materials

Article
A Modelling Study for Predicting Life of Downhole
Tubes Considering Service Environmental
Parameters and Stress
Tianliang Zhao 1,2 , Zhiyong Liu 1,2, *, Cuiwei Du 1,2 , Jianpeng Hu 1,2 and Xiaogang Li 1,2,3
1 Corrosion and Protection Center, University of Science and Technology Beijing, Beijing 100083, China;
ustb_tlzhao@163.com (T.Z.); dcw@ustb.edu.cn (C.D.); ustbck0804@163.com (J.H.);
lixiaogang99@263.net (X.L.)
2 Key Laboratory for Corrosion and Protection (MOE), Beijing 100083, China
3 Ningbo Institute of Material Technology & Engineering, Chinese Academy of Sciences,
Ningbo 315201, China
* Correspondence: liuzhiyong7804@126.com; Tel.: +86-10-6233-3975

Academic Editors: Alex A. Volinsky and Lijie Qiao

Received: 15 July 2016; Accepted: 25 August 2016; Published: 2 September 2016

Abstract: A modelling effort was made to try to predict the life of downhole tubes or casings,
synthetically considering the effect of service influencing factors on corrosion rate. Based on the
discussed corrosion mechanism and corrosion processes of downhole tubes, a mathematic model
was established. For downhole tubes, the influencing factors are environmental parameters and
stress, which vary with service duration. Stress and the environmental parameters including water
content, partial pressure of H2 S and CO2 , pH value, total pressure and temperature, were considered
to be time-dependent. Based on the model, life-span of an L80 downhole tube in oilfield Halfaya, an
oilfield in Iraq, was predicted. The results show that life-span of the L80 downhole tube in Halfaya
is 247 months (approximately 20 years) under initial stress of 0.1 yield strength and 641 months
(approximately 53 years) under no initial stress, which indicates that an initial stress of 0.1 yield
strength will reduce the life-span by more than half.

Keywords: mathematic model; life prediction; downhole tube; environmental parameter; stress

1. Introduction
Failure of the downhole tube and casing is a problem constantly disturbing oil production.
Once the tube or casing, especially the tube, is corroded to perforation or cracking, the oil well may
face production suspension or may even be discarded. Since the downhole environment is severely
corrosive and complex, it is hard to protect tubes and casings from aggressive corrosion. However,
we will have more choices if we can foreknow when they fail. With knowing service life of tubes and
casings, engineers can use different strategies regarding material selection, operation optimization and
periodic maintenance [1,2]. Thus, their values are made full use of during their service time so that
unnecessary cost can be avoided. Accidental risk can also be controlled effectively [3]. Therefore, it
is really of great significance to carry out an investigation on the service life prediction of downhole
tubes and casings.
Over the past decades, much effort has been devoted to prediction, and researchers have proposed
many models for prediction. The way of establishing model is various and can be divided into data
mining models, failure-mechanism-based models, and combination of the two [4–11]. The data mining
models do not care about the mechanism and processes of corrosion. They are mainly based on
statistics from vast amounts of field data; for example, Hu et al. [4] proposed a cross-scale life-time
prediction model for oil tubes mainly based on statistical theory and used Monte-Carlo method as

Materials 2016, 9, 741; doi:10.3390/ma9090741 www.mdpi.com/journal/materials

Materials 2016, 9, 741 2 of 17

the numerical simulation method. Caleyo et al. [5] also used Monte-Carlo method to predict the time
evolution of pit depth on underground pipelines. Yang and Wang [6] proposed a model based on
the grey system theory to predict the residual life of submarine tubes. These models usually give
relatively accurate prediction. However, they are of poor operability as they rely on large amounts
of field data and experience. The models based on failure mechanism overcome this shortcoming.
They require a scientific understanding of corrosion mechanism and processes. For example, Zhang [7]
developed a residual life prediction model based on elastic-plastic fracture mechanics. He took
the combined effects of internal pressure and axial force into consideration as the main influences,
but ignored their variation with service time in his model. Then, Parkins [8] further applied stress
corrosion crack growth kinetics to the prediction. For stainless steel, Song [3] developed a mathematical
model based on the film rupture and repassivation mechanism. However, the above studies mainly
focused on the individual process of crack propagation, which accounts for only a small part of the
whole service life. Although these models seem theoretically reasonable, the results are always away
from practice. Models combining data mining and failure mechanism, which overcome their respective
disadvantages, have also been proposed. Krouse et al. and Laycock et al. [9–11] established a model
characterizing the relationship between maximum deepness of local corrosion and time with method
of extreme value statistic. Melchers [9] also attempted to model variation of pit depth with elapsed
exposure time and influencing factors. His model was based on reasonable assumptions from corrosion
science and field observation and gave results through statistics of maximum pit depth. Although his
work is widely recognized, his model ignored the effect of influencing factors on pit depth.
Corrosion rate, based on corrosion mechanism and usually being applied as corrosion losses or
rate of pit growth, is one main type of data used in modelling [12]. It is not easy to establish a model
combining corrosion data and corrosion mechanism, especially when the effect of time and other
influencing factors are taken into consideration. Development of such model requires a combination of
scientific understanding of corrosion processes and sound approaches to mathematical modelling [13].
For easier use in practice, models are usually over-simplified using a time-independent corrosion rate,
such as initial corrosion rate or average corrosion rate, or ignoring the effect of environmental and
stress evolution [14]. Actually, corrosion rate mostly act as a non-linear functions of time and other
influence factors [15]. Models of irrational simplicity usually result in serious low predicted life-span
or error judgment about SCC critical conditions. The former will lead to uneconomic material-selection
and the later will lead to serious accidents [16].
For downhole casings and tubes, water content, partial pressure of H2 S and CO2 , pH value,
total pressure, temperature, and the axial stress induced by self-weight are the main influencing
factors of corrosion rate. The present work proposes a new prediction model or method synthetically
considering those factors mentioned above. This model or method is closer to practice and avoid the
above-mentioned problems. Two criteria for failure judgment, the thickness criterion and the strength
criterion, aiming at the failure caused by wall thickness thinning and strength loss, respectively, have
also been proposed.

2. Criteria for Failure Judgment

It is generally known that there are mainly two types of integrity loss of downhole tubes [17].
One type is perforation and leakage induced by local corrosion of tube wall. In this form, lives of the
tubes are limited by thickness of local corrosion position, namely tubes are assumed to fail when the
wall is corroded to a certain thickness. The well-known ASME B31G Standard assumes that tubes fail
when deepness of the defect is greater than 80% of the wall thickness [18]. Thus, failure of downhole
tubes can be judged with the following equation.

