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Journal of Natural Gas Science and Engineering 34 (2016) 185e196

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Journal of Natural Gas Science and Engineering

journal homepage: www.elsevier .com/locate/ jngse

Correlations for estimating natural gas leakage from above-groundand buried urban distribution pipelines

A. Ebrahimi-Moghadam a, M. Farzaneh-Gord a, M. Deymi-Dashtebayaz b, *

a Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iranb Faculty of Mechanical Engineering, Hakim Sabzevari University, Sabzevar, Iran

a r t i c l e i n f o

Article history:Received 7 April 2016Received in revised form23 June 2016Accepted 26 June 2016Available online 28 June 2016

Keywords:Natural gas distribution pipelinesLeakage holeAmount of leakageBuried pipeNumerical simulation

* Corresponding author.E-mail address: meh_deimi@yahoo.com (M. Deym

http://dx.doi.org/10.1016/j.jngse.2016.06.0621875-5100/© 2016 Elsevier B.V. All rights reserved.

a b s t r a c t

A numerical method is developed to investigate leakage in above-ground and buried urban distributionnatural gas pipelines. The main aim is to develop a few equations to estimate leakage form above-groundand buried urban natural gas pipelines. The equations are developed by considering the impact of variousparameters such as the pipeline and hole diameters. A computational model for steady, compressibleturbulent flow is built to model leakage. The natural gas as working fluid is treated as an ideal gas andsoil considered as a porous zone. The results indicate that for holes with small diameters, dischargespeed reaches to the sound speed and at the so-called, choking occurs in the flow. Also based on theresult, the volumetric flow rate of leaked gas have a linear relation, second order relation and fourthorder relation with pressure of initial point, diameter of hole and ratio of the hole diameter to the pipediameter, respectively. In the case of buried pipes, permeation depth of gas into soil at the small diameterholes is more than large holes but volumetric rate of leaked gas is lower. Also after permeation of naturalgas into the soil, and hitting the soil particles and the air moving through soil, a pair of vortex is createdinside the soil. Finally two new correlations have been proposed to calculate the natural gas leakage froma small hole located on the lateral surface of the above-ground and buried distribution gas pipelines.Results show that, the percentage of relative error between simulation results and correlation values isbelow 5% which implies high accuracy of the presented correlations.

© 2016 Elsevier B.V. All rights reserved.

1. Introduction

Pipelines are the common and comfortable method of trans-porting and distributing fuels and dangerous gases such as naturalgas (NG) (Parvini and Gharagouzlou, 2015). Pipelines are oftensubjected to various damages such as third party activities, corro-sion, mechanical or material failure and natural hazards(Sklavounos and Rigas, 2006). Annually, a significant amount ofgases are wasted under the title of Unaccounted for Gas (UAG) andit’s one of the main issues in the management and control of NGtransmission networks (Costello, 2014). The UAG is defined as dif-ference between the volume of gas injected into a distributionsystem and the gas measured at customers’ meters (Arpino et al.,2014).

The factors which affect UAG can be identified as the gas leak-ages from pipes and fittings, measurement errors, third party

i-Dashtebayaz).

damage and theft of the gas (Haydell, 2011). Due to the major shareof leakage among these factors, it is one of the most importantfactors in UAG. Leakage is the action of unmeasured gas that releasefrom a transmission or distribution pipeline to the surroundings.Gas leakages can be occurred on pipe walls, at welded and flangedconnections, valves and other gas equipment. Leakage can bearising by various ingredients such as internal friction, corrosionand external accidents (such as underground work, dealing withbulldozer, etc.) (Montiel et al., 1996).

When a hole is created on the pipe surface, an expansion wavemoves into the pipe and it makes the gas flow toward the damagedsection (Nouri-Borujerdi, 2011). Due to the high gas flow velocity inthe damaged pipe, commonmeasurements are not able to calculatethe amount of released gas; therefore the laws of gas dynamicsshould be used to calculate the amount of released gas.

There are many researches which studied gas leakage. Theseresearches can be categorized into two main groups: leakagedetection location and calculation its amount. In the case of leakagedetection, some methods and studies for gas pipeline leak detec-tion have been developed in recent studies (Buerck et al., 2003;

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196186

Batzias et al., 2011; Meng et al., 2012; Murvay and Silea, 2012;Zhang et al., 2015).

