Article
citation information:
Droździel, P., Vitenko, T.,
Zhovtulia, L., Yavorskyi, A., Oliinyk, A., Rybitskyi, I., Poberezhny, L.,
Popovych, P., Shevchuk, O., Popovych, V. Non-contact method of estimation of
stress-strain state of underground pipelines during transportation of oil and
gas. Scientific Journal of Silesian
University of Technology. Series Transport. 2020, 109, 17-32. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2020.109.2.
Paweł DROŹDZIEL[1], Tetiana VITENKO[2], Liubomyr ZHOVTULIA[3], Andrii YAVORSKYI[4], Andrii OLIINYK[5],
Ihor RYBITSKYI[6],
Liubomyr POBEREZHNY[7],
Pavlo POPOVYCH[8], Oksana SHEVCHUK[9],
Vasyl POPOVYCH[10]
NON-CONTACT
METHOD OF ESTIMATION OF STRESS-STRAIN STATE OF UNDERGROUND PIPELINES DURING
TRANSPORTATION OF OIL AND GAS
Summary. Development and
implementation of contactless methods for determining the stress-strain state
of pipelines in the process of transportation of energy hydrocarbons is
important for ensuring its safe operation. The authors developed a method for
determining the change in the stress-strain state of the underground part
of the main oil and gas pipelines according to the data about the displacement
of a certain set of points of the axis of the pipeline. This study was
conducted on a linear section of the main gas pipeline, where a landslide in
2010 created a force pressure on the pipeline, resulting in a pipeline rupture.
Keywords: underground pipelines, stress-strain state,
methodology, risk assessment, mathematic model, axis coordinates
1. INTRODUCTION
It is known that some main pipelines have been operating
for more than 20 years and about a quarter for more than 30 years. The high
level of technical requirements for the reliability and efficiency of their
operation requires the improvement of the accuracy of the evaluation of their
properties during operation [7,14]. The accumulation of any damage changes the
reliability, material characteristics and stability of pipeline materials to
static and dynamic loads. It is important to establish changes in the
stress-strain state and the physical and mechanical properties of the pipe
material. Particular attention should be paid to the determination of the stress-strain state since its changes in the degradation of pipeline material
contribute to the propagation of damage, in particular, the origin and growth
of microcracks. It is known that during the operation of pipeline transport,
the properties of the pipe material deteriorate. It is proved that cyclic
loading increases the intensity of damage, which causes a higher defect of the
metal of the main pipeline [11,12,15,16]. In addition, there are some known
works wherein a large effect of structural defects on the properties of
materials is revealed [6,8,11,13]. Interest in such phenomena is caused by the
specific nature of changes in the mechanical and physical properties of the
material after prolonged operation, when the anisotropy of the material is very
clearly manifested [11,12]. For this reason, the stress-strain state under cyclic loading of pipeline transport and the
structure and properties of the material to be exploited differ significantly
from those in the initial state. The problem of safe and reliable
transportation of energy hydrocarbons to the end consumer is one of the
priority areas of any state to be solved. Most pipeline failures are due to
mechanical and corrosion factors. Depressurisation and release of transported
product into the environment pose significant environmental risks [1,2,25,28].
Therefore, it is necessary to regularly monitor the state of the pipeline [20,25,26]
and the condition of its insulating coating. In recent years, the problem of
ensuring reliable and long-term mechanical stability of long-length engineering
structures is increasingly being considered in view of assessing and prognosis
the processes that take place in the earth's crust. According to the statistics of pipeline accidents [3,5], 14.9% of the accidents are because of geodynamic processes (damage
to pipelines as a result of the earth's surface activity: landslides, mudflows,
etc.).
When erosion forms are crossed with oil and gas pipelines,
bends (crimps) are formed in those areas, in particular, in the vertical plane
at short distances. With increased geodynamic stresses and external influences
in such areas, loads can locally grow, stimulating violations of the tightness
and integrity of the pipeline. Moving the axis of the pipeline results in the
stress-strain state change, the critical values of which cause the destruction
of the metal.
