Research on relationship between leakage of incompressible fluid and pressure change in pipeline

YANG Li,ZENG Zhoumo,FENG Hao,HE Yongfang,GAO Chao

(1. State Key Laboratory of Precision Measurement Technology and Instruments，Tianjin University，Tianjin 300072，China； 2. Tianjin Jingyijingce Technology Co.,Ltd，Tianjin 300000,China)

Abstract：In practical engineering,only pressure sensors are allowed to install to detect leakage in most of oil transportation pipelines,while flowmeters are only installed at the toll ports.For incompressible fluid,the leakage rate and amount cannot be accurately calculated through critical pressure conditions.In this paper,a micro-element body of the pipeline was intercepted for calculation.The relationship between radial displacement and pressure of pipe wall was studied based on the stress-strain equation.Then,the strain response of pipeline volume with pipeline pressure was obtained.The change in volume expansion of pipeline was used to characterize leakage of incompressible fluid.Finally,the calculation model of leakage amount of incompressible fluid was obtained.To verify the above theory,the pipeline expansion model under pressure was established by COMSOL software for simulation.Both simulation results and deduction equations show that the volumetric change has a quadratic parabolic relationship with the change of pipeline pressure.However,the relationship between them can be approximately linear when the pressure change is not too large.In addition,the leakage of incompressible fluid under the pressure of 0 MPa-0.8 MPa was obtained by experiments.The experimental results verify the linear relationship between leakage of incompressible fluid and the change of pipeline pressure.The theoretical and experimental results provide a basis for the calculation of leakage of incompressible fluid in the pipeline.

Key words：incompressible fluid；volumetric strain；finite element simulation；pressure gradient；leakage

0 Introduction

In recent years,the transmission pipelines scale of oil,natural gas and other energy has been expanding.High-pressure transmission method is mostly adopted[1].Pipeline rupture or perforation caused by corrosion often occurs[2],and leakage is an important basis for consequence evaluation of pipeline leakage accident[3].

At present,the small hole model and the pipe model are widely used in calculation of leakage.However,the applicable conditions of these two models are relatively harsh,otherwise it is hard to obtain accurate results[4].

The small hole model is more suitable for perforation leakage of pipeline,which refers to the leakage through small holes in pipeline due to corrosion or other reasons.In pipeline leakage accidents,there are many leakages caused by perforation.When the small hole model is established,it is assumed that the leakage orifice is a circular hole.If the shape of leakage hole section is irregular,the longest diameter is taken as the parameter for calculation.For perforation leakage,the diameter of leakage hole is small,and in this case,it is generally considered that the pressure distribution in pipeline is not affected by leakage.In addition,the friction between ejected gas and leakage hole can be ignored.The gas diffusion process can be considered as an isentropic process[5].From this perspective,it is believed that the jet velocity of the gas at leakage hole does not change with time.This model is applicable to the leakage problems with leakage diameter or equivalent diameter less than 2 cm[6].

When the diameter of leakage hole or its equivalent diameter is greater than or equal to the pipeline diameter,the pipe model instead of the small hole model is applicable.The pipe model is suitable for calculation of pipeline leakage caused by pipeline rupture[7].For the study of pipe model,experts and scholars have also done a lot of specific work.Bi[8]used computational fluid dynamics (CFD)method to simulate and analyze the flow field of high-pressure pipeline leakage.Venturino et al.[9]studied gas leakage through cracks in girth welds.

Up to now,many experts and scholars have also done numerous researches on the derivation and application scope of leakage models.However,the research results have some limitations.One of the most obvious problems is that pipeline leakage calculation methods mostly aim at the leakage of compressible fluid,such as natural gas[10-11].While there are few studies on the leakage of incompressible fluid such as oil or water,and the flow rate equation of compressible fluid based on critical pressure is not applicable to incompressible fluid.The leakage of incompressible fluid can be estimated by flowmeter in theory.However,in practical engineering,most oil transportation pipelines are installed with pressure sensors to detect leakage,while flowmeters are only installed at billing nodes,which makes it impossible to use flowmeters to estimate the leakage amount.