In the case of leakage calculation, there are many publishedbooks and manuscripts about the viscous compressible flow insidethe pipe around two adiabatic and isothermal flow states(Churchill, 1980; Crowl and Louvar, 2002; Levenspeil, 1984; Crane,1988; Farina, 1997).

One of the first studies about natural gas leakage modeling wasdone by Olorunmaiye and Imide (Olorunmaiyeat and Imideb,1993).In their study, a mathematical model of natural gas leakage frombroken end of a pipe was developed based on unsteady isothermalflow. The results of their study predicted that gas release rate islower than that of adiabatic flow theory by about 18%. It agreesquite well with the results of research that done by Lang andFannelop (1987) in the isothermal flow state. Woodward andMudan (1991) developed a model for computing liquid and gasdischarge rates through holes in process vessels, that takes intocalculate the decrease in pressure with changing in the time.

Montiel et al. (1998). developed a mathematical model of acci-dental gas release. They developed one dimension model for gasdistribution system operating at medium and low pressure withIdeal gas assumption (cp ¼ cte). A mathematical modeling of gasrelease through holes in high-pressure pipelines was developed byYuhu et al. (2003). based on constant compressibility factor(Z ¼ 0.9). They concluded that gas leaks from small holes can beconsidered as a steady state process. Lu et al. (2014). analyzed thetheory of the one-dimensional natural gas leakage in urban me-dium pressure pipelines, and examined their proposed model forboth steady and transient (unstable) states. In another investigationJo and Ahn (2003) developed a simple and definite model tocalculate leaked gas from a hole in a pipe containing dangeroushigh-pressure gas using the ideal gas equation.

Some researches were done by using numerical method ofcharacteristics to solve governing equations, such as investigationwhich done by Oke et al. (2003). They introduced a transientoutflow model for pipeline puncture for real natural gas flowconsists of several hydrocarbons. They concluded that because ofthe returned gas flow from downstream,modeling of the pipe like areservoir that gas flow discharge from end of it, is not a suitablemodel.

Kostowski and Skorek, (2012) investigated the flow in thedamaged pipes in the natural gas distribution network. Theyconsidered the damaged pipe as open side wall in their proposedmodel. They simulated the flow for two isothermal and adiabaticflow states and checked both of these states for ideal gas and realgas. They also analyzed the discharge coefficient effect in flow ofthe gas output.

In the case of buried pipe, a model based on finite elementmethod was provided for a leaking species migration from a heatsource buried into a fluid saturated porous medium by Nithiarasu,(1999) The results of this research showed that, the size of theleaking hole has a great effect on the dischargedmass in addition tothe Rayleigh number.

Almost in all previous researches, flow modeling has been doneas one-dimensional and in most of them the pipe modeled as aclosed reservoir. In other word, the length of the pipe after the holehas not been considered. The models, which not considered thelength of the pipe after the hole, are inappropriate and causesunreal results for cases with larger holes. Also most of researcheshave been done on the field of leakage calculation for above-groundpipelines and reliable investigations are very limited in the field ofburied pipelines. Consequently, the purpose of this study is toinvestigate the natural gas leakage for both above-ground andburied pipes cases in the urban distribution gas pipelines. Theinvestigation is carried out by using numerical simulation of a two-

dimension turbulent flow. Also the length of the pipe after the holeis considered and pipe divided into two down-stream and up-stream parts. Finally two simple, useful and comprehensive cor-relations have been presented to estimate the amount of naturalgas leakage from a small hole located on the lateral surface of theabove-ground and buried distribution gas pipelines.

2. Theoretical analysis

As it motioned, to calculate the amount of pipelines leakage, twomodels can be identified as:

a) Hole modelb) Pipe model

In the situation of (a), there is a hole on the lateral surface of thepipe, hole diameter is smaller than pipe diameter (d<D) and gasdischarges from the hole. But situation of (b) is a case that thediameter of the hole is equal or larger than pipe (d�D). In the otherwords, the pipe model is used when rupture occurs in the pipe.Fig. 1 shows schematic of the hole and pipe models that used in themost previous researches. It should be mentioned, the length ofpipe after the hole is not considered in most previous studies.

The type of flow in the hole depended to a dimensionlessparameter that called critical pressure ratio (CPR) and calculatedfrom Eq. (1) (Montiel et al., 1998). CPR indicates for each case, thetransition from sonic flow to subsonic flow.

