3. Previous research work: analytic prognostic and linear damage accumulation for buried petrochemical pipelines

In this section a comprehensive summary of a part of my previously published PhD thesis [16] and of the formerly published IFAC conference paper [14] will be done, and the results that this current chapter needs will be just cited.

#### 3.1 A brief introduction to the adopted methodology

The objective of my earlier research study, which will be enhanced in the present chapter and will be linked to CPP, was to develop an analytic linear model of prognostic capable of predicting the remaining useful lifetime and the degradation D curves of a buried petrochemical pipeline system subject to fatigue starting from an initial known damage and under a given environment [14, 16]. This shows to be beneficial for many reasons which are fewer pipe bending; reduced plant congestion, wind, and other loads; and protection from ambient temperature changes. This work is restricted here to normal service loads that consist of only soil action and internal pressure.

Petrochemical pipelines are systems that are used to transport natural gas and oil between sites. We believe that pipeline tubes are a major element in petrochemical industries. As a matter of fact, the prognostic of their life is essential in this industry since their availability has decisive and critical consequences on the cost of exploitation. Fatigue, which is due to internal pressure-depression variation along time, is the major failure cause of these systems. These pipelines are typically devised for ultimate limit states (resistance). Additionally, due to soil aggression influences, buried pipelines are subject to corrosion. Pipelines are designed as cylindrical tubes of thickness e and radius R.

A target failure probability of about 10<sup>5</sup> for pipelines is suggested by the DNV 2000 rules. Their major failure causes are soil settlements, seismic ground waves, deformations, buckling, stress concentration in welding and fitting, internal and external corrosion, pressure fluctuation over a long period, and vibration and resonance. Moreover, crack detection tools detect the crack propagation caused by failures due to fatigue.

An important part of the main pipes is exposed to external cracking, which is a dangerous setback for the industry of pipes, for example, in the USA, Canada, and Russia. External crack identification is accomplished using diverse nondestructive evaluation (NDE) methods. If cracks were detected during inspection, we should evaluate their influence on the remaining useful lifetime of the pipeline in order to select the action of maintenance that should be applied: do nothing/repair/replace. We judge the integrity of pipes by assuming that some defects after in-line inspection (ILI) can be still undetected; detected, but not measured; detected and measured.

Moreover, the objective in my publications was to assess the evolution of the lifetime of a system at each instant. Consequently, and for this purpose, the trajectories of degradation had been utilized in terms of the time of operation or cycles' number. Hence, we deduce the RUL variations from these trajectories of degradation. Thus, I have considered many industrial illustrations in the simulation of my

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

model in these earlier publications and work to prove the effectiveness of my model [1–6, 13–19]. Three case studies of pipelines were taken into consideration: buried, unburied, and subsea (offshore pipes). Each one of these situations necessitates different physical parameters like friction and soil pressure, corrosion, and atmospheric and water pressure. One of these cases is elaborated here which is the system of buried petrochemical pipes where three modes of pressure profiles (mode 1 = high, mode 2 = middle, and mode 3 = low-pressure conditions) were examined and simulated. My model showed that it presented a useful tool for a prognostic analysis and that it is very convenient in such industrial systems. Furthermore, it proved that it is less expensive than other models that require a huge number of measurements and data.

#### 3.2 Fatigue crack growth

The stress intensity factor was introduced to calculate the correlation between the crack growth rate, da/dN, and the stress intensity factor range, ΔK. The Paris-Erdogan's law [7] allows to evaluate the rate of propagation of the crack length a after its detection. This damage growth law is expressed by the following equation:

$$\frac{da}{dN} = C(\Delta K)^m \tag{1}$$

where

illustrations considering various continuous and discrete probability distributions in my 12 previous research papers verify this hypothesis and novel paradigm [20–31]. The extended Kolmogorov axioms (EKA for short) or the complex probability paradigm can be summarized and shown in the following figure (Figure 2).

