be chosen formodels allowing one to take into account decisionsof experts as well as historical considerations. For ex-ample, if in a panel of five experts, three think that itis better to use a constant-strength value to expressthe strength of the pillar, the strength model usedin the computation should have a probability of 3/5(0.6) to be this one. In the Lorraine region the tribu-tary area theory was originally used by miners to de-sign the underground workings. It thus seems logical

Fig. 6. Computed safety factors, taking into account the existenceof different models.

Fig. 7. Distribution of the safety factor when uncertainties existboth in input parameters and in the models.

to preferably use this formula to assess the risk of pil-lar failure. Fig. 6 shows the results obtained from thesimulation, assuming that the input parameters areperfectly known. It has been decided here to assigna confidence degree of 75% for the tributary areatheory while the models for the pillar strength areequally “trusted.”

It can then be shown in this case that the combi-nation that gives the lowest safety factor (1.07) is theone that has been presented before.

Fig. 7 presents the distribution function obtainedfor the safety factor when uncertainties in input pa-rameters and models are simultaneously taken intoaccount. In comparison with the previous results, theprobability of “design failure” decreases from 35 to24%.

5. ANALYSIS OF VARIANCE

Engineers and experts ask today for efficienttools that can help them to solve the tough dilemmathey face. They indeed have to give quick and accu-rate answers to problems full of uncertainties. One of


Table II. Relative Influence of theDifferent Sources of Uncertainty on the

Computed Safety Factor

Variance of theComputed Safety Factor

Variance/Source of Concerned Maximal Normal Uniform TotalUncertainty Parameters Error/Mean Distribution Distribution Variance (%)

Map inaccuracy L, l, b, h ∼0.10 13.03e-3 16.73e-3 35Use of database D 0.07 1.81e-3 2.32e-3 5Expert-made γ , K0, μ, E ∼0.10 6.88e-3 6.75e-3 15

hypothesesBack analysis R, k 0.19 13.65e-3 13.47e-3 31Models (–) (–) 5.92e-3 14

their greatest needs is thus to reduce the uncertaintyof the results they provide (in this study, the stan-dard deviation of the calculated safety factor (Fig. 7)is about 0.21) in a rapid and cheap way. For time andeconomical reasons, it is obvious that engineers can-not study all the sources of uncertainty but can onlytry to focus on some of them. The study presentedherein provides a tool that makes it possible to iden-tify the parameters whose uncertainty has the great-est impact on the result.

Table II presents the relative influences of fivedifferent sources of uncertainty on the total varianceof the computed safety factor. The map inaccuracyand the use of a database of expert-made hypothesesor of models (as back analysis) to estimate param-eter values may have impacts on the input param-eters. But, as presented in Section 4, the choice ofspecific models to deal with a problem may also haveimpacts on the result. In order to obtain the results inTable II, each source of uncertainty has been inves-tigated separately. It should be noted here that inputparameters have been considered to be independentof each other and that pillar strength and stress mod-els have been equally “trusted.” Normal and uniformdistributions have been used in the Monte Carlo sim-ulations to fit the input parameters. As the varianceof the computed safety factor is only slightly big-ger when uniform distributions have been chosen forall the input parameters (in this case, pf = 23%)than when normal distributions only have been used(pf = 21%), it appears that, in our study, the choiceof a specific distribution function has only a weak in-fluence on the result (a difference of 2% being con-sidered as a little value in geotechnical engineering).It can also be highlighted that in our case of inter-est, almost one-third of the total variance of the com-puted safety factor is due to the map inaccuracy andone other third of the total variance is due to thedifficulty of calibrating model parameters from backanalyses. The practical meaning of this is that if the

expert wants to reduce the variability of the result,and then to be more precise in his or her analysis,he should concentrate on providing a more accurateestimation of pillar dimensions (e.g., through in situmeasurements) and of model parameters (throughthe adding of new data to the studied database).

6. INTERPRETATION OF THE RESULTS

Another question that arises from the analysisconcerns the interpretation of the results presentedas a probabilistic number. What does it indeed meanfor a pillar to have a probability of “design fail-ure” of 24 or 35%? Hoek considers that for a slopestability analysis, a probability of failure (computedin the same way as our probability of “design fail-ure”) of 6.4% “means that, for the combination ofslope geometry, shear strength, water pressure andearthquake acceleration parameters assumed, 64 outof 1000 similar slopes could be expected to fail atsome time during the life of the slope. Alterna-tively, a length of 64 m could be expected to failin every 1000 m of slope.”(18) He thus interpretsthis probabilistic figure as a temporal or a spatialfrequency.

