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Is it necessary to install a downhole safety valve in a subsea oil/gas well?
Tested
barrier 1
Tested
barrier 2
R (t) = R (t) + R (t) - R (t) ⋅ R (t) = e
Figure 5-6 The parallel structure of two tested barriers
s
1
2
1
2
+ e
− e
−λ
1t
−λ2t
−(
λ1
+ λ2
) t
The reliability function is applied to Equation 5-1 and determines the unavailability.
∨ 1 τ
1 τ
−λ1t
MFDTS
( t)
= Q1(
t)
= 1-
∫ RS
( t)
dt = 1-
∫ ( e + e
τ 0
τ 0
1 −τλ
1
1
1
−τλ2
1−
( ( 1−
e ) + ( 1−
e ) − ( 1−
e
λ τ
λ τ
( λ + λ ) τ
1
2
1
2
−λ2t
− e
−(
λ1
+ λ2
) τ
−(
λ1
+ λ2
) t
))
) dt =
Equation 5-6
The {MV, WV} cut set is calculated with λWV =1.7E-6 hours and λMV=2.0E-6 hours. Applying
these values in Equation 5-6 obtain the unavailability, 2.16E-5
Ex. 2 Unavailability of two non-tested barriers
Unavailability of two non-tested barriers is calculated by the use of Equation 5-3 and Equation
5-5. The unavailability of barriers 1 and 2 is determined by the use of Equation 5-3. Than the
results are applied in Equation 5-5.
−λ1t
q ( t)
= (1-
e ) ,
1
q ( t)
= (1-
e
2
−λ2t
)
∨
Q(
t)
= (1-
e
− 1t
) ⋅ (1-
e
λ −λ
2t
)
Equation 5-7
The {AMVEXL, Tub} cut set is calculated with λAMVEXL =0.6E-6 hours and λTub=0.4E-6
hours. Applying these values to Equation 5-7 obtain the unavailability, 3.88E-3.
Ex. 3 Unavailability of a combination of two tested and one non-tested barriers
A combined cut set of two tested and two non-tested barriers is calculated by the use of
∨
Equation 5-5. Q ( ) represents the unavailability of the two tested components, calculated in
1 t
ex.1 and 2( ) represents the non-tested barrier. The unavailability of the combined cut set is
calculated
t Q∨
∨
comb(
∨
i
∨
1
∨
2
i∈K
j
Q
t)
= ∏ Q ( t)
= Q ( t)
⋅Q
( t)
Equation 5-8
Reliability data is inserted to the {DHSV, MV, SWAB} cut set, where λDHSV =2.8E-6 hours
and λMV=2.0E-6 hours and λSWAB=2.2E-6 hours. Applying these values to Equation 5-8 obtain
the unavailability, 8.92E-6.
5.4.4 CARA calculation results
The results of the unavailability calculations done in CARA are presented here. The
unavailability of the ‘TOP’-event for a well with and without a DHSV is presented in Table
5-1. Calculations of the well with and without an x-mas tree are also included. The calculations
of a well without an x-mas tree reflect a situation where the x-mas tree is unavailable as a
Diploma thesis, NTNU 2002
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