In this section a comprehensive summary of a part of my previously published PhD thesis [16] and of the formerly published IFAC conference paper [14] will be

The objective of my earlier research study, which will be enhanced in the present chapter and will be linked to CPP, was to develop an analytic linear model of prognostic capable of predicting the remaining useful lifetime and the degradation D curves of a buried petrochemical pipeline system subject to fatigue starting from an initial known damage and under a given environment [14, 16]. This shows to be beneficial for many reasons which are fewer pipe bending; reduced plant congestion, wind, and other loads; and protection from ambient temperature changes. This work is restricted here to normal service loads that consist of only soil

Petrochemical pipelines are systems that are used to transport natural gas and oil between sites. We believe that pipeline tubes are a major element in petrochemical industries. As a matter of fact, the prognostic of their life is essential in this industry since their availability has decisive and critical consequences on the cost of exploitation. Fatigue, which is due to internal pressure-depression variation along time, is the major failure cause of these systems. These pipelines are typically devised for ultimate limit states (resistance). Additionally, due to soil aggression influences, buried pipelines are subject to corrosion. Pipelines are designed as cylindrical tubes

A target failure probability of about 10<sup>5</sup> for pipelines is suggested by the DNV 2000 rules. Their major failure causes are soil settlements, seismic ground waves, deformations, buckling, stress concentration in welding and fitting, internal and external corrosion, pressure fluctuation over a long period, and vibration and resonance. Moreover, crack detection tools detect the crack propagation caused by

An important part of the main pipes is exposed to external cracking, which is a dangerous setback for the industry of pipes, for example, in the USA, Canada, and Russia. External crack identification is accomplished using diverse nondestructive evaluation (NDE) methods. If cracks were detected during inspection, we should evaluate their influence on the remaining useful lifetime of the pipeline in order to select the action of maintenance that should be applied: do nothing/repair/replace. We judge the integrity of pipes by assuming that some defects after in-line inspection (ILI) can be still undetected; detected, but not measured; detected and

Moreover, the objective in my publications was to assess the evolution of the lifetime of a system at each instant. Consequently, and for this purpose, the trajectories of degradation had been utilized in terms of the time of operation or cycles' number. Hence, we deduce the RUL variations from these trajectories of degradation. Thus, I have considered many industrial illustrations in the simulation of my

3. Previous research work: analytic prognostic and linear damage

done, and the results that this current chapter needs will be just cited.

accumulation for buried petrochemical pipelines

3.1 A brief introduction to the adopted methodology

action and internal pressure.

Fault Detection, Diagnosis and Prognosis

of thickness e and radius R.

failures due to fatigue.

measured.

70

da dN, the crack growth rate = the increase of the crack length a per cycle N. <sup>Δ</sup>K að Þ¼ Y að ÞΔ<sup>σ</sup> ffiffiffiffiffiffi π a p , the intensity factor of the stress. Y að Þ, the component's crack geometry function. Δσ, the range of the applied stress in a cycle.

m and C, the constants of materials obtained experimentally; 2ð Þ ≤ m ≤4 and ð Þ 0<C ≪ 1 .

#### 3.3 The modeling of linear cumulative damage

To do the prognostic of a degrading element, my approach was to evaluate and to predict the end of life of the element by modeling and tracking the function of degradation. My model of damage, whose progress is up to the macro-crack initiation point, is illustrated in Figure 3 by the damage linear rule of Palmgren-Miner.

As a matter of fact, this law [7] is used to calculate the cumulative damage di of different stress levels σ<sup>i</sup> (i = 1, i = 2, ..., i = k) applied for ni cycles. Knowing that Ni is

Figure 3. Palmgren-Miner's linear rule of damage.

the total cycle's number of stress σ<sup>i</sup> to be applied and that lead to failure. The linear cumulative damage corresponding to the applied stresses (i = 1 to k) is provided by

$$D\_k = \sum\_{i=1}^k d\_i = \sum\_{i=1}^k \frac{n\_i}{N\_i} \tag{2}$$

We will consider three different levels of internal pressure to take into account the diverse states of pressure conditions which are low, middle, and high. Moreover, as the stress load is a function of the cycles N or of time t, then we can draw the trajectories of degradation of D(N) or D(t) in addition to the trajectories of RUL(N) or RUL(t) in terms of the total number of loading cycles N or in terms of the pressure time t. Therefore, my developed model of linear damage will be applied in order to

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

We will consider in our current work a pipeline transporting natural gas of radius R = 240 mm and of thickness e = 8 mm. The parameters in this case are <sup>C</sup> = 1.3 � <sup>10</sup>�<sup>14</sup> (under soil, buried pipelines) and <sup>m</sup> = 3 (metal). The initial crack length is considered to be a<sup>0</sup> = 0.02 mm. The crack length aC at the failure cycle time tC was assumed in the model to be equal to e=8 for justified reasons [16].