L0 − ∆L ≥ nL0 (1)
Materials 2016, 9, 741 3 of 17

Equation (1) can be called the thickness criterion (TC). L0 is the initial thickness of tube wall. ∆L is
the corroded thickness during service and can be expressed as the functions of service time t. n is the
coefficient for thickness safety, the value of which is 0.2 according to ASME B31G Standard [18].
The other type is cracking induced by stress concentration. In this form, the tubes are assumed to
fail when the suffered stress exceeds the allowable maximum stress (Sc ). According to the residual
strength criterion [19], failure of downhole tubes can be judged with the following equation.

L0
mS0 ≤ Sc (2)
L0 − ∆L

where m is the coefficient for strength safety, the value of which can be set as 1.5 empirically. S0 is
the initial stress posed on tube wall. It is determined by the initial service status. Equation (2) can be
called the strength criterion (SC). It is noted that the SC is suitable both for general corrosion and local
corrosion. The corrosion form will significantly affect the value of Sc . This will be discussed in detail
in Section 6.1.

3. Expression of Life Prediction Model

Naturally, the tube wall reduces as service time goes on. Thus, ∆L is the function of service time, t.
Z t
∆L = Cdt0 (3)
0

where C is the corrosion rate of oil tube and a function of service time t. Its influence factors includes
the suffered stress (S), water content (the proportion of water in oil–water mixture, W), temperature (T),
the partial pressure of H2 S (PH2 S ), the partial pressure of CO2 (PCO2 ), pH value and the concentration
of Cl− (CCl− ). Therefore, C can be expressed as below.


C = F S, W, T, PH2 S , PCO2 , pH, CCl− , · · · (4)

where S, W, T, PH2S , PCO2 , pH, CCl− etc. are also functions of t. Their dependences on t can be obtained
through monitoring on service environment over time. They are expressed as follows:

S = f 1 (t) (5)

W = f 2 (t) (6)

T = f 3 (t) (7)

PH2 S = f 4 (t) (8)

PCO2 = f 5 (t) (9)

Therefore, Equation (4) is transformed to the below equation.

C = G (t) (10)

Plugging Equation (10) into Equation (3), dependence of ∆L on t can be obtained as follows.
Z t
∆L = G (t0 )dt0 = H (t) (11)
0

Combining Equations (1), (2) and (11), service life of oil tube can be obtained. The life prediction
model is mainly expressed by Equation (11). It provides a new way of life-span prediction of downhole
tubes, namely, synthetically considering the evolution of service environment and stress levels.
Thus, it is more in accordance with the engineering practice. Simultaneously, this also makes it
Materials 2016, 9, 741 4 of 17

have some disadvantages inevitably. The specific expression of Equation (4) becomes difficult to work
out when it is a multivariate function. However, we can still approach it through the method of
multivariate
Materials 2016, 9,function
741 interpolation described in literature [20]. 4 of 17

4. Experimental
4. Experimental

4.1. Material and

4.1. Material and Medium
Medium
Specimens
Specimens used used in
in this
this work
work were
were made
made of of L80
L80 tubing
tubing steel
steel with
with chemical composition (wt
chemical composition (wt %):
%):
0.32
0.32 C,
C,0.19 Si, Si,
0.19 1.351.35
Mn, Mn,
0.24 Cr,
0.240.033
Cr, Cu, <0.10
0.033 Cu,Mo, 0.015Mo,
<0.10 S, 0.0088
0.015P S,
and Fe balance.
0.0088 P andItsFemicrostructure
balance. Its
is
microstructure is shown in Figure 1. It reveals that L80 steel is mainly made of fineL80
shown in Figure 1. It reveals that L80 steel is mainly made of fine bainite. That makes steel have
bainite. That
amakes
prettyL80
good mechanical performance: yield strength
steel have a pretty good mechanical performance: (σ s ) of 675 MPa, ultimate tensile strength (UTS)
yield strength (σs) of 675 MPa, ultimate
of 797 MPa,
tensile elongation
strength (UTS) of(δ797
0 ) ofMPa,
22.3%elongation
and reduction-in-area
(δ0) of 22.3% (RA) of 69.5%.
and reduction-in-area (RA) of 69.5%.

Figure 1. Microstructure of L80 steel.

Figure 1. Microstructure of L80 steel.

The mediums were oil–water mixtures prepared in different proportions of oil and mineralized
The mediumsthe
water to simulate were oil–water
water content mixtures prepared
of oilfield Halfayainatdifferent
differentproportions
service time.of oil and mineralized
Halfaya is a typical
water
oilfields in Middle East. Dependence of its water content on service time is shown inis Figure
to simulate the water content of oilfield Halfaya at different service time. Halfaya a typical2.
oilfields in Middle East. Dependence of its water content on service time is
Accordingly, the proportions were set at 5 wt %, 30 wt %, 50 wt %, 80 wt %, and 100 wt %. Linear shown in Figure 2.
Accordingly, the proportions
fittings corresponding were set
to different at 5 wtstages
service %, 30 were
wt %,also50 wtgiven.
%, 80 The
wt %, and 100 wtwater
mineralized %. Linear
was
fittings
prepared according to chemical composition shown in Table 1. The oil is crude oil fromprepared
corresponding to different service stages were also given. The mineralized water was oilfield
according
Halfaya. The to oil
chemical
and thecomposition
mineralizedshown in Table
water were 1. and
mixed, The stirred
oil is crude
for 12oil from
h to formoilfield Halfaya.
an oil-in-water
The oil and the mineralized
or water-in-oil emulsion. water were mixed, and stirred for 12 h to form an oil-in-water or
water-in-oil emulsion.
Practical monitoring120 result of oilfield Halfaya shows that partial pressure of H2 S and CO2 ,
pH value, total pressure, and temperature
I 85 II
downhole 285be steadyIIIsoon after the oilfield is put into
will
production. Although100 pH value is generally supposed to have a great effect on corrosion rate, it is
Water content, W (wt%)

not concluded in the influencing 2

factors because acidification of mineralized water is neutralized by
y=-59.0021+1.0146x-0.0018x
constantly injected water. 80 Thus, water content and stress is assumed to be only two variables during
y=0.0132x+83.6576
the whole service period. The partial pressure of H2 S and CO2 , total pressure and temperature are
approximately 0.15 MPa, ◦
601.1 MPa, 10 MPa and 80 C, respectively. The experimental conditions were
set in accordance to these results. Water content of oilfield Halfaya
Fitting curve at the initial stage
40 Fitting curve at the middle stage
Fitting curve at the later stage
20
y=0.0809x+6.6269

0
0 100 200 300 400 500
Service time, t (month)
oilfields in Middle East. Dependence of its water content on service time is shown in Figure 2.
Accordingly, the proportions were set at 5 wt %, 30 wt %, 50 wt %, 80 wt %, and 100 wt %. Linear
fittings corresponding to different service stages were also given. The mineralized water was
prepared according to chemical composition shown in Table 1. The oil is crude oil from oilfield
Halfaya. The 9,oil741and the mineralized water were mixed, and stirred for 12 h to form an oil-in-water
Materials 2016, 5 of 17
or water-in-oil emulsion.