In this paper,a cylindrical pipe is taken as the research object.Under pressure,the pipeline will be deformed and the leakage of incompressible fluid is approximately equal to the change of pipe volume[12-14].According to the equation derivation of solid stress and strain[15-16],the relationship between volumetric change and pressure is obtained.And then the formula for calculating leakage of incompressible fluid by pressure is developed.The solid mechanics finite element simulation is carried out by COMSOL software,and the volumetric strain response of pipeline caused by pressure is simulated and analyzed.Then the theoretical results are verified and analyzed through experiments.The research in this article provides theoretical basis and technical support for accurate calculation of incompressible fluid leakage.

1 Theoretical modeling

1.1 Pipeline strain model

In engineering,high-pressure transportation method is often used for long-distance energy transportation,where the fluid fills the entire pipeline.When the pipeline transports incompressible fluid leaks,the pressure in the pipeline will drop.Since incompressible fluid always fills the pipe,the leakage amount is equal to the change of pipeline volume.Therefore,it is necessary to calculate the change of pipeline volume accurately.

The research object of this paper is the volume response of long-distance cylindrical pipeline affected by internal pressure.For convenience,a section of the pipeline with both ends closed is selected as the research object.

Assume that the energy transportation pipeline is a cylinder.The inner radius of the cylinder isRiand the outer radius isRo.The internal pressure isPiand the external pressure isPo.The directions of force are shown in Fig.1.The directions of external pressures are radially inward,and the directions of internal pressures are radially outward.

Fig.1 Schematic diagram of force on cylindrical pipe

The micro-element body is taken at a certain pointAof the cylindrical pipe.A small fan-shaped hexahedron is obtained by cutting with two concentric cylindrical surfaces separated from dr,a longitudinal section with an angle of dθand two horizontal planes separated from dz.As shown in Fig.2(a),the length of infinitesimal segmentABis dr,the length of micro-arcADisrdθ,and the distance between two horizontal planesACis dz.

(a)Micro-element body of cylinder

According to the definition of strain,the positive strain of segmentABisεr.Similarly,the positive strains of lineACandADareεzandεθrespectively.

(1)

Assume that the pressure on the cylinder is uniform,and the geometry of the cylinder does not change along withzaxis.The cross section perpendicular tozaxis is still flat after deformation.So the change ofuonly depends onr,and the change ofwonly depends onz.The strain of micro-element body satisfies the spatial symmetry geometric equation,which can be represented as

(2)

The equilibrium equation of force is expressed as

(3)

whereσr,σzandσθare the radial,axial and circumferential stress of the micro-element body respectively.

The physical equation of the stress-strain relationship is expressed as

(4)

whereEis elastic modulus andμis Poisson ratio.

When the cylinder is subjected to uniform internal and external pressure,its boundary conditions are defined as

(5)

whereσriandσroare the stresses on the inner and outer walls of the pipeline respectively.

For the cylinder sealed at both ends,the axial equilibrium condition is expressed as

(6)

(7)

1.2 Pipeline volumetric strain model

The volume change of cylinder pipe originates from its change of inner diameter,which can be expressed as

(8)

When the cylinder is only subjected to internal pressure (Pi≠0,Po=0),Eq.(7)can be simplified as

(9)

For a fixed pipe,the diametersRoandRiare both constants.Since the environment of long-distance pipeline does not change much,it is assumed that the elastic modulusEand Poisson ratioμof the pipeline do not change with temperature,and they are considered as constants.Thenuis the proportional function ofPi.Eq.(9)represents the inner diameter offset of the pipe when the value ofrisRi.The following assumptions are made.

ui=KiPi,

(10)

The oil pipelines used in project are connected by girth welds.The length of each pipeline is about 12 m.The Steel AlSl 4340 steel pipe is taken as an example,with outer radius of 219 mm and wall thickness of 6 mm.

Based on the above parameters,Matlab is used to calculate the radial displacement of this pipe.The change of inner radius of the pipeline varies as the gradually increasing internal pressure,Fig.3 shows that the change of inner radius is proportional to internal pressure.