<sup>¼</sup> <sup>0</sup>:<sup>02</sup>

and gas content is Wp = 203.27 kg/m. The specific gravity of the pipe material and of the natural gas are, respectively, γpipe = 7850 kg/m<sup>3</sup> and γgas = 600 kg/m<sup>3</sup>

The internal pressure Pj is modeled following a triangular form and distribution in order to be similar to the real case of pipeline operating condition (pressure-

We will consider three maximal levels of Pj which are P<sup>0</sup> = 3, 5, and 8 MPa and with a period of repetition T. This repetition period varies depending on the conditions of exploitation; it is considered to be equal to 20 h. We note that these three levels are supposed to be the extreme conditions of the pipeline exploitations and are mean estimations of the real and actual random period and pressure rates. A trajectory of degradation D(N) is inferred at each of these three levels in terms of the cycle number N or pressure cycle time t. When Dt or DN attains the unit value, therefore the corresponding t = t<sup>C</sup> or N=NC is the lifetime of the pipeline in the

For the purposes of simulations, in Table 1, the values of pressure Pj are considered to be equal to the maximal values P0. The analytic linear prognostic model

depth of the pipe is taken as H = 7R, and the friction coefficient interval is

ð Þ� <sup>8</sup>=<sup>8</sup> <sup>0</sup>:<sup>02</sup> <sup>¼</sup> <sup>0</sup>:<sup>02</sup>

<sup>0</sup>:<sup>98</sup> <sup>¼</sup> <sup>0</sup>:<sup>020408163</sup>

. The weight per linear meter of pipe

. The

compute the pipeline system prognostic.

DOI: http://dx.doi.org/10.5772/intechopen.90157

Hence, from Eqs. (5) and (6), we get

aC � a<sup>0</sup>

<sup>D</sup><sup>0</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup>

0.5 ≤ μ ≤ 0.7 [16].

fatigue case.

Figure 4.

73

Triangular variation of internal pressure.

depression) (Figure 4).

3.5 The three levels of internal pressure simulations

<sup>¼</sup> <sup>a</sup><sup>0</sup> ð Þ� e=8 a<sup>0</sup>

The soil specific weight is γ = 9.843 kN/m<sup>3</sup>

The initial detectable crack a<sup>0</sup> at the cycle N0, the crack length aN at any cycle N, and the crack length aC at the failure cycle NC are estimated by a sensor, and their values are included in the model of damage prognostic in the equation of damage. It is expressed in my model by the resulting relation

$$D\_N = \frac{\mathfrak{a}\_N}{\mathfrak{a}\_C - \mathfrak{a}\_0} \tag{3}$$

Or in terms of the pressure cycle time t, the relation is given by

$$D\_t = \frac{a\_t}{a\_C - a\_0} \tag{4}$$

To simplify the study, it is suitable to adopt a measurement of damage denoted by D ϵ [0, 1] which is computed by the Palmgren-Miner's law of linear cumulative damage. The damage level in a system at a specific cycle which is due to fatigue is illustrated by a scalar function of damage denoted by D(t) or D(N). "No damage" corresponds to the value D = 0, and "total damage" or the appearance of the first macro-crack corresponds to D = 1.

#### 3.4 An expression for degradation

Therefore, my general prognostic analytic linear model function, which is a recursive relation for the sequence of D, is given by [16]

$$\begin{split} D\_{N} &= D(\mathbf{N}) = P\_{\text{reg}}(a\_{N}) \\ &= \frac{a\_{N-1}}{a\_{C} - a\_{0}} + \frac{\mathbf{C}}{a\_{C} - a\_{0}} \times (\pi a\_{N-1})^{3/2} \times \left[ \mathbf{0.6} \times \frac{\mathbf{1} + 2(a\_{N-1}/\varepsilon)}{\left(\mathbf{1} - a\_{N-1}/\varepsilon\right)^{2}} \right]^{3} \times \left( P\_{\text{j}} \mathbb{R}/\varepsilon \right)^{3} \end{split} \tag{5}$$

where C, the environment parameter; e, the pipe thickness; R, the pipe radius; a0, the initial crack length at the cycle N0; aN�1, the crack length at the load cycle N–1; ac: the crack length at the failure cycle NC. It was assumed in the model that aC ¼ e=8 for justified reasons [16]; Pj: the pipe internal pressure.