120
I 85 II 285 III
100

Water content, W (wt%)

2
y=-59.0021+1.0146x-0.0018x
80 y=0.0132x+83.6576

60
Water content of oilfield Halfaya
Fitting curve at the initial stage
40 Fitting curve at the middle stage
Fitting curve at the later stage
20
y=0.0809x+6.6269

0
0 100 200 300 400 500
Service time, t (month)
Figure
Figure2.2.Dependence
Dependenceofofwater
watercontent
contentofofoilfield
oilfieldHalfaya
Halfayaon
onservice
servicetime.
time.

Table 1. Chemical composition of mineralized water in the medium (g·L−1 ).

NaCl NaHCO3 Na2 SO4 CaCl2 MgCl2 ·6H2 O pH

236.5 1.01 0.64 26.64 12.68 6

4.2. Potentiodynamic Polarization Measurement

Considering that water content and stress would have an impact on the corrosion behaviour of
L80 steel, potentiodynamic polarization measurement was conducted on the specimens under different
water contents and stresses. A thermostatic autoclave with conventional three-electrode system was
used to perform the measurements under different water contents. The specimens were cut into
plates with sized of 10 mm × 10 mm × 3 mm and sealed with epoxy resin, leaving a working square
of 10 mm × 10 mm exposed. The prepared medium was added into the autoclave and experimental
conditions were applied with reference to Section 4.1.
The measurements under different stresses were performed with a CORTEST slow strain rate test
system (CORTEST, Willoughby, OH, USA), which also has a three-electrode system. The specimens
were commonly-seen tensile plates. They were sealed with high-temperature silicone, leaving a square
of 5 mm × 10 mm in the working segment exposed. Prior to the measurements, the specimens were
preloaded to different stress levels, i.e., 0.5 σs , 0.8 σs and 1.0 σs (here it is assumed that σs is equal
to the proof stress). The medium of 80 wt % water content was then added into the autoclave and
experimental conditions were applied with reference to Section 4.1. The polarization measurements
would not start at these particular stress levels until corrosion potential of the specimen was stable.
All tests were carried out with a potential scanning rate of 1 mV/s. The medium was
de-oxygenated by purging nitrogen gas for 2 h before test. The L80 steel specimen was used as
working electrode, SCE as reference electrode and a platinum plate as counter electrode. The entire
specimen was ground sequentially to 1000 grit emery paper.

4.3. Immersion Test

Immersion test was performed on specimens with different stress levels. Before immersion,
specimens were cleaned and weighed. They were then preloaded to different stress levels with a
WDML-30KN (LETRY, Xi’an, China) electron-tensile tester and fixed with assembling jigs and nuts.
The levels were 0 σs , 0.5 σs , 0.8 σs and 1.0 σs . Size of the specimen and the jig are shown in Figure 3.
The jig was made from Hastelloy alloy, which has excellent rigidity and corrosion resistance. Ceramic
 Materials 2016, 9, 741 6 of 17

gaskets were used to ensure the specimen insulated from the jig. Both ends of the specimen, the nuts
and the arc transition sections were coated with temperature-resistant silica gel, leaving the necking
sections exposed. The prepared medium and specimens were then added into a thermostatic autoclave
and the medium was de-oxygenated by purging nitrogen gas for 2 h. The experimental conditions
were applied
Materials 2016, 9, 741with reference to Section 4.1. The immersion time lasted for 720 h. 6 of 17

Hastelloy alloy jig

Ceramic
gasket

Figure 3. Sizes of (a) the specimen and (b) the assembling jig used for preloading.
Figure 3. Sizes of (a) the specimen and (b) the assembling jig used for preloading.

After
After immersion,the
immersion, the specimens
specimens were
werecleaned
cleanedwith acetone
with andand
acetone distilled waterwater
distilled and then
andweighed.
then
Corrosion rates corresponding to different preloading stresses were calculated according
weighed. Corrosion rates corresponding to different preloading stresses were calculated according to the weight
loss. Corrosion morphologies of the necking sections were observed with VHX 2000
to the weight loss. Corrosion morphologies of the necking sections were observed with VHX 2000 (Keyence, Osaka,
Japan) stereomicroscope. Then tensile tests were performed on those corroded specimens.
(Keyence, Osaka, Japan) stereomicroscope. Then tensile tests were performed on those corroded
specimens.
4.4. Tensile Tests after Immersion
Tensile
4.4. Tensile Teststests
afterwere conducted on other parallel immersed specimens according to ASTM E8M-09
Immersion
(ASTM, West Conshohocken, PA, USA) [21]. The tensile rate was 0.007/min and 0.05/min before and
Tensile tests were conducted on other parallel immersed specimens according to ASTM E8M-09
after the specimen yielded, respectively. Then, the stress–strain curves, elongations and reductions in
(ASTM, West Conshohocken, PA, USA) [21]. The tensile rate was 0.007/min and 0.05/min before and
area were obtained.
after the specimen yielded, respectively. Then, the stress–strain curves, elongations and reductions
in area were obtained.
5. Results

5. Results
5.1. Potentiodynamic Polarization Curves
Figure 4 shows the potentiodynamic polarization curves of L80 steel measured under different
5.1. Potentiodynamic Polarization Curves
water contents and preloading stresses. It is seen that both water content and preloading stress
Figure
affect 4 shows the
significantly potentiodynamic
the electrochemicalpolarization
polarizationcurves of L80ofsteel
behaviour L80measured under different
steel. However, shape of
water contents and preloading stresses. It is seen that both water content and preloading
the potentiodynamic polarization curves does not change with the increasing of water content stress affect or
significantly
preloadingthe electrochemical
stress. Generally, the polarization
smooth shape behaviour
indicatesof that
L80anodic
steel. and
However, shape
cathodic of theare
processes
potentiodynamic
controlled by electrochemical reaction step. Increasing the water content or preloading stress doesornot
polarization curves does not change with the increasing of water content
preloading
affect the stress.
corrosion Generally, theinsmooth
type of L80 shapeexcept
test medium indicates thatcorrosion
for the anodic and cathodic
potential processescurrent
and corrosion are
controlled bywas
density. It electrochemical
also found in reaction
previousstep.
studiesIncreasing the the
[22–24] that water content
anodic and or preloading
cathodic stress
processes ofdoes
carbon
notsteel
affect the corrosion type of L80 in test medium except for the corrosion potential and
in similar environment (the high pressure H2 S/CO2 environment) are activation controlled. It is corrosion
current density.
also noted thatItcathodic
was alsoreduction
found in previous
reactions studies [22–24]
(hydrogen that the
reduction) anodic
are and cathodic
promoted processes of
by both increasing
of water
carboncontent
steel inandsimilar environment
preloading stress (see(the
the high pressure
cathodic H2S/CO
polarization 2 environment)
segment in Figure are activation
4a,b). Increasing
controlled. It is also noted that cathodic reduction reactions (hydrogen reduction) are promoted by
both increasing of water content and preloading stress (see the cathodic polarization segment in
Figure 4a,b). Increasing of water content increases the exposed area of steel to mineralized water,
thereby promoting the cathodic reactions. The cathodic reactions increase with increasing of
preloading stress, which can be explained by local additional potential model (LAPM) [25]. In elastic
Materials 2016, 9, 741 7 of 17

