Most oil storage tank failures
result from defects like corrosion pits and flaws on the bottom
plates.^{1} Since oil storage tanks in the petroleum
and petrochemical industries are usually
made from welded steel plates, defects in the tanks should be
detectable during the welding process or in storage service
phases.^{2} But even with efforts at advance flaw
detection, there is still much complexity and randomness
associated with tankbottom corrosion. This makes it necessary
to use probability methods in the reliability analysis and
life/mean time between failures (MTBF) calculation for tank
bottoms with corrosion pits.
Probability model.
The structure of a steel tank
bottom can be considered as a series system from the point of
view of the failure mechanism, which means that all plates must
be in a normal state in order for the bottom to function
properly. The tank fails when any one of its bottom plates has
the first perfected pit. This model is called a series model or
weakest link model.
Then, for a tank bottom made of
m plates with the reliability
R_{i}(t) for the ith plate, its
reliability, R_{T}(t), is the probability that
all plates simultaneously survive to time t and can be
expressed under the independence assumption as
(1)
The failure or hazard rate for a
series system can also be expressed by an exponential
distribution, where l_{T}(t) is the
failure rate of the tank bottom,
l_{i}(t) is the failure rate of the
ith plate.^{3}
(2)
In another way, the reliability of
a tank mostly depends on the corrosive statuses on its bottom
plates. When one plate has defects, the applied stresses or
corrosives will increase the size of these defects and,
ultimately, failure occurs when the size of any one defect in
the plate reaches a critical value. Usually, the defect growth
is the main cause of failure and a defect with the least
resistance to the applied stress or corrosives will be the
first to fail. In this case, the reliability of the tank bottom
of a steel tank will be
(3)
where R_{i} is the
reliability of the ith plate of the tank bottom. It
shows that it is an extreme value distribution.
Therefore, the reliability of a
steel tank could be calculated by combining an extreme value
distribution and an exponential distribution in the
weakestlink or series model.
Calculating tankbottom reliability.
According to extreme value theory,
the tank bottom will fail as soon as only one defect penetrates
the plate of the tank bottom. When there are n
corrosive pits or defects in a tank bottom, the
corrosiveresistance function can be expressed as
(4)
where Z is the surplus
thickness of any one of the tankbottom plates,
T_{b} is the original thickness of the bottom
plates, a_{i} is the depth of the ith
corrosive pit or defect, t is the time before failure
of the tankbottom, t_{i} is the time to
failure of the ith defect.
Applying the concept of smallest
extreme value distribution,^{3} the probability of
failure for the tankbottom is
(5)
where F(z) or
F(t) is the cumulative distribution of defects. For
simplicity, assume that n r ∞ and then:
(6)
It is evident that the time of the
plate penetration is proportional to the difference between the
plate thickness and the initial depth of defects and there
is:
(7)
where T_{b} is the
thickness of tankbottom plates, mm; k is the
corrosion rate of the tank bottom, mm/year and
a_{i} is the original depth of the
ith defect.
Assume that the probability density
function for the depth of corrosive defects be
(8)
where a is the depth of
defects, mm, and ā is the average depth of all
defects in a tank bottom, mm.
The probability for the failure of
the corrosive defects is
(9)
Thus, the reliability of a tank
bottom is
(10)
The previous formula shows that the
reliability of a tankbottom has a correlation with several
factors—such as the number and mean depth of defects, the
rate of corrosion and the original thickness of the bottom
plates—and that it declines with the tank’s service
time or MTBF in a double exponential function.
Analyzing tank bottom reliability.
The reliability of a tank bottom
can be shown and perceived more directly by visualizing its
calculation in MATLAB.^{4} By substituting the
different values of the number and the mean depth of defects,
the corrosion rate and the plate thickness of the tank bottom
into Eq.10, the reliabilities of the tank bottom with very
different states of defects and plates can be calculated and
shown in Fig.1 through Fig. 4. These figures clearly show that
reliability drops rapidly with the increase of the
defectnumber and the corrosive rate as well as the mean depth
of defects. This is especially true when the service time is
more than 20 years (as shown in Fig.1, Fig. 2 and Fig. 3). It
should be noted that an increase of the bottom plate’s
thickness of just 1 millimeter or 0.5 millimeter will greatly
upgrade the reliability as well as prolong the service time or
MTBF of a tank. From the curves in Fig. 4, it can be seen that
an increase or decrease of 0.5 millimeter of the bottom
plates’ thickness makes the reliability vary greatly at
the same service time of the tanks.

Fig. 1. Reliability
examined with different
numbers of defects in a tank bottom.


Fig. 2. Reliability
considered with different
tankbottom corrosion rates.


Fig. 3. Reliability examined with
different
average depth of the defects in a tank
bottom.


Fig. 4. Reliability
comparison of different
thicknesses for tankbottom plates.