Or in terms of the pressure cycle time t, the recursive relation for the sequence of D is given by

$$\begin{split} D\_t &= D(t) = P\_{\text{reg}}(a\_t) \\ &= \frac{a\_{t-1}}{a\_C - a\_0} + \frac{\text{C}}{a\_C - a\_0} \times (\pi a\_{t-1})^{3/2} \times \left[ 0.6 \times \frac{1 + 2(a\_{t-1}/e)}{(1 - a\_{t-1}/e)^{\frac{3}{2}}} \right]^3 \times \left( P\_{\text{j}} \text{R}/e \right)^3 \end{split} \tag{6}$$

Consequently, the previous recursive relation leads to a sequence of Dt values with N<sup>0</sup> ≤ N ≤ NC or t<sup>0</sup> ≤t≤tC whose limit is DC = 1:

$$D\_0 = \frac{a\_0}{a\_C - a\_0}; \ D\_1 = \frac{a\_1}{a\_C - a\_0}; \ D\_2 = \frac{a\_2}{a\_C - a\_0}; \ \cdots \ \vdots \ D\_{t-1} = \frac{a\_{t-1}}{a\_C - a\_0}; \ \quad D\_t = \frac{a\_t}{a\_C - a\_0} \tag{7}$$

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

We will consider three different levels of internal pressure to take into account the diverse states of pressure conditions which are low, middle, and high. Moreover, as the stress load is a function of the cycles N or of time t, then we can draw the trajectories of degradation of D(N) or D(t) in addition to the trajectories of RUL(N) or RUL(t) in terms of the total number of loading cycles N or in terms of the pressure time t. Therefore, my developed model of linear damage will be applied in order to compute the pipeline system prognostic.

#### 3.5 The three levels of internal pressure simulations

We will consider in our current work a pipeline transporting natural gas of radius R = 240 mm and of thickness e = 8 mm. The parameters in this case are <sup>C</sup> = 1.3 � <sup>10</sup>�<sup>14</sup> (under soil, buried pipelines) and <sup>m</sup> = 3 (metal). The initial crack length is considered to be a<sup>0</sup> = 0.02 mm. The crack length aC at the failure cycle time tC was assumed in the model to be equal to e=8 for justified reasons [16]. Hence, from Eqs. (5) and (6), we get

$$D\_0 = \frac{a\_0}{a\_C - a\_0} = \frac{a\_0}{(e/8) - a\_0} = \frac{0.02}{(8/8) - 0.02} = \frac{0.02}{0.98} = 0.020408163$$

The soil specific weight is γ = 9.843 kN/m<sup>3</sup> . The weight per linear meter of pipe and gas content is Wp = 203.27 kg/m. The specific gravity of the pipe material and of the natural gas are, respectively, γpipe = 7850 kg/m<sup>3</sup> and γgas = 600 kg/m<sup>3</sup> . The depth of the pipe is taken as H = 7R, and the friction coefficient interval is 0.5 ≤ μ ≤ 0.7 [16].

The internal pressure Pj is modeled following a triangular form and distribution in order to be similar to the real case of pipeline operating condition (pressuredepression) (Figure 4).

We will consider three maximal levels of Pj which are P<sup>0</sup> = 3, 5, and 8 MPa and with a period of repetition T. This repetition period varies depending on the conditions of exploitation; it is considered to be equal to 20 h. We note that these three levels are supposed to be the extreme conditions of the pipeline exploitations and are mean estimations of the real and actual random period and pressure rates. A trajectory of degradation D(N) is inferred at each of these three levels in terms of the cycle number N or pressure cycle time t. When Dt or DN attains the unit value, therefore the corresponding t = t<sup>C</sup> or N=NC is the lifetime of the pipeline in the fatigue case.

For the purposes of simulations, in Table 1, the values of pressure Pj are considered to be equal to the maximal values P0. The analytic linear prognostic model

Figure 4. Triangular variation of internal pressure.

the total cycle's number of stress σ<sup>i</sup> to be applied and that lead to failure. The linear cumulative damage corresponding to the applied stresses (i = 1 to k) is provided by

> di <sup>¼</sup> <sup>X</sup> k

The initial detectable crack a<sup>0</sup> at the cycle N0, the crack length aN at any cycle N, and the crack length aC at the failure cycle NC are estimated by a sensor, and their values are included in the model of damage prognostic in the equation of damage. It

aC � a<sup>0</sup>

aC � a<sup>0</sup>

To simplify the study, it is suitable to adopt a measurement of damage denoted by D ϵ [0, 1] which is computed by the Palmgren-Miner's law of linear cumulative damage. The damage level in a system at a specific cycle which is due to fatigue is illustrated by a scalar function of damage denoted by D(t) or D(N). "No damage" corresponds to the value D = 0, and "total damage" or the appearance of the first