Control of the stressed-deformed state of the pipeline is
important for ensuring its working capacity. It is especially important for long-term operated pipelines [6,8,12,22,24], as well as in difficult sections
of the route (saline soils, landslides, marshy areas). There, stress corrosion
cracking [6,8,10,19,22,27], pitting corrosion [6,7] and the like may occur due to
the strengthening of the corrosion factor by mechanical stresses. Worthy of
note is the influence of the induced current on the areas near the transmission
line [4,6,8,9,22].
An analysis of existing methods for determining the
stress-strain state of oil and gas pipelines under conditions of geological
risk [25] made it possible to evaluate their advantages and disadvantages. The
main obstacle is in the difficulties of underground oil and gas pipelines
availability for contact diagnostic methods. Based on this, the urgent task is
to create a system to prevent accidents of pipelines laid in severe engineering
and geological conditions. To solve this problem, for the process of further
research, it is necessary to outline the changes influence in operating
conditions and operating parameters concerning the strength and stability of
the pipeline, as well as to find potentially dangerous sections.
2. METHOD OF INVESTIGATION
As a result of
theoretical studies, a method was developed to determine the change in the
stress-strain state of an underground section of an oil and gas pipeline using
the data on the movement of a certain set of points [25], based on a developed
mathematical model of the deformation process at the underground section of a
pipeline under the influence of soil weight and its movement.
As input data for
determining stresses, the displacement values of a pipeline axis certain set of
points are used. For this purpose, the specific and design spatial position of
the oil and gas pipeline is compared. The spatial coordinates of the oil and
gas pipeline axis are determined by the non-contact method, using modern locating
detectors and global positioning tools.
The modelling of
deformation process at the underground sections of the main pipelines according
to the data on the change of the spatial configuration of their axis is based
on the approach proposed in [18] for the aboveground sections of the pipelines.
In this case, the geometric configuration of the pipeline axis is determined
with some accuracy at the control point in time using the experimental methods
[17].
For this research, a
linear section of the "Pasichna-Dolyna" gas pipeline D 500 per km 5.1
was selected. This is where the landslide, which occurred in 2010, caused a
force pressure on the pipeline, resulting in a pipeline rupture.
The initial position of
the pipeline was taken as a geodetic survey conducted by PJSC
"Prykarpattransgaz" (Fig. 1) in the form of a topographic plan with
the pipeline route and the coordinates of the pipeline axis. Surveying was
carried out after repair works as a result of the landslide in 2010.
By imposing on the
primary profile of the coordinate route, the measured true position of the
pipeline axis, the data necessary to determine the magnitude of the
stress-strain state of the pipeline by the developed method were obtained.
The initial data for the
calculation are the values of displacements of a certain set of points of the
pipeline axis, which are expressed as functions that describe the change of the
studied area geometry in the radial, transverse and longitudinal directions,
respectively: ρ(s, ϕ, r, t);
ω(s, ϕ, r, t); Ψ(s, ϕ, r, t). They are either given or
expressed during problem solving.
Fig. 1.
Fragment of a topographic map of pipeline position geodetic survey
3. MATHEMATICAL MODELLING OF THE UNDERGROUND
SECTION DEFORMATION PROCESS
While modelling the process of main pipelines underground
sections deformation, based on data of change in the spatial configuration of
their axis, the approach suggested for the on-surface pipeline part is applied.
In such a case, using the experimental methods [21,29], the geometrical
configuration of the pipeline axis is determined, including the specific
accuracy at the control moment of time. It is assumed that the initial position
of the pipeline axis is known (for example, according to the design
documentation). Thus, for the radius vector of the pipeline point, the
following relationship is written.