CPR ¼ pap2cr

¼�

2gþ 1

� gg�1

(1)

where pa and p2cr indicate the atmosphere pressure and criticalpressure of the point 2, respectively (the indexes in this equationare according to the Fig. 1). According to Eq. (1), when the pressureat point 2 is larger than critical pressure (p2>p2cr), choked flowoccurs and the discharge speed of the gas reaches the sound speed(Ma¼ 1). But when p2<p2cr a subsonic flow in the discharge sectionis occurred (Montiel et al., 1998; Yuhu et al., 2003; Lu et al., 2014; Joand Ahn, 2003).

Fig. 2 is a schematic of the model considered in this work. As itcan be seen, the length of pipe before and after the hole areconsidered as l1 and l2 respectively.

In the handbook of the gas engineering (GPSA) (EngineeringData Book and Tw, 2004), there is an equation to calculate theamount of the gas leakage based on the law of the orifices. In thisway, an orifice is installed between a pair of flanges at the outlet of avertical pipe nipple which calculates the amount of gas dischargedinto the atmosphere. An approximate flow rate of gas which dis-charged from the small orifice into the atmosphere can be calcu-lated from Eq. (2) (Engineering Data Book and Tw, 2004).

Qd ¼ 16330�1þ b4

�d2orf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHð29:32þ 0:3HÞ

pFtf Cg (2)

b ¼ dorfDp

(3)

Ftf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

520460þ Tm

s(4)

Cg ¼ffiffiffiffiffiffiffi0:6G

r(5)

where Qd is the approximate volumetric rate of the discharged gas

Fig. 1. Schematic diagram of the studied model used in most previous studies, (A) hole model, (B) pipe model (Montiel et al., 1998; Yuhu et al., 2003; Lu et al., 2014; Jo and Ahn,2003).

Fig. 2. Schematic diagram of the considered model.

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(ft3/day), dorf is orifice diameter (in), Dp is pipe diameter (in), H ispressure (in Hg) and Tm is the average temperature of the gas (�F).

3. Simulation analysis

In this section, firstly, the attention has been paid to introducethe physical model. Then, the numerical simulation which used tocalculate the amount of leakage is presented.

A two dimensional, steady, compressible turbulent flow hasbeen built to calculate the amount of natural gas leakage from ur-ban distribution pipelines. Since the flow equations in the pipe andsoil are different, the equations are divided into two distinct parts.

Fig. 3. Solution domain (above-ground pipe).

3.1. Governing equations of above-ground pipe

The governing equations for natural gas flow inside the pipe areexpressed as follows:

Conservation of mass (continuity) (Versteeg and Malalasekera,2007; Lu et al., 2016)

v

vxiðruiÞ ¼ 0 (6)

Conservation of momentum (Versteeg and Malalasekera, 2007;Lu et al., 2016)

v

vxj

�ruiuj

� ¼ �vpvxi

þ vtijvxj

þ v

vxj

� rui

�uj�

!þ rgi þ Fi (7)

Conservation of energy (Versteeg and Malalasekera, 2007; Luet al., 2016)

v

vxi½uiðrE þ pÞ� ¼ v

vxj

"keff

vTvxj

þ ui�tij�eff

#(8)

Equation of state (EOS) (Moean et al., 2011)

pv ¼ RT (9)

3.2. Governing equations of buried pipe

Since the flow equations in the pipe and soil are different, theequations are divided into two distinct parts:

a) Equations of flow inside the pipe: equations (6)e(9);b) Equations of flow within the soil: equation (10)e(12).

To simulate the soil above the pipe, the definition of a porous

Fig. 4. Solution domain (buried pipe).

Fig. 5. Mesh generation around the hole in the state of buried pipe.

Table 1Effect of the computational cells number on the mass flow rate of leaked gas.

Number of the cells (cells) Mass flow rate of leaked gas (kg/s)

240,000 0.175530,00 0.1721,070,000 0.1591,260,000 0.158

Table 2Thermal properties of the soil (Itoh, 1981).

Property Unit Value

Permeability (m2/s) 1.6 � 10�6

Conductivity (W/mK) 2.9Heat capacity (J/kgK) 732.69Density (kg/m3) 2650

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zone is used. Equations of the flow in the porous zone are definedand solved independently from the flow inside the pipe. Accord-ingly, the governing equations for this case are containing Eqs.(6)e(12) (Ewing et al., 1999).

v

vxiðYruiÞ ¼ 0 (10)

Table 3Type of the boundary conditions.