Therefore, my general prognostic analytic linear model function, which is a

<sup>3</sup>=<sup>2</sup> � <sup>0</sup>:<sup>6</sup> �

where C, the environment parameter; e, the pipe thickness; R, the pipe radius; a0, the initial crack length at the cycle N0; aN�1, the crack length at the load cycle N–1; ac: the crack length at the failure cycle NC. It was assumed in the model that

Or in terms of the pressure cycle time t, the recursive relation for the sequence of

<sup>3</sup>=<sup>2</sup> � <sup>0</sup>:<sup>6</sup> �

Consequently, the previous recursive relation leads to a sequence of Dt values

aC � a<sup>0</sup>

; <sup>D</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup>

1 þ 2ð Þ aN�<sup>1</sup>=e ð Þ <sup>1</sup> � aN�<sup>1</sup>=<sup>e</sup> <sup>3</sup>

1 þ 2ð Þ at�<sup>1</sup>=e ð Þ <sup>1</sup> � at�<sup>1</sup>=<sup>e</sup> <sup>3</sup>

; <sup>⋯</sup> ; Dt�<sup>1</sup> <sup>¼</sup> at�<sup>1</sup>

" #<sup>3</sup>

2

aC � a<sup>0</sup>

" #<sup>3</sup>

2

� PjR=<sup>e</sup> � �<sup>3</sup> (5)

� PjR=<sup>e</sup> � �<sup>3</sup> (6)

; Dt <sup>¼</sup> at

aC � a<sup>0</sup> (7)

i¼1

ni Ni

(2)

(3)

(4)

Dk <sup>¼</sup> <sup>X</sup> k

Or in terms of the pressure cycle time t, the relation is given by

is expressed in my model by the resulting relation

Fault Detection, Diagnosis and Prognosis

macro-crack corresponds to D = 1.

3.4 An expression for degradation

DN ¼ D Nð Þ¼ Progð Þ aN

Dt ¼ D tðÞ¼ Progð Þ at

þ

C aC � a<sup>0</sup>

; <sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>1</sup>

with N<sup>0</sup> ≤ N ≤ NC or t<sup>0</sup> ≤t≤tC whose limit is DC = 1:

aC � a<sup>0</sup>

<sup>¼</sup> at�<sup>1</sup> aC � a<sup>0</sup> þ

C aC � a<sup>0</sup>

<sup>¼</sup> aN�<sup>1</sup> aC � a<sup>0</sup>

D is given by

<sup>D</sup><sup>0</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup>

72

aC � a<sup>0</sup>

recursive relation for the sequence of D, is given by [16]

� ð Þ πaN�<sup>1</sup>

aC ¼ e=8 for justified reasons [16]; Pj: the pipe internal pressure.

� ð Þ πat�<sup>1</sup>

i¼1

DN <sup>¼</sup> aN

Dt <sup>¼</sup> at


Table 1.

Characteristics of each internal pressure mode.

(Eqs. (5) and (6)) simulation is achieved for each internal pressure level (low, middle, and high).

A huge amount of pressure simulations of the order of hundreds of millions are required to estimate the real system lifetime; hence, we have used an approximated model of lifetime simulation of the order of 10,000,000 iterations. Accordingly, we have considered for this purpose a high-capacity computer system: a workstation computer with parallel microprocessors, a 64-Bit operating system, a 64 GB RAM, as well as a 64-Bit MATLAB version 2019 software.

#### 3.6 RUL computation

The evaluation of the remaining useful lifetime of the system is the major objective in a prognostic study. Since the RUL is the complement of the damage curve D(t), it can be deduced from it. Accordingly, at each time t, the required RUL is the length from cycle time t to the critical cycle time tC that corresponds to the threshold D = 1. The entire RUL is inferred using the following relation:

$$RUL = t\_C - t\_0 \tag{8}$$

examples from several and other prognostic studies [7, 8]: <sup>C</sup> = 5.2 � <sup>10</sup>�<sup>13</sup> (free air, unburied pipelines), <sup>C</sup> = 1.3 � <sup>10</sup>�<sup>14</sup> (under soil, buried pipelines), <sup>C</sup> = 2 � <sup>10</sup>�<sup>11</sup>