However, as state Baecher and Christian,(7) thereal meaning of the result “depends on how the mod-elling assumptions are made; specifically, on how thetotal uncertainty is divided between aleatory (tem-poral or spatial) and epistemic (parametric). To theextent that all the uncertainty is assumed aleatory,the probability refers to a temporal or spatial frac-tion. To the extent that all the uncertainty is assumedepistemic, the probability refers to a chance of com-plete failure.” In our mining layout, all the uncertain-ties are related to a lack of knowledge: all the pa-rameters have a fixed and constant value but we donot know them exactly. Therefore, the uncertaintyin the result is also related to a lack of knowledgeand no other considerations—as Hoek’s frequency


Fig. 8. Two distributions of safety factor having the same proba-bility of “design failure.”

interpretation—are valid. Probabilities in this caseare just a tool to quantify the epistemic uncertainty.

The practical interpretation of the probability of“design failure” is rather delicate. This number mustbe considered only as a complement to the informa-tion brought by the deterministic value of the safetyfactor. Fig. 8 presents, for example, the results of twodifferent analyses that have been undertaken in twodifferent mining areas. As the distribution presentedin Fig. 7, the “dark” distribution has a mean safetyfactor of 1.16 and a probability of design failure of24%. The “lighter” distribution has also a probabil-ity of “design failure” of 24% but it has a mean safetyfactor of 2. From a practical point of view, the con-clusions an engineer can make will be very differentin the two cases. A safety factor of 2 associated witha significant probability of “design failure” will high-light the fact that, even though the pillar might bestable, we are not very confident in the result. Thedata are so imprecise that we have to keep in mindthat a chance of complete failure still exists. In thiscase, more accurate data have to be integrated in themodel in order to better define the result. On theother hand, when the safety factor is very close to 1and the probability of “design failure” is low, we haveto balance the benefits that more accurate data maybring to the accuracy of the result against the cost ofsuch precisions.

7. CONCLUSIONS AND PRACTICALRECOMMENDATIONS

This article provides a logical way to take intoaccount several kinds of uncertainties that can be en-countered in a risk analysis study. Defining statistical

distribution functions for input parameters may helpengineers to deal with the problem of “data uncer-tainties” inherent to the study of a natural system.It can also be pointed out here that the definition ofthe characteristics of those distribution functions re-quires the expert to make some choices (e.g., eval-uation of mean values and maximal errors), so thatthe “expertise uncertainty” is also involved. Choiceshad also to be made regarding the nature of thedistribution functions fitting the parameters. In ourstudy, both normal and uniform distributions havebeen used and it has been shown that such a choicehas only a weak influence (around 10%) on the es-timation of the probability of failure, compared touncertainty usually considered or accepted in suchgeotechnical problems, which may often reach 50%in deterministic analysis. More generally, this workmay guide engineers through their analysis process ingiving them a methodology to follow. It thus allowsto break free from the individual ability or from thedeterministic choices of the expert in charge of theanalysis.

As has been said in Section 1, the use of prob-abilistic methods in geotechnical engineering hasbecome more and more common in the last fewdecades. However, confusion still exists on the realmeaning of the results of such studies, and we are of-ten tempted to interpret a probabilistic number thatquantifies a lack of knowledge as a spatial or tem-poral variability. In this study, which only deals withepistemic uncertainty, practical tools have been in-troduced that allow experts to highlight situationswhere more accurate data have to be provided inthe analysis. The taking into account of uncertain-ties, through the use of probabilities, thus allows oneto concentrate on the scatter of the data. Therefore,it has to be considered as a communication tool be-tween the engineer and the stakeholders. It is now upto them to decide whether the situation is acceptableor not, and whether uncertainty in the result has tobe reduced or not.

However, the interpretation of the probabilisticresults as spatial frequencies can be possible in somecases. We can assume, for example, in a large ho-mogeneous mining area consisting of a large num-ber of similar pillars, that the pillar dimensions andthe physical parameters are affected by spatial vari-ability. Such assumptions may be justified by themining extraction methods and the geological con-ditions. The probabilistic result will then also rep-resent “partly” a natural variability. The problem isthat, as Baecher and Christian state, “almost always,


the uncertainty is apportioned between aleatory andepistemic, so the probability itself is a mixture.”(7) Inevery risk analysis, important care has to be takenwith both the hypotheses that are made and the in-terpretations, sometimes too fast, that we can makeabout our results.

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Modeling Uncertainties in Mining Pillar Stability Analysis

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