Page 120 - Reliability and Maintainability of In service Pipelines

Time-Dependent Reliability Analysis 109

Let dtðÞ denote the deterioration at time t, t $ 0, and let the probability density
function of dðtÞ, in accordance with the definition of the gamma process, be given
by

f dtðÞ dðÞ 5 Ga d αðtÞ; λÞ ð4:11Þ

with mean and variance as follows:
α tðÞ
Ed tðÞÞ 5 ð4:12Þ
ð
λ
αðtÞ
Var d tðÞÞ 5 ð4:13Þ
ð
λ 2
A pipe is said to fail when its corrosion depth, denoted by dtðÞ, is more than a
specific threshold (a 0 ). Assuming that the threshold a 0 is deterministic and the
time at which failure occurs is denoted by the lifetime T. Due to the gamma dis-
tributed deterioration, Eq. (4.11), the lifetime distribution can then be written as:
ð
a 0 Γðα tðÞ; a o λÞ
FtðÞ 5 Pr T # tÞ 5 Pr dtðÞ $ a 0 Þ 5 f dtðÞ dðÞd d 5 ð4:14Þ
ð
ð
0 Γðα tðÞÞ
where Γ ν; xÞ 5 Ð N ν21 2t
t
e dt is the incomplete gamma function for x $ 0 and
ð
t5x
ν . 0:
To model corrosion in a pipe, in terms of a gamma process, the question that
remains to be answered is how its expected deterioration increases over time. The
expected corrosion depth at time t may be modeled empirically by a power law
formulation (Ahammed and Melchers, 1997):
α tðÞ 5 ct b ð4:15Þ
for some physical constants c . 0 and b . 0.
Because there is often engineering knowledge available about the shape of the
expected deterioration in terms of the exponential parameter b in Eq. (4.15), this
parameter may be assumed constant. The typical values for b from some exam-
ples of expected deterioration according to a power law are presented in
Table 4.1.
The reliability analysis approach which is developed in this section by using
the gamma process concept is entitled the “Gamma Distributed Degradation,
GDD” model.
In the event of expected deterioration in terms of a power law (i.e., Eq. 4.15),
the parameters c and λ can be estimated by using statistical estimation methods.
The estimation procedure is discussed for the two scenarios including a case with
available corrosion depth data and a case of unavailability of corrosion depth
data.