(1)
where:
s, ϕ, r -
related to the investigated area of underground pipeline, which is simulated as
the curved cylindrical body with coordinates respectively:
s - along the
pipeline axis;
ϕ - by vectorial angle;
- radius vector of the point on the upper
generatrix of the pipeline;
D - exernal diameter of the pipeline;
ρ(s, ϕ, r, t); ω(s, ϕ, r, t); Ψ(s, ϕ, r,
t) - functions that describe the geometry change of the investigates site,
respectively, in radial, transverse and longitudinal directions and are either
given or those that are expressed in the process of solving the problem;
; ; - vectors of the tangent binormal and
normal to the upper generator. At the initial time, when the pipeline is
considered an object with a rectilinear axis, the dependence (1) in coordinate
form can be written as follows:
(2)
where:
R1, R2 - respectively inner and outer radii of the
pipeline;
L - length of the investigated area.
In controlled time moment the dependence (1) is written as:
(3)
where positions of s, ϕ, r
gain the same meaning as in (2), ; ; - points position of the upper simulated area, D - pipeline
diameter; ; ; - positions of
normal vector to the upper generatix; ; ; - position of
binormal vector.
When constructing the (3), the following suppositions were used.
Thus, the only output information concerning the geometry change of the
underground sections are the coordinates of its’ deformed axle, then in
(1), it is assumed that:
(4)
That is due to the fact, that, coordinates of the
upper generatrix are experimentally defined and are set in the form of points
position , and for origination of () interpolation or
approximation procedures [10,19,21] are used, while there is no information
about the behaviour of ; та , which makes their record
in such a form in which it was written for undeformed areas. If the
representation of (3) leads to physically unrealistic results, these functions
are simulated by the techniques, where the cross-section configuration change
is counted towards the various types of its presentation; ellipticity,
pear-shape, ellipticity parameter spacing of axle degree of deformability,
thus, the mentioned methods are justified for surface areas when the
information concerning the cross-section damages is available at least
visually. In the case of the underground sections, the representation of (3),
is justified with the lack of information concerning the cross-section
deformation. This explains the choice of , as taking into account the
underground areas, it is also not impossible to perform the visual inspection
of the hypothesis justification concerning the flat sections. If the same
methods are used as for the investigation of both underground and surface
areas, it is thus, at different ways of setting ; ; , there is one more problem;
it is difficult to make the balance equation for underground areas, as it is
impossible to take into account in these equations, the action of mass forces
(weight of the pipe, weight of the product, weight of the soil acting on each
section of the pipeline).
Thus, given (2) and (3), the following sequence of
calculations is performed:
1.
In the controlled and initial moment of time, the
vectors of local basis are defined in each point of the simulated area [17,22]:
(5)
,
where is
calculated according to (2), а - according to (3).
Calculating the derivatives are carried out by direct differentiation of
(2) and (3) corresponding to the coordinates.
2.
Based on (4), the components of metric tensor are
defined:
, (6)
,
3.
Components and form matrix, and
for the correctness of the calculations the hypothesis should be carried out:
(7)
Performing
of (6) based on (5) allow to provide calculations of matrix contravariant
components and as matrix
components, inverse to these:
(8)
It is
obvious, that according to (2)
(9)
Thus,
in case of small strains, the statements of (6) are performed, as in this case:
, ,
4.
Components of tensor strain are calculated by the [20]:
(10)
5.
Based on (4) - (9), the tensor strain components
are defined according to Hook’s Law by applying the linear elastic theory
device [21]:
(11)
The
listed calculations can be performed within the model of an anisotropic body:
(12)
where - tensor components of material elastic
modulus, although (11), is used only if the pipeline material is substantially
anisotropic, and coefficients are known. For engineering calculations,
(10) is typically used, where and - material Lame parameters, related to
Young's modulus and Poisson coefficient of the material in the following way:
(13)
As for pipeline steels, it is generally taken
Е=210000 MPa= 0,3.
In represented (10) function is the first strain
invariant and is calculated by the formula:
(14)
Where is calculated by
(9), and - according to
(7).