of water content increases the exposed area of steel to mineralized water, thereby promoting the
cathodic reactions. The cathodic reactions increase with increasing of preloading stress, which can
be explained by local additional potential model (LAPM) [25]. In elastic stress region, local stress
concentration may occur at micro-defects, such as twins, micro-cracks, inclusions, etc. Dislocation
slip could
Materials thus
2016, occur at these sites. Dislocation emergence points and slip steps introduce active
9, 741 7 of 17
sites on the steel surface, and electrons would flow and concentrate at these sites to result in a local
sites to result
charging effect. inAsaa local
result,charging effect. As potential
a local additional a result, (LAP,
a localgenerally
additional potential
positive) (LAP, generally
is generated when
the steel surface is exposed in a solution. With an increasing of stress, the LAP increases, resulting the
positive) is generated when the steel surface is exposed in a solution. With an increasing of stress, in
LAP
the increases,ofresulting
increasing cathodicin the increasing
reaction current.ofTherefore,
cathodic reaction
with thecurrent. Therefore,
above two factors with the above
increasing, risktwo
of
factors increasing,
hydrogen brittlementrisk of hydrogen
or hydrogen brittlement
induced crackingormay
hydrogen inducedMechanical
be increased. cracking may be increased.
properties of L80
Mechanical properties of L80 after immersed in
after immersed in the test medium need to be investigated. the test medium need to be investigated.

(a) 30wt% H2S:0.15 MPa

0
50wt% CO :1.1 MPa
2
80wt%
-200 100wt% Total pressure: 10 MPa
Temp: 80ºC
-400 Preloading stress: 0
E(mVSCE)

-600

-800

-1000
Note: polarization curve corresponding to 5% water content cannot be obtained
-1200 -5 -4 -3 -2 -1 0
10 10 10 10 10 10
-2
Current(mA·cm )

(b) -100 0 σs H2S: 0.15 MPa

-200
0.5 σs CO2: 1.1 MPa
-300
0.8 σs Total pressure: 10 MPa
-400 Temp: 80ºC
1.0 σs
-500 Water content: 80%
E (mVSCE)

-600
-700
-800
-900
-1000
-1100
-5 -4 -3 -2 -1 0 1
10 10 10 10 10 10 10
-2
Current (mA·cm )

Figure4.4. Potentiodynamic
Figure Potentiodynamic polarization
polarization curves
curves of
of L80
L80steel
steelmeasured
measuredunder
underdifferent:
different:(a)
(a)water
water
contents;
contents;and
and(b)
(b)preloading
preloadingstresses.
stresses.

5.2. Corrosion Morphology

Figure 5 shows corrosion morphologies of L80 steel after immersed in the medium of 80 wt %
water content for 720 h under preloading stresses of: Figure 5a 0 σs; Figure 5b 0.5 σs; Figure 5c 0.8 σs;
and Figure 5d 1.0 σs. It can be seen that many shallow pits are uniformly distributed on the surface
and no crack nucleates. Those shallow pits are not deepened, but gradually connected to each other
along with the increasing of preloading stress. The connected pits transform into a relatively
Materials 2016, 9, 741 8 of 17

5.2. Corrosion Morphology

Figure 5 shows corrosion morphologies of L80 steel after immersed in the medium of 80 wt %
water content for 720 h under preloading stresses of: Figure 5a 0 σs ; Figure 5b 0.5 σs ; Figure 5c 0.8 σs ;
and Figure 5d 1.0 σs . It can be seen that many shallow pits are uniformly distributed on the surface
and no crack nucleates. Those shallow pits are not deepened, but gradually connected to each other
along with the increasing of preloading stress. The connected pits transform into a relatively uniform
corrosion. The corrosion morphology partially confirms the results of potentiodynamic polarization
measurement, i.e., L80 steel will corrode uniformly in downhole environment, even with stress
of 1.0 σs2016,
Materials . 9, 741 8 of 17

Figure 5. Corrosion morphology of L80 steel after immersed in the medium of 80 wt % water content
Figure 5. Corrosion morphology of L80 steel after immersed in the medium of 80 wt % water content
for under preloading stresses of: (a) 0 σs; (b) 0.5 σs; (c) 0.8 σs; and (d) 1.0 σs.
for under preloading stresses of: (a) 0 σs ; (b) 0.5 σs ; (c) 0.8 σs ; and (d) 1.0 σs .

5.3. Tensile Properties after Immersion

5.3. Tensile Properties after Immersion
Figure 6 shows the stress–strain curves of L80 steel tested after immersion in the test medium
Figure 6 shows the stress–strain curves of L80 steel tested after immersion in the test medium
for 720 h under different water contents and preloading stresses. It is seen from them that both water
for 720 h under different water contents and preloading stresses. It is seen from them that both water
content and preloading stress have little influence on the mechanical properties of L80 steel, except
content and preloading stress have little influence on the mechanical properties of L80 steel, except
slightly decreasing the elongation. The yield strengths in all conditions remain the same as that in
slightly decreasing the elongation. The yield strengths in all conditions remain the same as that in air.
air. It can be assumed that Sc is equal to the yield strength in air.
It can be assumed that Sc is equal to the yield strength in air.
Figure 7 shows the SCC susceptibility of L80 steel varying with water content and preloading
900
(a) that loss of elongation, loss of reduction-in-area and loss of strength are all less than
stress. It can be seen
12%. It indicates that 800
L80 steel have little tendency towards stress corrosion in downhole environment.
700
600
Stress  (MPa)

No immersion
500 5wt% H2S:0.15 MPa
400 30wt%
50wt% CO2:1.1 MPa
300
80wt% Total pressure: 10 MPa
200 100wt% Temp: 80ºC
100 Preloading stress: 0.5 σs
 5.3. Tensile Properties after Immersion
Figure 6 shows the stress–strain curves of L80 steel tested after immersion in the test medium
for 720 h under different water contents and preloading stresses. It is seen from them that both water
content and preloading stress have little influence on the mechanical properties of L80 steel, except
slightly
Materials decreasing
2016, 9, 741 the elongation. The yield strengths in all conditions remain the same as that in9 of 17
air. It can be assumed that Sc is equal to the yield strength in air.

