Environmental Engineering Reference
In-Depth Information
distribution is then fitted to a collection of similar pipes. The pipe group can then be aged to
assess what the likely future costs of replacement would be. Data stratification is required in
this analysis (Pelletier at al. 2003). For example, it can be necessary to identify the time to
failure between the installation and the first break, between the first and second break, etc. in
order to distinguish between the subsequent break occurrences. This analysis relies heavily
on a proper estimate of the lifetime and is highly dependent on the individual pipe
characteristics.
In the
aggregated (regression) models
, pipes with the same intrinsic properties are grouped
together and then use linear regression to establish a relationship between the age of the pipe
and the number of failures. Shamir and Howard (1979) proposed an exponential model at
which the pipe failure is increased with time (also quoted by Engelhardt et al., 2000):
A
(
t
−
t
)
λ
(
t
)
= λ
(
t
)
e
2.3
o
o
where,
λ(t)
is the average annual number of failures per unit length of the pipe surveyed at
year
t
,
t
0
is the base year for analysis, and
A
is the growth rate coefficient between year
t
0
and
t
. This approach does not provide any information about the variability that may exist
between individual pipes in general. A number of researchers have used the multiple
regressions to improve the above equation to relate the environmental and intrinsic properties
of the pipe. Su et al. (1987) have obtained a regression equation that correlates the failure rate
λ
and pipe diameter
D
using the data from the 1985 St. Louis Main Break Report (also quoted
by Gargano and Pianese, 2000):
0
6858
2
7158
2
7685
2.4
λ
=
+
+
+
0
042
3
26
1
3131
3
5792
D
D
D
where,
D
is the pipe diameter in inches and
λ
is the failure rate in breaks/mile/year.
Probabilistic predictive models
try to predict the probability that a pipe will burst at a
particular moment. This probability can then help to calculate the economic life of the pipe
and therefore predict when it should be replaced. Andreo et al. (1987) proposed the use of the
Cox Proportional Hazard Model
to consider the hazard function to this kind of model. The
basic form of this model is:
zb
h
(
t
:
z
)
=
h
(
t
)
e
2.5
o
where,
h(t:z)
is the failure rate at time
t
related to factor
z
,
h
0
(t)
baseline hazard function,
z
is
a vector of explanatory variables (diameter, soil, etc.), and
b
is a vector of regression
coefficients. These models are more sophisticated but typically lack statistical rigour in their
information and suffer from a reliance on complete long-term failure records.
2.4.2 Lifetime Distribution Models
The failure rates can also be analysed using the
lifetime distribution models
. For a well-
designed and well-installed water distribution network, the series of pipes can be considered
repairable as the cost of failure is small in comparison to the replacement. The main reasons
for this are that firstly, the pipes are mostly constructed under the ground and possible
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