Boundary location Boundary type

Initial of the pipe Pressure inletEnd of the pipe Pressure outletHole In above-ground pipe Pressure outlet

In buried pipe On pipe: pressure outletOn soil: interface

Lateral surface of the pipe Wall (adiabatic)Ground surface Pressure outlet

2500

3000

3500

Present study (without L after the hole)Present study (with L after the hole)Montiel' s study [21]Eq. (2) [28]

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196 189

v

vxj

�Yruiuj

� ¼ �Yvpvxi

þ v�Ytij

�vxj

þ Yrgi ��m

aþ C2r

2juij�ui (11)

v

vxi

hui�rf Ef þ p

�i¼ v

vxj

"keff

vTvxj

þ ui�tij�eff

#(12)

In the last term of Eq. (11), the first part (ma) and second part (C2r2 ),

represents the viscous and inertial drag forces. The effective con-ductivity, keff, defines as Eq. (13).

keff ¼ Ykf þ ð1� YÞks (13)

3.3. Turbulence modeling

According to the Reynolds number, turbulent flow is considered.Tomodel turbulence in this study, the standard k�εmethod is used.Also for modeling near-wall treatment, standard wall function isused which is common high-Reynolds flows. In the k�ε method,two additional transport equations are solved (these two equationscalled k and ε equations) and turbulent viscosity (mt) is obtained as afunction of k and ε. The k and ε equations in the standard k�ε

method are as (14) and (15) equations (Jianwen et al., 2015).

v

vxiðrkuiÞ ¼

v

vxj

"�mþ mt

sk

�vk

vxj

#þ Gk þ Gb � YM � rε (14)

v

vxiðrεuiÞ ¼

v

vxj

"�mþ mt

sε

�vε

vxj

#þ C1ε

ε

kGk � C2εr

ε2

k(15)

In these equations, Gk is the generation of turbulence kineticenergy due to the mean velocity gradients and it can be used Eq.(16) to evaluate Gk in a manner consistent with the Boussinesqhypothesis (Jianwen et al., 2015).

Gk ¼ 2mt

"�vuvx

�2þ�vv

vy

�2þ�vwvz

�2#þ�vuvy

þ vv

vx

�2

þ�vuvz

þ vwvx

�2þ�vwvy

þ vv

vz

�2(16)

For two-dimensional case, Eq. (16) becomes as follows:

Gk ¼ 2mt

"�vuvx

�2þ�vv

vy

�2#þ�vuvy

þ vv

vx

�2(17)

Fig. 6. Convergence of the outlet mass flow rate from the hole.

The effect of buoyancy on turbulence appears as so-called“turbulence kinetic energy generation due to buoyancy” and isdefined as Eq. (18). With ideal gas assumption, Eq. (18) reduces toEq. (19) (Jianwen et al., 2015).

Gb ¼ bgimtPrt

vTvxi

(18)

Gb ¼ �gimtrPrt

vr

vxi(19)

In the flow with high Mach number value, compressibility af-fects turbulence through dilatation dissipation, which is thecontribution of the fluctuating dilatation in compressible turbu-lence to the overall dissipation rate (YM). This effect is normallyneglected for incompressible flow modeling (Wilcox, 1998). Also incompressible flow, this parameter is modeled according Sarkar(Sarkar and Balakrishnan, 1990) proposal as Eq. (20).

YM ¼ 2rεk

gRT(20)

In Equations (14)e(19), C1ε ¼ 1.44 and C2ε ¼ 1.92 are constants;sk ¼ 1.0, sε ¼ 1.3 and Prt ¼ 0.85 are the turbulent Prandtl numbersfor k, ε and energy, respectively. Also mt is the eddy viscosity coef-ficient which is calculated from Eq. (21).

mt ¼ rCmk2

ε

(21)

where Cm is a constant and its value for standard k�ε method isCm ¼ 0.09.

