**3. Model for estimating non-destructive testing**

Problems of item content analysis are everywhere for any oil and gas systems in their life cycle. Pipes and pipelines, the equipment (e.g., fountain armature, columned heads and welded tanks), monolithic walls of buildings and the constructions, to be checked in the presence of emptiness, can be considered as such items—see **Figure 6**.

For solving some problems of item content analysis, the existing probabilistic model "information faultlessness after checking" may be used by renaming input and output [6]. For example, for estimating non-destructive testing, the probability of soundness of the checked

**Figure 6.** Examples of item content analysis.

allows a customer to formulate better justified requirements and specifications, a developer to implement them rationally without wasted expenses and a user to use the system's potential

In a general case, a probabilistic space (*Ω*, *B*, *P*) for the evaluation of system operation processes should be proposed, where *Ω* is a limited space of elementary events; *B* is a class of all subspaces of the *Ω* space, satisfied to the properties of *σ* algebra; *P* is a probability measure

The descriptions for some from the proposed probabilistic models and methods for their transformations, adaptations, applications and result interpretations are the following.

Problems of item content analysis are everywhere for any oil and gas systems in their life cycle. Pipes and pipelines, the equipment (e.g., fountain armature, columned heads and welded tanks), monolithic walls of buildings and the constructions, to be checked in the pres-

For solving some problems of item content analysis, the existing probabilistic model "information faultlessness after checking" may be used by renaming input and output [6]. For example, for estimating non-destructive testing, the probability of soundness of the checked

*}* is limited, there is enough to establish

in the most effective way [1–12].

62 Probabilistic Modeling in System Engineering

→*pk = P(ω<sup>k</sup>*

a reflection *ω<sup>k</sup>*

on the space of elementary events *Ω*. Because *Ω = {ω<sup>k</sup>*

*)* like *pk 3 0* and ∑ *k pk* <sup>=</sup> 1.

**Figure 5.** The proposed way to support making-decisions in quality and safety.

**3. Model for estimating non-destructive testing**

ence of emptiness, can be considered as such items—see **Figure 6**.

item (renamed) may be estimated instead of the probability of information faultlessness during the required term (according to referenced model [6]). A soundness of the checked item means the zero of defects (or anomalies) after non-destructive testing during the given term.

What about the effectiveness of non-destructive testing methods for some technical items?

Example 1: Let an application of some instruments of non-destructive testing be planned in the applications to check 10,000 conditional items (the items can be meters of pipes, square meters of walls in storehouses and so on). The operator using instruments forms a system for non-destructive testing. Speed of testing equals 5000 items a day. Taking into account the human factor, a frequency of first-type errors (when the absence of defect [anomaly] is accepted as defect [anomaly]) equals one error a week. The mean time between second-type errors for the system (when real defect [anomaly] does not come to light) is equal once a month. The non-destructive testing is performed permanently for 10 days. It needs to estimate the maximum density of defects (anomalies) for which the probability of soundness of the checked 10,000 conditional items is more than 0.90.

Results of probabilistic modeling have shown that the required density is about 0.02%, that is, 2 defects (anomalies) on 10,000 items. In addition it is expedient to notice that since density of defects about 1%, the probability of soundness is stabilized at level 0.88. It does not fall as less, because first-type and second-type errors seldom occur in example 1.

Example 2: Continuing example 1, it needs to prove minimum speed of non-destructive testing, the checked volume for which the probability of soundness of 10,000 conditional items will exceed 0.95 at continuous work within 8 h of working hours.

The results of probabilistic modeling are reflected in **Figure 7**.

The analysis shows that the found rational speed is about 1100 items per hour. And the part of defects after the control in the checked-up volume of 10,000 items will be 0.0008% against the primary 0.02%. It can be interpreted: at the checked volume of 1,00,000 items (i.e., in 10 times more primary 10,000, when quantity of defects is 20), the average residual quantity of defects

**Figure 7.** The way for rationale speed of non-destructive testing.

will not exceed 1. It means that under the second example conditions, 19 from 20 defects will be revealed in time with probability 0.95 and more.

combination). In case of detecting a danger source an operator recovers system integrity. The ways of integrity recovering are analogous to the ways of technology 1. Faultless operator's actions provide the neutralization of a danger source trying to penetrate into a system. When operators alternate a complex diagnostic is held. A penetration of a danger source is possible only if an operator makes an error but a dangerous influence occurs if danger is activated before the next diagnostic. Otherwise the source will be detected and neutralized during the

**Figure 8.** Some accident events for technology 2 (left – "Correct operation", right – "a loss of integrity" during Treq.).

It is supposed for technologies 1 and 2 that the used diagnostic allows to provide necessary system integrity recovery after revealing danger source penetration into a system or consequences of influences. Assumption: for all time input characteristics, the probability distribution function (PDF) exists. Thus, the probability of the correct system operation within the given prognostic period (i.e., the probability of success) may be computed as a result of the use of models. For identical damage risk, to lose integrity is an addition to 1 for the probability

There are possible next variants for technologies 1 and 2: variant 1 in the given prognostic period Treq is less than the established period between neighboring diagnostics (Treq < Tbetw. + Tdiag); variant 2 in the assigned period Treq is more than or equals to the established period between

The main output of integration modeling is the probability of the correct system operation or risk to losing system integrity during the given period of time. If probabilities for all

Tbetw. + Tdiag). Here, Tbetw. is the time between the end of the diag-

Probabilistic Modeling Processes for Oil and Gas http://dx.doi.org/10.5772/intechopen.74963 65

next diagnostic.

of correct system operation, R = 1−P [3–4].

3

**4.2. Integration of probabilistic models for complex structures**

nostic and the beginning of the next diagnostic, Tdiag is the diagnostic time.

neighboring diagnostics (Treq