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

In this section, the novel complex probability paradigm will be presented after

It is very well-known that in systems engineering, the remaining useful lifetime and the degradation prediction is profoundly linked to many aspects (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.) that usually have a stochastic and chaotic behavior which reduces the degree of our certain system knowledge [32–35]. Consequently, the lifetime of the system becomes a random variable and is computed by the arbitrary time tC which is evaluated when sudden failure occurs due to these stochastic causes and chaotic factors. We can deduce from the CPP that we can foretell the exact probabilities of RUL and D with certitude in the whole set C ¼ R þM if we add to the probability measure of a random variable in the real set R the corresponding imaginary counterpart M since Pc = 1 perpetually and constantly. In fact, prognostic is based on the forecast of a system remaining useful lifetime at any cycle N or instant t and during the system operation. Therefore, we can make use of this novel idea and procedure to do the prognostic analysis of the

Let us consider a system degradation trajectory D(t) where we study a specific instant (or cycle) tk. The system age is measured by the number of years and by the variable tk (Figure 5). From the illustrated figures (Figures 5a and 5b), we can infer that at the system age tk of the prognostic study must give the prediction of the failure instant tC. Therefore, the RUL predicted here at the instant tk has the

As a matter of fact, at tk = 0 (at the beginning) (point J), the system is intact, then the failure probability of the system is Pr = 0, the chaotic factor in our prognostic is null (MChf = 0) because no chaos exists yet, and our knowledge of the unharmed and

RULð Þ¼ 0 tC � tk ¼ tC � 0 ¼ tC:

If tk = tC (point L), the system is completely damaged, then RUL(tC) = tC � tC = 0, and therefore the failure probability of the system is one (Pr = 1). Failure occurs at this point. Thus, our knowledge of the totally worn-out system is perfect (DOK = 1)

If 0 < tk < tC (point K, where J < K < L), the probability of occurrence of this instant and the probabilities of prediction of RUL and D are both less than 1 and are imperfect in R (0 < Pr < 1). This is the result of non-zero chaotic factors influencing the system (MChf > 0). The system degree of our knowledge which is subject to chaos is thus uncertain and is consequently less than one in R (0.5 < DOK < 1).

undamaged system is complete and certain (DOK = 1); consequently,

and the harmful task of chaos has finished; hence it is no more applicable

RUL tð Þ¼ <sup>k</sup> tC–tk (9)

4. The complex probability paradigm applied to prognostic

(for offshore pipelines), and m = 3 (metal).

DOI: http://dx.doi.org/10.5772/intechopen.90157

4.1 The basic parameters of the new model

system RUL and degradation prediction and evolution.

applying it to prognostic.

following value:

(MChf = 0).

75

where tC is the necessary cycle time for the appearance of the first macro-cracks that means to reach failure, and t<sup>0</sup> is the initial cycle time considered in general to be equal to 0.

Consequently, my prognostic model computes the RULs for the three internal pressure modes that can be now simply inferred from these three curves at any instant t or at any active cycle N in this manner:

For mode 3, RUL3(t) = tC<sup>3</sup> – t. For mode 2, RUL2(t) = tC<sup>2</sup> – t. For mode 1, RUL1(t) = tC<sup>1</sup> – t.

#### 3.7 The effects of environment in the suggested prognostic model

Two parameters which are C and m embody the effects of the environment. These two parameters are associated to the material environment. C and m depend on the initial crack length, on the geometry and size of the specimen, and on the testing conditions (such as the loading ratio σ). These two parameters affect the performance of the material during the process of fatigue through the crack propagation. The influencing parameters on this fatigue process, like humidity, temperature, material nature, geometry dimensions, applied load location, corrosion, water action, soil pressure and friction, atmospheric pressure, etc., can be stochastic and can be also embodied by C and m. Furthermore, it is crucial to note here that these two parameters can be as well random variables and hence can be represented by probability distributions materializing the environment stochastic and chaotic influences on the system. It is also important to mention that these two parameters are computed by the mean of experiments in real conditions. We give here some

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

examples from several and other prognostic studies [7, 8]: <sup>C</sup> = 5.2 � <sup>10</sup>�<sup>13</sup> (free air, unburied pipelines), <sup>C</sup> = 1.3 � <sup>10</sup>�<sup>14</sup> (under soil, buried pipelines), <sup>C</sup> = 2 � <sup>10</sup>�<sup>11</sup> (for offshore pipelines), and m = 3 (metal).