The inner radius of pipeline is the key factor affecting calculation of pipeline leakage.Substituting Eq.(10)into Eq.(8),the equation of volume change is obtained as

(11)

Fig.3 Relationship between change of inner radius and internal pressure

The volume change of the pipe with both ends closed due to pressure is calculated by Matlab,and the results shown in Fig.4 indicate that the relationship between volumetric strain and internal pressure is quadratic parabola.When the internal pressure is not so big,as shown in the small graph of Fig.4,the volume change of pipeline is linear with the internal pressure,which is due to the large value of elastic modules,resulting in the quadratic coefficient is too small.

Fig.4 Relationship between volume change and internal pressure

1.3 Estimation model of incompressible fluid leakage

In practical engineering,most of incompressible fluid is transported by high pressure,so the fluid always fills the pipeline.The volume of incompressible fluid (such as oil)can be considered as the change of pipeline volume.During normal transportation,the internal pressure of the pipeline is a stable valueP1.When the pipeline leaks,the internal pressure of the pipeline will drop and quickly stabilize atP2.The drop of pressure will lead to decrease in the expansion volume of the pipeline.The difference of the expansion volume of pipeline under two pressures before and after leakage is the leakage of incompressible fluid in the pipeline.Then the leakage calculation equation of incompressible fluid in the pipeline is

(12)

2 Simulation research

COMSOL is a powerful multi-physics finite element simulation software.In this paper,COMSOL software is used to simulate the expansion process of closed vessels under pressure.

2.1 Definition of simulation model

In the process of building the simulation model,the 2D axisymmetric geometry modeling method and the steady-state research are selected,and the physical field of solid mechanics is added.

The axisymmetric solid model is created inr-zplane,and the geometric model of 1/4 part of the cylindrical pipe is established as shown in Fig.5.In order to compare theoretical results,simulation results and experimental results more intuitively,the simulation uses the same materials and parameters as the experimental platform.Setting up arc bends at both ends of the pipeline is closer to actual situation.

Fig.5 Axisymmetric geometry model

In the simulation,Steel AlSl 4340 material is selected for the pipeline,with elastic modulus of 2.05×1011Pa and Poisson ratio of 0.28.

2.2 Simulation results

In the simulation,a boundary load with the pressure of 1 MPa is added inside the pipe.The magnitude and direction of the pressure on the pipe surface are shown in Fig.6(a).Fig.6(b)shows the stress distribution on the surface of the pipe.The stress on the pipe surface ranges from 5 MPa to 40 MPa.It is about 14.6 MPa without considering the two ends of the pipe.

(a)Magnitude and direction of boundary load

In the simulation results,the point atr=0 andz=-6 is selected to calculate the displacement of inner wall.Since the volume expansion of the pipeline is mainly caused by radial strain,only theRcomponent of displacement field is considered.As shown in Fig.7 below,there is a linear relationship between radial displacementuand boundary loadP.When the pressure reaches 1 MPa,the radial displacement of inner wall is about 0.008 mm.

Fig.7 Relationship between radial displacement and boundary load

The radial displacement of the pipeline brings the volume change.The relationship between volume change and boundary load is calculated as shown in the Fig.8.When the pressure in the pipeline reaches 1 MPa,the volume change of the pipeline is about 61.8 cm3.With the increase of pipeline pressure,volume change and the applied pressure load present a quadratic parabolic relationship.

Fig.8 Relationship between volumetric change of pipeline and boundary load

2.3 Analysis of simulation results

The simulation results are compared with the theoretical values of the calculation equation.From Table 1,it can be seen that the simulation results by COMSOL are basically consistent with the theoretical values.The slight differences may be related to the accuracy selection and mesh division.Mesh division needs to balance calculation accuracy and calculation speed during simulation,which may produce errors.

Table 1 Comparison of simulation results and theoretical values

3 Experimental verification

3.1 Experimental platform

In order to verify the correctness of theoretical derivation and simulation,a pipeline experimental platform was built.The AISI 4340 steel pipeline was used in the experiment.The length of the pipeline was 12 m,the outer diameter was 219 mm,and the wall thickness was 6 mm.Fig.9(a)shows the schematic diagram of the pipeline installation,and the field pipeline experimental platform is shown in Fig.9 (b).

