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Is it necessary to install a downhole safety valve in a subsea oil/gas well?
If the unavailability of a barrier i is expressed by qi(t), which states the probability that basic
event i occurs at time t. Cut set unavailability can be calculated by the use of Equation 5-5:
∨
∏
i∈K
j
Q ( t)
= q ( ) ,
i t
Equation 5-5
Equation 5-5 is also applied when combining tested and non-tested barriers in a cut set.
The approximation formula 4-2
The approximation formula of Equation 5-2 is not always valid. Figure 5-5 illustrates the
relationship between the approximation of Equation 5-2 and the general unavailability formula
in Equation 5-1. By using the approximation formula an error will be generated. This error
increases as λ⋅τ increases (see Figure 5-5). For a λ⋅τ-value of 10 -2 the generated error is 1.67E-
5. The difference will continue increasing as λ⋅τ increases. When λ⋅τ is set to 0.1, which is the
case for some of the barriers in this report, the unavailability difference between the two is
1.67E-3. The use of the thumb rule is applied in the CARA-calculations, but not in the
calculations done by hand.
Approximation of λ⋅τ
2
The approximation
The general formula
Figure 5-5 A plot comparing the approximation and the general formula of unavailability.
Calculation uncertainties
In reliability studies of technical system one always has to work with models of the system. As
a rule to model construction, the models should be sufficiently simple to be handled by
available mathematical and statistical methods [8]. The reliability models are constructed by
applying generic data from existing systems. Models deduced from different formulas and
methods are never 100% correct. The value of expressing reliability values by decimals is
therefore limited. Failure rates and methods applied to determine reliability values have many
uncertainties and exact values are unlikely to be found.
Diploma thesis, NTNU 2002
λ⋅τ
27