900
(a)
800
700
600

Stress  (MPa)
No immersion
500 5wt% H2S:0.15 MPa
400 30wt%
50wt% CO2:1.1 MPa
300
80wt% Total pressure: 10 MPa
200 100wt% Temp: 80ºC
Materials 2016, 9, 741 9 of 17
100 Preloading stress: 0.5 σs
(b) 9000
0 5 10 15 20 25
Materials 2016, 9, 741 800 9 of 17
Strain 
700
(b) 900
600
800
Stress  (MPa)

No immersion
500
700 0 s
H2S: 0.15 MPa
400
600 0.5 s
CO2: 1.1 MPa
Stress  (MPa)

300 No immersion
500 0.8s
0 s Total pressure: 10 MPa
200 1.0s H2S: 0.15 MPa
400 0.5 s Temp: 80ºC
100 CO2: content:
Water 1.1 MPa80%
300 0.8s
0 Total pressure: 10 MPa
200 0 5 1.010s 15 80ºC 20 25
Temp:
100 Strain 
Water content: 80%
Figure 6. Dependence 0 of stress–strain curve of L80 steel on: (a) water content; and (b) preloading
0 the test medium
stress after immersion in 5 10h.
for 720 15 20 25
Strain 
Figure 7 shows the SCC susceptibility of L80 steel varying with water content and preloading
Figure 6. Dependence
Dependence of stress–strain
of stress–strain curve
curve of of L80
L80 steel
on:on:
(a)(a) water content; and(b)
(b)preloading
preloadingstress
Figure
stress. It6.can be seen that loss of elongation, loss ofsteel water
reduction-in-area content;
and lossand
of strength are all less
after stress after immersion
immersion in the testinmedium
the test medium
for 720 for
h. 720 h.
than 12%. It indicates that L80 steel have little tendency towards stress corrosion in downhole
environment.
Figure 7 shows the SCC susceptibility of L80 steel varying with water content and preloading
stress. It can be seen that loss of elongation, loss of reduction-in-area and loss of strength are all less
12
than 12%. It indicates that L80 steel Losshave little tendency towards stress corrosion in downhole
of elongation
environment.
10 Loss of reduction-in-area
Loss of strength
12
(%)

8 Loss of elongation
SCC susceptibility

10 Loss of reduction-in-area
6 Loss of strength
(%)

8
4
SCC susceptibility

6
2
4
0
2 A B C D E
Experimental conditions
0
Figure
Figure 7. Variation
7. Variation of SCC
of SCC susceptibility
susceptibility ofof L80steel
L80 steelafter
afterimmersion
immersion for
for 720
720 h
h with
with water
watercontent
content and
and preloading stress (A: 5 wtA%, 0 σs; B: 80B wt %, 0 σC D0.5 σs; D: 80E wt %, 0.8 σs; and E:
s; C: 80 wt %,
preloading stress (A: 5 wt %, 0 σs ; B: 80 wt %, 0 σs ; C: 80 wtconditions
Experimental %, 0.5 σs ; D: 80 wt %, 0.8 σs ; and E: 80 wt %,
80 wt %, 1.0 σs).
1.0 σs ).
Figure 7. Variation of SCC susceptibility of L80 steel after immersion for 720 h with water content
5.4. Corrosion Rates
and preloading stress (A: 5 wt %, 0 σs; B: 80 wt %, 0 σs; C: 80 wt %, 0.5 σs; D: 80 wt %, 0.8 σs; and E:
Table 2 1.0
80 wt %, lists
σs).the corrosion rates of L80 steel under different water contents and different
preloading stresses. Those data are replotted in Figure 8. It is seen that corrosion rate (in the form of
5.4. Corrosion
average Ratesrate) of L80 steel increases linearly with increasing water content when the
thinning
preloading
Table stress
2 lists isthe
constant. Relationship
corrosion expressions
rates of L80 between
steel under corrosion
different water rate and water
contents content
and different
Materials 2016, 9, 741 10 of 17

5.4. Corrosion Rates

Table 2 lists the corrosion rates of L80 steel under different water contents and different preloading
stresses. Those data are replotted in Figure 8. It is seen that corrosion rate (in the form of average
thinning rate) of L80 steel increases linearly with increasing water content when the preloading
stress is constant. Relationship expressions between corrosion rate and water content under different
preloading stresses are labelled on the corresponding fitting lines.

Table 2. Corrosion rates of L80 steel under different water contents and different preloading stresses.

Corrosion Rate Average Corrosion

Water Content (wt %) Preloading Stress (σ s ) No.
(mm/a) Rate (mm/a)
1 0.0132
0 0.0243
2 0.0354
1 0.0403
0.5 0.0311
2 0.0219
5
1 0.0663
0.8 0.0562
2 0.0461
1 0.1941
1.0 0.2069
2 0.2197
1 0.0329
0 0.0384
2 0.0439
1 0.1219
0.5 0.1331
2 0.1443
30
1 0.3201
0.8 0.3015
2 0.2829
1 0.7584
1.0 0.7405
2 0.7226
1 0.0507
0 0.0431
2 0.0355
1 0.2484
0.5 0.2608
2 0.2932
50
1 0.5986
0.8 0.5813
2 0.5640
1 1.3550
1.0 1.3322
2 1.3094
1 0.0920
0 0.0797
2 0.0674
1 0.3111
0.5 0.3845
2 0.4580
80
1 0.8316
0.8 0.8569
2 0.8822
1 2.1980
1.0 2.2529
2 2.3078
1 0.1002
0 0.0951
2 0.0900
1 0.4526
0.5 0.4455
2 0.4384
100
1 1.0161
0.8 0.9870
2 0.9579
1 2.9331
1.0 2.9558
2 2.9785
Materials 2016, 9, 741 11 of 17
Materials 2016, 9, 741 11 of 17

Materials 2016, 9, 741 11 of 17

3.0 0s linear fit of 0.0 s
0.5s linear fit of 0.5s
2.5
3.0

C (mm/a)
0
0.8s linearfitfitofof0.8
linear 0.0
s y=0.0282x
s s
0.5s
1.0 linearfitfitofof1.0
linear 0.5