Life/MTBF for steel storage tanks.
The life/MTBF of a steel tank can
be calculated by rearranging Eq. 10, which results in
(11)
The previous equation indicates
that the life or MTBF for a tank bottom is a complicated
function of reliability, which is affected by the number and
depth of defects and the thickness of the bottom plate, while
there is a simple linear relation between the life or MTBF and
the corrosive rate of the tank bottom.
Ordinarily, the thickness of
tankbottomplates can be designed and the reliability
predicted in advance. The corrosion rate of a tank bottom can
be evaluated by its surrounding conditions, such as the
tankbase humidity and the water content of oil stored. Thus,
the life of a tank bottom will be decided mainly on the number
of defects, which can be investigated in periods during the
inspection of tanks.
According to the sizes of most oil
storage tanks in the industry, set the reliability at 95% and
the bottom plate’s thicknesses and corrosive parameters
into two groups: one with thinner bottom plates and another
with thicker ones, which respectively are:
Group A: T_{b} = 5 mm, 6 mm,
7 mm; k = 0.10, 0.15 mm / a; a = 1.0
mm;
Group B: T_{b} = 8 mm, 10
mm, 12 mm; k = 0.15, 0.2 mm / a; a =
2.0 mm.
With this established, the life or
MTBF of these tankbottoms can be calculated and visualized
with different corrosive parameters as shown in Fig. 5 and Fig.
6. The curves show the MTBF descends at the initial of the
defects arising in the tankbottom plates with whatever
thicknesses much more than at the later time. It is during the
initial arising of several defects in the bottomplates that
for the number of defects increase by one or two will cause
MTBF to reduce greatly.

Fig. 5. MTBF of tank
bottoms with thinner
plates.


Fig. 6. MTBF of tank
bottoms with thicker
plates.

Further, the case that the life/MTBF of a tankbottom change
with its reliability required can be demonstrated in the
following examples of calculations. Take two examples: one is
the tank with T_{b} = 5 mm, k = 0.1
mm / year, ā = 0.5 mm and another is the
tank with T_{b} = 10 mm, k = 0.15
mm / year, ā = 1.5 mm. The life/MTBF
function of the tankbottoms related with variables of the
reliability and the number of defects can be visualized in Fig.
7.

Fig. 7. The tankbottom
life/MTBF with
different numbers of defects.

The previous calculation results show that the life or MTBF of
the tank bottom with a different number of defects will be very
different and will decrease with the increasing number of
corrosive pits. The higher reliability choice is the selection
of thicker bottom plates. This is a very effective method to
increase the life/MTBF and the reliability as well as the
resistance to corrosion. At the same time, we can also take
other measures to protect the bottom plates from being
corroded, such as coating them and keeping them away from
water. In fact, this calculation and analysis are much more
important for planning the inspection and maintenance of a tank for safe
storage than for knowing its accurate life/MTBF.
Bringing it all together.
There is a functional relation
between the life/MTBF and reliability of a steel storage tank
with defects in its bottom plate, which is also dependent on
the parameters of tankbottom structure and corrosion (such as
the corrosion rate and the thickness of the tankbottom plate).
So we can increase the life/MTBF and reliability of steel
storage tanks by selecting thicker bottom plates and
strengthening the resistance to corrosion. The other key
component is to reduce and/or eliminate the factors that cause
or accelerate bottom corrosion pits and defects.
The MTBF of a steel storage tank
will depend mainly on the defects in its bottom. There seems to
be no difference regarding the size of the tankbottom area in
the previous life calculation. However, usually there are more
defects in a larger tank bottom. So, a larger tank should be
designed with the thicker bottom plates at the same corrosive
environment and reliability level. The larger tanks should also
be inspected in shorter periods than the smaller ones in order
to maintain proper reliability.
Through the analysis of life/MTBF
and reliability, a more reasonable plan for the future
inspection and maintenance of a tank bottom can be
made. This will greatly increase cost efficiency and avoid both
unneeded work and losses that come from not finding the defects
that can lead to failure in time. HP
LITERATURE
CITED
^{1} JiYi, F.,
Analysis on a thousand cases of accidents in oil
depots, Sino Petrochem Publishing House, pp. 228 and 355,
Beijing, 2005.
^{2} Guangchen, G. and Z. Zhang, Design and
management of petroleum depots, Petroleum University
Publishing House, pp. 221227, Dongying, Shandong, 1991.
^{3} ShuHo, D. and M. Wang, Reliability analysis
in engineering applications, Van Nostrand Reinhold, pp.
32–35, pp. 359–361, New York, 1991.
^{4} Cheng, W., et al, MATLAB 5.3 essentials and
programming with advanced application, China Machine
Press, pp. 82–87.
The authors 


Lichuan
Liu is a professor in the petroleum engineering
department of Logistic Engineering University in China.
She engages in research on petroleum storage and
transportation system design and reliability. 


Tianqi Liu
is a professor in the electric engineering school of
Sichuan University in China. She engages in research on
electricity power system design and reliability. 