Determination of components allows
to identify the most dangerous investigating areas, concerning the sector
stress state changes, and if at the initial time the pipeline stress is equal
zero, then (10) allows to assess the true strain values. Stress acceptance
criterion may be elastic strength value , or yield strength , in case when the given values are different for various
types of piping steels and are determined from the reference literature. It
should be mentioned that the described approach to the assessment of the
underground stress state is integral, and it does not require the detailed
information on loading and stress, the impact of which on the areas is due to
displacement measurements. In case, when some tensions (for instance, due to
pressure impact, temperature changes, etc.) are acquainted, it is possible to
use the superposition principle of elastic theory:
(15)
where - tensions,
determined by (10), - acquainted
tensions, - tensions of
unknown nature.
4. ACCURACY EVALUATION FOR SPATIAL ATTITUDE
INTERPOLATION OF ABOVE-GROUND PIPELINE DEFORMED AXLE
For realisation of the method of stress-strain state
assessment, expressed by the dependencies (1)-(13), it is necessary, by means
of experimentally measured upper generatrix points positions (, ), to get the expression
for radius vector for every generatrix position as , where and are continuous
functions. For this purpose, the widely known interpolation device is used,
applying interpolative cubic spline [19,21] or
interpolative cubic spline with test data smoothing. For interpolative cubic
spline, the interpolation grid settings are set [12,22],
characterised by the relationships between the minimum and maximum distances
between interpolation nodes:
(16)
where - accuracy necessary for
interpolating of the function by spline , value of sets the accuracy level of function value
assignment at interpolation nodes; - norm of function at given metric space [10]. Dependence
(14) can be written in a more compact form, bearing in mind that for the main
pipelines, the radius of axis curvature should meet the hypothesis:
(17)
where - pipeline diameter, - the constant given by the value , - radius of pipeline curvature, which for
engineering calculations can be written in the form:
(18)
Taking into account (16) and (17), the dependence (15)
for equally spaced grid for nodes coordinates measuring in increments can be written in
the form:
(19)
(20)
For pipeline section with the length , pipe diameter with measurement accuracy level 1 cm,
step value, with which it is necessary
to measure the coordinates of the points of the upper generatrix, with step metres, which is quite acceptable in
critical. The interpolation cubic spline peculiarity is the following: at
its’ development, the accuracy of the interpolation setting affects
significantly the axis interpolation accuracy. As a rule, the significant deviation
from real data yields results that do not correspond to the actual physical
picture of the process. The way out is possible by means of implementation of
two approaches:
1.
Applying the other implementation methods
(Lagrange, Chebyshev and Hermite polynomial) or approximation by the LS method,
with the resulting curves can differ significantly from real in some cases
(insufficient number of interpolation nodes, their inappropriate placement,
etc.);
2. Use of
approaches, related to embedding the smoothing spline device, allowing to
reduce the error of points position measuring by means of some correction
coefficients that depend on the accuracy of measuring these points position by
testing methods. While embedding the smoothing splice device, the desired smoothing
function minimises on class integrated on function interval
with their square of functional in the form of [23]:
(21)
Formula (18) requires detailed explanation: - positions of
actually measured points; - points
positions on the curvature, describing the spline; - weighting
coefficients. Minimisation problem (18) is solved for different values. In extreme cases, if for any , then the constructed spline will not actually be a
smoothing, it will pass through all nodes with point positions . If, then the actually obtained line will be straightforward
since it delivers the extremum of a functional in the form of.
(22)
which,
obviously, will have a minimum for - that is, - straightforward
line. With the knowing of performed measurements accuracy, it is possible to get the values , where in the function configuration will, on the one
hand, smooth the effect of measurements error, and on the other hand, will not
allow to lose the features of the real section configuration. This can be
depicted as a simulation in the following way (Fig. 2).
I – absolute smoothing; II –
non-smoothed spline; III – smoothed curvature
Fig. 2.
