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on the subjective expert opinion of a group of geologists, using qualitative observations of the rate of cavity collapse in paleokarst, and reasoning from rock mechanics principles. The study concluded that the probability of tailings release due to embankment failure is of the order 5 10 7 , and due to liner failure, of the order 2.5 10 6 . Thus, the incremental probability of failure of the tailings embankment due to karst is small compared
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18 15 12 Frequency 9 6 3 0 0 5 10 15
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Length (m) 40
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0 0 30 60 Length (m) 90 120 150
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Figure 22.7 Exponential distribution for Karst analysis. (Van Zyl, D., Miller, I., Milligan, V. and Tilson, W. J., 1996, Probabilistic Risk Assessment for Tailings Impoundment Founded on Paleokarst, Uncertainty in the Geologic Environment, Madison, WI, pp. 563 585, reproduced by permission of the American Society of Civil Engineers.)
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with commonly accepted probabilities of failure of other water retaining structures such as earth dams (i.e. 10 4 ).
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22.5 Simulation Approach to System Reliability
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Event and fault trees approach system reliability by developing a logical structural model of the system built around events. A geotechnical structure such as a dam, foundation,
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Rate of Aqueous Radionuclide Exposure at Waste Package Inner Container Failure Description
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Figure 22.8 Factors contributing to repository simulation. (Miller, I. and Kossik, R., 1996, Probabilistic Simulation of Geologic Waste Disposal Facilities Using the Repository Integration Program (RIP), Uncertainty in the Geologic Environment, Madison, WI, pp. 944 964, reproduced by permission of the American Society of Civil Engineers.) (Also Miller et al. 1992.)
SYSTEM RELIABILITY ASSESSMENT
System Parameters
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Figure 22.9 Interaction diagram for repository simulation. (Miller, I. and Kossik, R., 1996, Probabilistic Simulation of Geologic Waste Disposal Facilities Using the Repository Integration Program (RIP), Uncertainty in the Geologic Environment, Madison, WI, pp. 944 964, reproduced by permission of the American Society of Civil Engineers.) (Also Miller et al. 1992.)
or tunnel is represented not through models of engineering mechanics but in an abstract relationship of events. Probabilities are assigned to these events in one manner or another, and then a calculation is made combining these probabilities with the logic-tree structure assumed to apply to the events (i.e. the event or fault tree), and an analytical probability of system failure is calculated. A limitation of this approach is that, for complex systems, the logic tree becomes large and messy. Similarly, the analyst may overlook important interrelationships or functional dependencies among events not obvious by inspection, even to the trained observer. Simulation approaches to system reliability use a mathematical model of the physics (in our case, engineering mechanics) of the system as a vehicle for experimentation. The simulation is subject to a wide range of parameter values and boundary and initial conditions and is run many times to assess system performance. Incorporating randomness in the parameter values and boundary and initial conditions leads to variability in system response. A statistical sampling approach is used to draw conclusions about the reliability of the system. Often, uncertainties in the models used to calculate performance (i.e. model
0.01 0.01 0.008 0.006 0.004 0.002 0 1000 0 200 400 600 800 1000 0 200 0 400 600 800 0.002 0.004 0.006 0.008 0.01
(Note: probability density bars are shaded to show confidence bounds. The middle bar is the best estimate of the density). 1.0 0.8 0.6 0.4 0.2 0 1000 0 200 400 600 800 1,000 Realizations 1000 0.8 0.6 0.4 0.2 0 0 200 400 600 800 10,000 Realizations 1000 1.0
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400 600 800 100 Realizations
Figure 22.10 Output of repository simulation. (Miller, I. and Kossik, R., 1996, Probabilistic Simulation of Geologic Waste Disposal Facilities Using the Repository Integration Program (RIP), Uncertainty in the Geologic Environment, Madison, WI, pp. 944 964, reproduced by permission of the American Society of Civil Engineers.) (Also Miller et al. 1992.)