2.0
2.5 s s
s

(mm/a)
0.8s linear fit of 0.8s y=0.0282x

rate Crate 1.5

2.0 1.0 s linear fit of 1.0s
y=0.0107x
1.0
Corrosion

1.5
y=0.0107x
y=0.0048x
0.5
1.0
Corrosion

-4
y=1.593810 x
y=0.0048x
0.0
0.5
-4
y=1.593810 x
-10 0 10 20 30 40 50 60 70 80 90 100 110
0.0
Water content (%)
-10 0 10 20 30 40 50 60 70 80 90 100 110
Figure 8. Corrosion rates of L80 steel and their fittings as a function of water content under different
Figure 8. Corrosion rates of L80 steel and their
preloading stress.
Water content
fittings (%) of water content under different
as a function
preloading stress.
Figure 8. Corrosion rates of L80 steel and their fittings as a function of water content under different
Figure 9 shows
preloading stress.the dependence of corrosion rate at 80% water content on preloading stress.
Figure 9 shows the
Data are fitted with dependence
exponential of corrosion
function. rate at 80%
The expression water content on preloading stress. Data
is as follows:
are fitted Figure
with exponential
9 shows thefunction. Theofexpression
dependence is asatfollows:
corrosion rate 80% water content on preloading stress.
y  g  S   a exp  b  S  (12)
Data are fitted with exponential function. The expression is as follows:
where fitting values of a and b are 0.0311y = and
g (S) = aexp (b · S) Relationship between corrosion (12)
y  g  S4.2697,
 respectively.
 a exp b  S  (12)
rates at other water content and preloading stress can be also expressed by Equation (12).
wherewhere
fittingfitting
values of a and
values b areb 0.0311
of a and andand
are 0.0311 4.2697, respectively.
4.2697, Relationship
respectively. Relationshipbetween
betweencorrosion
corrosion rates
at other
rateswater content
at other 2.5 preloading
waterand
content stress can
and preloading stressbecan
also
beexpressed by Equation
also expressed (12).
by Equation (12).
80% water content
2.5
2.0 Fiting of the above
C (mm/a)

80% water content

2.0
1.5 Fiting of the above
(mm/a)

1.5
1.0 y=g(S)=0.0311exp(4.2697 S)
rate Crate

y=g(S)=0.0311exp(4.2697 S)
Corrosion

1.0
0.5
Corrosion

0.5
0.0

0.0
-0.5
0.4 0.0
0.6 0.80.2 1.0
S (s)
-0.5
0.0 0.2 0.4 0.6 0.8 1.0
Figure 9. Dependence of corrosion rate of L80 steel at 80% water content on preloading stress.
S (s)
6. LifeFigure
Prediction of L80 Oil
9. Dependence Tube in rate
of corrosion Halfaya
of L80 steel at 80% water content on preloading stress.
Figure 9. Dependence of corrosion rate of L80 steel at 80% water content on preloading stress.
6.1. Suitability
6. Life of the of
Prediction Model
L80 Oil Tube in Halfaya
6. Life Prediction of L80 Oil Tube in Halfaya
Potentialdynamic polarization curves (Figure 4) and corrosion morphologies (Figure 5) in last
6.1. Suitability
section of theL80
reveal that Model
steel is corroded uniformly in simulated medium of Halfaya downhole
6.1. Suitability of the Model
environment. Generally, uniform corrosion
Potentialdynamic polarization means 4)that
curves (Figure andthe oil tubemorphologies
corrosion will not suffer great5)stress
(Figure in last
Potentialdynamic
concentration. It is polarization
reasonable to curves
assume (Figure
that the true4) and
stress corrosion
is equal to
section reveal that L80 steel is corroded uniformly in simulated medium of Halfaya morphologies
the nominal (Figure
stress (S n). Sn5)
downhole isin last
section reveal thatGenerally,
environment. L80 steeluniform
is corroded uniformly
corrosion means that in simulated
the oil tubemedium
will not of Halfaya
suffer great downhole
stress
concentration.
environment. It is reasonable
Generally, uniformto assume that the
corrosion true stress
means is equal
that the to thewill
oil tube nominal
not stress
suffer(Sgreat
n). Sn is
stress
concentration. It is reasonable to assume that the true stress is equal to the nominal stress (Sn ).
Sn is defined as the load (F) divided by the actual cross-sectional area (A). For oil tube, F is generally
Materials 2016, 9, 741 12 of 17

defined as the load (F) divided by the actual cross-sectional area (A). For oil tube, F is generally equal
to gravity of itself and A is time-dependent. When the true stress reaches the proof stress, which is
regarded as the
Materials 2016, 9, 741threshold of local instability and usually equivalent to the yield strength, Sn12also of 17
reaches the allowable maximum stress (Sc in Equation (2)). Here, Sc is equal to the proof stress, i.e.,
the yield strength.
equal to gravity
However, uniform and A is time-dependent.
of itselfcorrosion When the form
is not the only corrosion true stress
for thereaches
tubes.the proof
Local stress, which
corrosion and
stress corrosion also widely exist on the tubes. For local corrosion, such as pitting, it will inducealso
is regarded as the threshold of local instability and usually equivalent to the yield strength, S n a
reaches
stress the allowable
concentration maximum
around stress
itself. The (S c in Equation
local stress will(2)).
be Here, Sc is equal
obviously greater tothan
the proof stress,Sci.e.,
Sn. Here, the
is no
yield strength.
longer equal to the proof stress. Because the local stress will reach the proof stress before Sn does. Sc
However,
should be equal to uniform corrosion
the nominal stressis at
notthethe onlywhen
point corrosion form
the local for the
stress tubes.
reaches theLocal
proofcorrosion and
stress. It can
stress
be corrosion
obtained also widely
by methods exist element
of finite on the tubes. For local
modelling corrosion, such
or introducing as pitting, itcoefficient.
an appropriate will induceIfa
stress concentration around itself. The local stress will be obviously greater
stress corrosion induces the steel initiating crack before the local stress reaches the nproof stress,than S . Here, Sc is Snoc
longer equal to the proof stress. Because the local stress will
should be equal to the nominal stress at the point when the steel initiates crack. reach the proof stress before S n does.
Sc should bethe
In brief, equalmodelto the
cannominal stress
be suitable notatonly
the point when the
for uniform local stress
corrosion reachescorrosion)
(or general the proofbutstress.
alsoIt
local corrosion and stress corrosion by adjusting value of Sc according to the actual corrosion formIf
can be obtained by methods of finite element modelling or introducing an appropriate coefficient.
stress
and corrosion induces the steel initiating crack before the local stress reaches the proof stress, Sc
mechanism.
should be equal to the nominal stress at the point when the steel initiates crack.
In brief, theProcess
6.2. Mathematical model can be suitable not only for uniform corrosion (or general corrosion) but also
local corrosion and stress corrosion by adjusting value of Sc according to the actual corrosion form
andFigure 10 shows the variation of corrosion rate with service time under different preloading stress.
mechanism.
It is obtained by iterating ordinate function of Figure 2 into abscissa variable of Figure 10. Expressions
of6.2.
0.0Mathematical
σs and 1.0 σs Process
corresponding to the three stages are given as follows (Equations (13)–(18)).