Test data smoothing
Optimising methods (21) with parameters , which characterise the level of data smoothing depending on
the measurement accuracy, are well-known, and they are used for above-ground sections
[21], for this reason, their application for underground sections is
well-reasoned. In particular, the procedure of functional minimising (20) is
used, by implementation of iteration procedure, at each step of which the
coefficients are based on the formula:
(23)
which is realised until
the fulfilment of the condition is achieved
(24)
In formulae (22), (23) - iteration process step number; - accuracy of node points position
measurement, - smoothing coefficient value at
iteration process step , - smoothed positions of node point after minimising procedure (20) at
iteration process step under number ; - initial non-smoothed positions of this
node point. Test calculations within implementation of these method show, that
by the embedding this smoothing iteration procedure, the error of stress
evaluation is MPa for the operating
pipeline section, displacement measurement of which is carried out with the
accuracy of 1 mm for the pipeline section with the L=100м.
The following
assumptions were used in building the mathematical model:
·
since the only initial information about
the change in the geometry of the underground section is the coordinates of its
deformed axis, it is assumed in (1) that the coordinates of the upper
generatrix are determined experimentally and are given as coordinates of points
, and for obtaining () interpolation or approximation
procedures are used [25], whereas no information on the nature of the behaviour
; and is available, which causes them to be
recorded in the form in which it was done for an undistorted area. If the
presentation results in physically unrealistic data, these functions are modelled
according to the methods described in [21], which takes into account, the
change in the sections configuration with different types of its
representation; ellipticity, pear-shape, proportionality of the ellipticity
parameters to the axis deformation degree, however, the indicated approaches
are for open areas where the information on section deformation is available,
at least visually.
·
in the
case of underground sections, the representation of equations is reasonable
because of information limitation on the sections deformation. This explains
the choice of , since it is also impossible for
the underground section to check at least visually that the flat section
hypothesis is not possible. If while studying of the underground section, the
same approaches as for terrestrial are applied, then different ways of setting ;; cause one more problem; it is
difficult to build an equilibrium equation for an underground section since it
is almost impossible to account for the action of mass forces (pipe weight,
product weight, soil weight, acting on each section of the pipeline) in these
equations.
5. FIELD RESEARCH
The developed technique requires precise determination of
the pipeline axis coordinates. The soil layer above the pipeline is a
significant obstacle. Data on the position of the pipeline axis with maximum
accuracy can be obtained by pit sampling using geodetic positioning methods,
though it takes considerable time and resources. Currently, technologies are
available that allow determining the spatial position of the pipeline from the
ground with the sub-centimetre precision.
Fig. 3. Procedure for obtaining pipeline SSS
data
The procedure for data obtaining the pipeline SSS changes
is shown in Fig. 3, which is carried out in the following sequence:
A linear section of the “Pasichna-Dolyna” gas
pipeline of D 500 for 5.1 km, where a landslide
occurred in 2010 causing a force pressure on the pipeline and as a result, a
pipeline rupture occurred was selected for conducting industrial research.
A graph of the calculated values of changes in the pipework
metal stresses in evenly spaced generating points at intervals of 20 m is shown
in Fig. 3. These points define the cross-sections of the pipe section under
study, where the values of the change in stresses at 16 points uniformly
distributed along the circle of the cross-section were calculated. The graph
(Fig. 4) shows that anomalous changes in stresses are recorded at the section
"400 - 600 m" of the pipeline, which is further confirmed by the
measurements results of strain test stations STS1, STS2, and STS3.
Fig. 4. Graph of changes in the pipe stresses
at the investigated section of
the "Pasichna-Dolyna" gas pipeline D 500 with a length of 800 m
The basic load on the main pipelines is the internal
pressure (the pressure of the product being pumped). The underground pipeline
is in a difficult stress state, being influenced not only by internal pressure
but also by other loads that occur in special situations (mountainous areas,
swamps and deserts). Under the influence of transverse and longitudinal forces,
main pipelines, laid in mountainous areas, significantly change their initial
position which is determined by diagnostics using the developed methodology.
The spurious operating conditions of underground pipelines
in areas of abnormal behaviour (wetlands, karst cavities or technological
developments, places of subsidence and slipping of soil, zones of tectonic
faults, neotectonics or formation of terraces, seismic and mudflow hazardous
areas) require additional analysis. It should be noted that for pipeline
systems laid in mountainous areas, it is rather difficult for prognosis of the
mechanical load on the pipeline. This makes it partly impossible to use
existing models for estimating the SSS of pipelines in such anomalous areas.
