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International Journal of Rock Mechanics & Mining Sciences 38 (2001) 735744

Technical Note

Likelihood statistic for interpretation of the stability graph foropen stope design

F.T. Suorineni a, *, P.K. Kaiser a , D.D. Tannant ba Mirarco/Geomechanics Research Centre, Laurentian Uni versity, Sudbury, Ont., Canada P3E 2C6

b Department of Ci v il and En v ironmental En g ineerin g , School of Minin g and Petroleum En g ineerin g , Uni versity of Alberta, Edmonton, Alta.,Canada T6G 2G7

Accepted 20 May 2001

1. Introduction

The stability graph method for open stope design is anempirical method, and its interpretation is highlysubjective. Subjective interpretation of the stabilitygraph has resulted in unknown risks from human biasand inherent errors. Users of the stability graph methodfor open stope design are for example, given the wrongimpression that if a stope plots in the stable zone, thatstope is denitively stable and its performance in servicepresents no risk of instability. Statistical tools exist thatcan be applied to interpret the stability graph andsignicantly minimize the subjectivity in the stabilitygraph method without making it seem more rigorousthan it is currently perceived. This paper identies theBaysian likelihood method as a powerful tool for astatistical interpretation of the stability graph, and usesthe extended database based on the Potvin [1] calibratedstability graph factors to illustrate the method and its

benets. Mathews and his co-workers [2] in GolderAssociates introduced the stability graph method of open stope design in 1980. The stability graph is a plotof a stability number N against a shape factor HR . Thestability number and shape factor are dened in Eqs. (1)and (3) respectively:

N 0 Q 0ABC ; 1

Q 0 RQD

J n

J rJ a

J w ; 2

where Q 0 is the modied tunnelling quality index [2]. Q 0

is obtained from Barton et al. [3] rockmass quality index

Q by setting the stress reduction factor SRF to 1 asshown in Eq. (2). The water reduction factor J w is oftenset to one in Canadian conditions (dry hard rockunderground mines). In other mining environments,where this is not the case, the relevant J w values shouldbe applied. The parameters A, B and C are the stressfactor, joint orientation factor and gravity factorrespectively. The shape factor HR is dened as thehydraulic radius:

HR Area

Perimeter: 3

Fig. 1 is the stability graph. The stability graph isdivided into three zones labeled stable, unstable andcaving zones. These zones were originally dened byvisually tting boundaries between clusters of datapoints representing stable and unstable stope surfaces,and between the unstable and caved stope surfaces. Thezone enclosed by the boundaries is the transition zone.The transition zone contains a mixture of stable,unstable and caved stopes. A stope surface that plotsin this zone may be stable, unstable or cave in service.The transition boundaries in Fig. 1 are eye-balled.The problem with visual demarcation of zones is theirsubjective nature and reproducibility in future analysisand the unknown risk associated with inherent errors.

Since 1980, the boundaries originally dened byMathews et al. have changed considerably (see [4]).The changes are ascribed to the following factors:

* accumulation of more data with time,* re-denition of the stability number by various

authors,* calibration of the boundaries to specic local mine

site conditions,

*Corresponding author. Tel.: +1-705-675-1151; fax: +1-705-675-4866.

E-mail address: fsuorineni@mirarco.org (F.T. Suorineni).

1365-1609/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved.PII: S 1 3 6 5 - 1 6 0 9 ( 0 1 ) 0 0 0 3 3 - 8