CI  1.2886
Figure 10 shows the variation 105rate
of corrosion t  1.0562 103time
with service , t  under
85 different preloading (13)
stress.
It is obtained by iterating ordinate function of Figure 2 into abscissa variable of Figure 10. Expressions
of 0.0 σs and 1.0 σs correspondingto7 2the three stages4are given as follows
3 (Equations (13)–(18)).(14)
CII  2.8369 10 t  1.6170 10 t  9.4035 10 ,85  t  285
CI = 1.2886 × 10−5 t + 1.0562 × 10−3 , t ≤ 85 (13)
CIII  2.0996 106 t  0.0133, t  285 (15)
CII = −2.8369 × 10−7 t2 + 1.6170 × 10−4 t − 9.4035 × 10−3 , 85 < t ≤ 285 (14)
' 36
−
C =2.2800
CIII
I ×10
2.0996  10 t t+0.1869,
0.0133, tt 
> 85
285 (16)
(15)

'
0
CI = 2.2800
5 2
× 10−3 t + 0.1869, t ≤ 85 (16)
C  5.0196 10 t  0.0286t  1.6639,85  t  285
II 0
(17)
CII = −5.0196 × 10−5 t2 + 0.0286t − 1.6639, 85 < t ≤ 285 (17)

CCIII' III
0
=3.7139
3.7139×10
4
10−t4 t+2.3591,
2.3591,tt >285
285 (18)
(18)

3.0
I II III
2.5 0.0 s
Corrosion rate C (mm/a)

2.0 0.5 s
0.8 s
1.5 285
1.0 s
1.0 85

0.5

0.0
0 50 100 150 200 250 300 350 400 450 500
Service time t (month)
Figure 10. Corrosion rates of L80 steel as a function of service time under different preloading stress.
Figure 10. Corrosion rates of L80 steel as a function of service time under different preloading stress.
Materials 2016, 9, 741 13 of 17

As previously mentioned, temperature and partial pressure of H2S and CO2 are steady during
As time.
service previously mentioned,
Therefore, T, PH2S temperature and partial
and PCO₂ in Equation (4)pressure of H2as
can be seen S and CO2 are
constants andsteady during
C is mainly
service time. Therefore,
affected by S and W. In orderH2S T, P and P in Equation (4) can be seen as constants and
CO2 process feasible, Equation (4) is simplified as follows:
to make the C is mainly
affected by S and W. In order to make the process feasible, Equation (4) is simplified as follows:
C = F  S,W   f  S   g W  (19)
C = F (S, W ) = f (S) · g (W ) (19)
The initial stress will have a great influence on corrosion rate. Life-span of tubes will be far
different
The when
initial the initial
stress willstress
haveisazero
greatorinfluence
not. The initial stresses rate.
on corrosion that different
Life-spanparts of thewill
of tubes downhole
be far
tube or casing
different when suffer are illustrated
the initial stress is zeroin or
Figure 11. Generally,
not. The levelthat
initial stresses of the oil–water
different mixture
parts can reach
of the downhole
one-kilometre
tube high from
or casing suffer the bottom
are illustrated of the
in Figure 11.tube. Stresslevel
Generally, at level
of theofoil–water
the oil–water
mixturemixture
can reach is
approximatelyhigh
one-kilometre equal to the
from 0.1 bottom
σs, which is roughly
of the calculated
tube. Stress at levelfrom
of theweight of one-kilometre-long
oil–water tube.
mixture is approximately
The stress
equal to 0.1atσthe bottom
s , which is zero. Wall
is roughly thickness
calculated fromand diameter
weight of the downhole tube
of one-kilometre-long tube.are
The9 mm and
stress at 200
the
mm, respectively.
bottom is zero. Wall thickness and diameter of the downhole tube are 9 mm and 200 mm, respectively.

Figure
Figure 11. The
The initial
initial stresses
stresses that different parts of the downhole tube or casing suffer.
suffer.

6.2.1. When
6.2.1. When the
the Initial
Initial Stress is Zero
Stress is Zero
If the
If the initial
initial stress
stress is
is zero,
zero, the
the stress
stress that
that oil
oil tube
tube suffers
suffers in
in whole
whole service
service time
time is
is also
also zero.
zero.
Relationship between C and t should be subject to Equations (13)–(15). Therefore, ∆L can be obtained
Relationship between C and t should be subject to Equations (13)–(15). Therefore, ∆L can be obtained
by substituting
by substituting Equations
Equations (13)–(15)
(13)–(15) into
into Equation
Equation (3).
(3).
 R tt

  C dt0' , t0'  85
0RCI dt , t > 85
0 + t C dt 0 , 85 < t 0 ≤ 285
R 85
∆L =  085 CI dt t II ' (20)
 
85
  85 CI dt0C+I dt '285
L 
 R R  CIIC 0 dt
R t  t '  285
+ ,85 0 0 (20)
dt 285 CIII dt , t > 285
0 85 II
 85
0 85
285 t
 σsCI dt ' 

6.2.2. When the Initial Stress is 0.1
 0 85
CII dt '   CIII dt ' , t '  285
285

It is already known that C is the function of W and S, and W and S are the functions of t. Thus, C
can be expressed as follows.
6.2.2. When the Initial Stress is 0.1 σs
C = f ( S ) · g (W ) = f 1 ( t ) · f 2 ( t ) (21)
It is already known that C is the function of W and S, and W and S are the functions of t. Thus, C
Thus,
can be if only as
expressed follows. of f 1 and f 2 are obtained, the C as a function of t is achieved.
expressions

The Initial Stage (0–85th Months) C  f  S   g W   f  t   f  t  (21)

1 2
In the initial stage, there is a linear relationship between C and t when the preloading stress is
Thus, if only expressions of f1 and f2 are obtained, the C as a function of t is achieved.
constant. Thus, f 2 in the initial stage can be assumed as:
The Initial Stage (0–85th Months)
f 2 = At + B (22)
In the initial stage, there is a linear relationship between C and t when the preloading stress is
constant. Thus, f2 in the initial stage can be assumed as:
Materials 2016, 9, 741 14 of 17

where A and B are constants. Since the axial force that the oil tube suffers is equal to the gravity,
there is:
(S + dS) ( L0 − Cdt) = S0 L0 (23)

Equation (23) can be transformed into:

L0 dS = S0 Cdt = S0 f (S) f 2 (t) dt (24)

dS S
= 0 f 2 (t) dt (25)
f (S) L0
Equations (12) and (22) are substituted into Equation (25), and then there is:

dS S
= 0 (At + B) dt (26)
aexp (bS) L0

Both sides of Equation (26) are infinitely integrated. Then there is:

L0
exp (bS) = −   (27)
A 2
abS0 2t + Bt + C1

where C1 is a constant. Equation (27) is substituted into Equation (12), and then there is:

L0
f (S) = −   (28)
A 2
bS0 2t + Bt + C1

Then, Equations (28) and (22) are substituted into Equation (21).