Therefore, the solution to this problem can be the use of the developed model
for calculating the stress state and the corresponding value of the pipelines
deformation
As a result of the calculations, the pipework stresses
values in cross-sections with an interval of 15 m were obtained. In each section
of the pipe, the values of the stresses in uniformly distributed 16 points of
the cross-section were obtained. The stress distribution according to the
calculations results of the stress-strain state of one of the pipe
cross-sections is shown in Fig. 5.
Fig. 5. Distribution diagram of stress-strain state
changes in pipeline cross-section at
300 m of the investigated section
The
change in the stress-strain state of the transverse displacement pipeline is
shown in Fig. 5. This indicates the presence of lateral loads in this
section of the investigated area. These loadings are the basic reason for the
movement of the "Pasichna - Dolyna" underground gas pipeline, which
is confirmed by the measurements of the routing.
The
spurious operating conditions of underground pipelines in areas of abnormal
behaviour (wetlands, karst cavities or technological developments, places of
subsidence and slipping of soil, zones of tectonic faults, neotectonics or
formation of terraces, seismic and mudflow hazardous areas) require additional
analysis. It should be noted that for pipeline systems laid in mountainous
areas, it is rather difficult for prognosis of the mechanical load on the
pipeline. This makes it partly impossible to use existing models for estimating
the SSS of pipelines in such anomalous areas.
6.
CONCLUSION
The proposed method allows to identify the most dangerous
sectors of the investigated area in terms of changes in the stress state, and
if it is assumed that at the initial moment of time the pipeline stresses were
equal to zero, then the technique allows to estimate the real values of
stresses. The criterion of stress tolerance can be the value of the elastic
limit or the yield strength when the values are different for different types
of pipeline steels and are determined from the reference literature. It should
also be noted that the described approach to the assessment of the stress state
of underground pipelines is integral, it does not require detailed information
on forces and loads, the effect of which on this pipeline section is due to
displacement measurements. The reliability of the results of the applied
methodology is confirmed by the results of stress-testing measurements of
stresses in the pipe body.
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[1] Faculty Mechanical of
Engineering, Lublin University of Technology, Nadbystrzycka 36 Street, 20-618
Lublin, Poland. Email: p.drozdziel@pollub.pl. ORCID: https://orcid.org/0000-0003-2187-1633
[2] Ternopil Ivan Puluj
National Technical University, Ruska Street
56, 46001 Ternopil, Ukraine. Email: tetiana.vitenko@gmail.com. ORCID: https://orcid.org/0000-0003-4084-0322
[3] Ivano-Frankivsk
National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine. Email: tdm@nung.edu.ua. ORCID: https://orcid.org/0000-0001-5255-8522
[4] Ivano-Frankivsk National Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine. Email: tdm@nung.edu.ua. ORCID: https://orcid.org/0000-0002-5970-4286
[5] Ivano-Frankivsk National Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine. Email: pma@nung.edu.ua. ORCID: https://orcid.org/0000-0003-1031-7207
[6] Ivano-Frankivsk National Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine. Email: tdm@nung.edu.ua. ORCID: https://orcid.org/0000-0003-3596-3918
[7] Ivano-Frankivsk National Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine. Email: lubomyrpoberrezhny@gmail.com.
ORCID: https://orcid.org/0000-0001-6197-1060
[8] Ternopil Ivan Puluj National Technical University, Ruska Street 56, 46001 Ternopil, Ukraine. Email: ppopovich@ukr.net. ORCID: https://orcid.org/0000-0001-5516-852X
[9] Ternopil Ivan Puluj National Technical University, Ruska Street 56, 46001 Ternopil, Ukraine. Email: oksana_shevchuk84@ukr.net. ORCID: https://orcid.org/0000-0001-8283-4620
[10] Ivano-Frankivsk National Technical University of Oil and Gas,
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