(a)Schematic diagram

3.2 Analysis of experimental results

Based on the closed pipeline mentioned in the first section,several experiments have been carried out on the pipeline experiment platform.Due to the limitations of experimental conditions,the pressure in the pipeline changes only from 0 MPa to 0.8 MPa.However,it is found that there is a large gap between the experimental results and the theoretical values,as shown in Table 2.

Table 2 Comparison of experimental and theoretical values without considering bubbles

Through analysis of the experimental process,it is found that there are some air bubbles in the pipe during experiment.The following experiments are carried out for bubbles.

1)After the pipeline is filled with water,it continues to be filled slowly.Move the end closed to exhaust port up and down to expel some bubbles in pipeline.In the experiment on bubbles,the times in moving the end of pipe up and down was taken as a variable condition,because more times of movement can discharge more bubbles in the pipe.The results show that the more the times of movements,the fewer bubbles in the pipe,that is to say,the smaller the experimental errors.

2)The compression characteristic of air at normal temperature isPV=nR(Ris a constant),and the volume of a certain mass of air changes under pressure.At normal temperature,the volume change of a certain amount of air varies with pressure.The relationship can be expressed as

(13)

whereVa0is the volume of bubbles at normal temperature and pressure in the pipeline.The change trend of compressed air volume is the same as that of the experimental errors under different pressure changes,and both are the downward trends in growth rates.

3)Assume that some bubbles still exist in the pipeline after exhaust.At normal temperature and pressure,the volume of these bubbles isVa0.Some small invisible bubbles will be discharged at the leakage port during the pipeline leakage.It is assumed that the volume of bubbles discharged from the leakage port is proportional to the pressure difference between inside and outside the pipeline.The volume of bubbles in the pipeline at normal temperature and pressure can be expressed as

(14)

wherek0is constant.Combining Eqs.(13)and (14),the volume change of the bubble in the pipeline is

(15)

The volume change caused by the bubble is used as the compensation term of Eq.(12),and the final leakage calculation is

(16)

In order to reduce the random errors of this experiment,the results of multiple experiments are averaged.The theoretical and experimental leakage corresponding to the change of pressure in the pipeline is shown in Fig.10.It can be seen from Fig.10 that the experimental data fits well with the theoretical curve.The results prove that the errors between experimental and theoretical values become small after considering the bubble factor.

Fig.10 Comparison of theoretical and experimental results

The theoretical leakages are calculated and compared with the experimental data when the pressure changes.The specific comparison results are shown in Table 3.

Table 3 Comparison of experimental and theoretical values by considering bubbles

By comparing experimental and theoretical data,it can be found that the experimental leakages are very close to the theoretical values under the same pressure.However,when the pressure change is small,the difference between theoretical and experimental value is relatively larger.This is because the pressure change in the experiment starts from the standard atmospheric pressure,and part of the leaking fluid flows out in the form of osmosis when the internal pressure is small.It is difficult to collect the leaking fluid on the pipe wall,resulting in experimental value is much lower than theoretical value.

Due to the limitation of experimental equipment and experimental condition,the bubbles in the pipeline cannot be completely emptied.In addition,there are invisible bubbles dissolved in the water entering the pipeline during experiment,which are unavoidable and uncontrollable.Therefore,the errors produced by the experiments cannot be eliminated,and the volume of bubbles in the pipeline is also difficult to know exactly.The compensation of experimental errors by bubbles in the pipeline is difficult to be accurately expressed with specific formulas or correction items,and can only be given based on engineering experience.However,there are almost no bubbles in the pipeline in actual project,implying that such error can be reduced based on engineering experience.In such situation,the theoretical estimate of incompressible fluid leakage can reach the same order of magnitude as the experimental value.Moreover,the linear relationship between leakage of incompressible fluid and change of pipeline pressure is also of great significance in actual engineering.

4 Conclusions

In this paper,a cylindrical pipeline was taken as the research object.Through COMSOL simulation and experimental verification,the results show that leakage of incompressible fluid is proportional to pressure change in pipeline in actual project.In the process of calculating leakage,it is necessary to pay attention to the influence of bubbles in pipeline.However,the theoretical derivation in this paper is still available in practical engineering,which is very important to estimate the leakage of incompressible fluid.