L0 At + B
CI 0 = − · A 2
(29)
bS0 2t + Bt + C1

Thus, ∆L of the initial stage (∆L1 ) can be obtained according to Equation (3).

L0 At + B L0 A 2
Z
∆L1 = − · A 2
dt = − ln t + Bt + C1 + D (30)
bS0 2t + Bt + C1 bS 0 2

where D is a constant. When value of S0 is 1.0 (σs ), the equation below can be set up through combining
Equations (12), (16), (21) and (22).

C = (At + B) · aexp (b) = 2.28 × 10−3 t + 0.1869 (31)

By substituting value of a and b into Equation (31), the value of A and B are obtained:
A = 1.0253 × 10−3 , and B = 0.0840. When value of t is 0, value of S is 0.1. Thus, the equation
below can also be obtained through combining Equations (12), (21), (22) and (29).

10L0 B
C = B · aexp (0.1b) = − (32)
bC1

where the value of L0 is 9 (mm) in this work. Therefore, value of C1 can be obtained: C1 = −442.2359.
When value of t is zero, value of ∆L1 should be zero too. Therefore, D can be worked out according to
Equation (30): D = 128.4085.
where the value of L0 is 9 (mm) in this work. Therefore, value of C1 can be obtained: C1 = −442.2359.
When value of t is zero, value of ∆L1 should be zero too. Therefore, D can be worked out according to
Equation (30): D = 128.4085.

The Middle
Materials Stage
2016, 9, 741 (85th–285th Months) 15 of 17

In the middle stage, there is a quadratic relationship between C and t when the preloading
stress is constant. Thus, f2 in the middle stage can be assumed as:
The Middle Stage (85th–285th Months)
2
f 2 relationship
In the middle stage, there is a quadratic At  Bt between
C1 (33)
C and t when the preloading stress
is constant. Thus, f 2 in the middle stage can be assumed as:
As same as the process in the initial stage does, expression of ∆L of the middle stage (∆L2) can be
derived: f = A0 t 2 + B0 t + C0 (33)
2 1
L0 A 3 B 2
As same as the processinL2the
 initial ln t  expression
stage does, t  C1t of
D∆L
Eof the middle stage (∆L2(34)
) can
be derived:
bS 0 3 2
L A0 B0
where values of Aʹ, Bʹ, C1ʹ,∆L 2 =
Dʹ and− Eʹ0 can
ln bet3worked
+ t2 +outC10 tin+the + E0 similar to the initial stage:
D0 way (34)
bS 0 3 2
Aʹ = −2.2574 × 10 , Bʹ = 1.2862 × 10 , C1ʹ = −0.7483, Dʹ = −442.2359 and Eʹ = 129.9444.
−5 −2

where values of A0 , B0 , C1 0 , D0 and E0 can be worked out in the way similar to the initial stage:
A 0 = −2.2574 × 10−5 , B0 = 1.2862 × 10−2 , C 0 = −0.7483, D0 = −442.2359 and E0 = 129.9444.
6.2.3. The Last Stage (285th Month Onward) 1

6.2.3.As same
The Lastas the process
Stage in theOnward)
(285th Month initial stage does, expression of ∆L of the last stage (∆L3) can be
derived:
As same as the process in the initial stage does, expression of ∆L of the last stage (∆L3 ) can
be derived: L A
L3   L00 ln A00 2t 2  Bt  C1  D (35)
∆L3 = − bS 0ln 2 t + B00 t + C1 00 + D00 (35)
bS0 2
where values
values of
of A“,
Aʺ, B“,
Bʺ, C
C1ʺ“ and Dʺ are 3.7139 × 10−4−, 41.0609, −442.2359, and 115.2410, respectively.
where 1 and D“ are 3.7139 × 10 , 1.0609, −442.2359, and 115.2410, respectively.

6.3. Results of Life Prediction

Equations (20), (30), (34) and (35) were plotted as follows. Figure 12 shows the dependence of
the residual
residual wall
wall thickness
thicknesson
onservice
servicetime.
time.Failure
Failurepoints
pointsaccording to to
according thethe
TCTCand SCSC
and arearemarked
marked on
the curves. It can be known that life-span of the L80 downhole tube
on the curves. It can be known that life-span of the L80 downhole tube in Halfaya is in Halfaya is 247 months
(approximately 20
(approximately 20years)
years)under
underinitial
initialstress
stressofof0.10.1
σsσor
s or
641641 months
months (approximately
(approximately 53 53 years)
years) under
under no
no initial
initial stress,
stress, whichwhich indicates
indicates thatthat an initial
an initial stress stress
of 0.1ofyield
0.1 yield strength
strength reducesreduces corrosion
corrosion life bylife by
more
more than half.
than half.

9
under no initial stress
8
7
6 under initial stress of 0.1s
L0-L (mm)

5 (641, 1.8)Failure point

(247, 1.8)Failure point according to the TC
4
according to the TC
3
2
(250,1.35)Failure (671,1.35)Failure point
1 according to the SC
point according to the SC
0
0 100 200 300 400 500 600 700
t (month)
Figure 12. Variation
Figure 12. Variation ofof residual
residual wall
wall thickness
thickness of L80 steel
of L80 steel with
with service
service time
time under
under different
different initial
initial
stress: 0 and 0.1 σ , respectively.
σs , respectively.
s

7. Conclusions
A mathematic model was proposed for predicting life of the downhole tubes. As the premise
of this model, corrosion mechanism and corrosion processes of the downhole tubes under different
Materials 2016, 9, 741 16 of 17

conditions were discussed. Conditions including service environmental parameters and stress as
well as their effects on corrosion rate were thought to vary with service duration. Suitability of the
model was also discussed. With the model applied, life-span of L80 downhole tubes in Halfaya was
predicted. The results show that the life-span is 247 months (approximately 20 years) under initial stress
of 0.1 yield strength or 641 months (approximately 53 years) under no initial stress, which indicates
that an initial stress of 0.1 yield strength reduces corrosion life by more than half.

Acknowledgments: The authors wish to acknowledgement the financial support of the National Basic Research
Program of China (973 Program project, No. 2014CB643300) and the Natural Science Foundation of China
(No. 51471034 and No. 51131001).
Author Contributions: Zhiyong Liu and Tianliang Zhao conceived and designed the experiments;
Tianliang Zhao and Jianpeng Hu performed the experiments; Tianliang Zhao analysed the data; Cuiwei Du and
Xiaogang Li contributed reagents, materials, analysis tools; and Tianlaing Zhao wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.

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