3.4. Geometry and grid generation

The geometry under investigation is a pipe with 500 m long inwhich a hole has been made in the first 300 m from initial point ofpipeline. The leakage will be investigated for various amounts ofthe hole diameter and initial point pressure. Since the walls are amajor source of the vortex and turbulence and near the wallsquantities such as speed has a severe gradient, modeling the flow

(mm)

(Nm³/hr)

10 15 20 25 30

500

1000

1500

2000

Q

dFig. 7. Comparison of normalized release flow rate from hole between this study andothers.

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196190

near the wall properly has a very important effect on the success ofthe solution. Therefor to achieve this, the solution domain isdivided to 9 zones in the above-ground pipe state and in the buriedcase is divided to 18 zones. Figs. 3 and 4 show the solution domainin the above-ground and buried pipe state, respectively.

The grid structure employed for computation is shown in Fig. 5.The structured mesh (Square cells) is used for computation. Thefine grid is utilized near the walls as well as hole, where the highestchange is expected.

Table 1 shows the validity and independence of the number ofcells used in the computational domain (this table is as an examplefor a model with D¼ 163.6 mm, d¼ 15mm and p1¼3 bar). Becauseof the small size of the holes, the investigations show that themodels can be extended to other diameter holes, too. Themass flowrate of leaked gas difference betweenmodel with 530,000 cells and1,070,000 is about 7.5% while this difference for the models with1,070,000 and 1,260,000 cells is about 0.6%. Finally, based on theresults, the computational domain for all samples is consideredalmost 1.5 million computational cells.

3.5. Numerical simulation of flow inside the soil

Coefficients required for modeling porous medium resistance tofluid flow are defined in two ways of experimental or porous ma-terial properties. Viscous resistance and inertial resistance are co-efficients that are obtained from the properties of the porousmaterial used. In this research, soil is considered as a porous ma-terial with an isotropic porosity (same resistances in all directions).Viscous resistance is defined as the inverse of the permeability ofporous material (1/a).

Another way to determine the coefficients of viscous and iner-tial resistances is the use of experimental formulas; for example, anexperimental formula provided by Ergun, (1952) to calculating thecoefficients. In this formula, viscous and inertial resistances are asfunctions of porous particle diameter and the porosity amount.Also, the properties of the soil which used in this research are asTable 2.

3.6. Boundary conditions and fluid properties

The boundary conditions which are used in the computationaldomain contain pressure inlet, pressure outlet andwall boundaries.Type of the boundary conditions, which used for each of theselected boundaries, is shown in Table 3. Also for buried gas pipe,hole is a subscriber boundary between pipe and the soil and this

Fig. 8. Contour of the Mach number around the (

boundary should be disconnected between pipe and soil; afterdisconnecting, two boundaries are created and type of theseboundaries is according to Table 3. The pipe under investigation islocated in the urban distribution gas pipelines with low pressurelevel and relatively small pipe diameter and this means that naturalgas can be treated as an ideal gas (Montiel et al., 1998; Yuhu et al.,2003; Lu et al., 2014; Luo et al., 2006). Therefor there is a differentbetween current study and other models that investigated leakagefor high-pressure gas pipelines (Olorunmaiyeat and Imideb, 1993;Flatt, 1986), in which it should be take into account dispersion ofthe pressure along the pipe. Natural gas is mainly composed ofmethane, ethane and propane (Jarrahian and Heidaryan, 2014a).Since the major component of natural gas is methane (approxi-mately 75%e98% of natural gas is methane) (Jarrahian et al., 2015;Jarrahian and Heidaryan, 2014b), natural gas is assumed to containonly methane.

3.7. Numerical procedure

The governing equations are discrete by Upwind method. Thecoupling of the velocity and pressure phrases is done by usingSIMPLE algorithm. The Fluent software in pressure based solvermode is used for the current study.

Fig. 6 shows the convergence of the outlet mass flow rate fromthe hole (this figure is as an example for a model withD ¼ 163.6 mm, d ¼ 30 mm and p1 ¼ 4.5 bar in above-ground state).Our experience shows that for flow with compressibility impor-tance or high working pressure, for better convergence and thecorrect answer, it is suggested to discretize the governing equationswith first order firstly. After the solution reached convergence, thesecond order method of discretization can be employed.

In this study, the governing equations are discretized and solvedfor 8000 iterations by the first order Upwind method. As it can berealized from Fig. 6, after 8000 iterations, the solution is convergedbut the results are not accurate enough. Therefore, the termsincluding density phrases are discretized and solved with the sec-ond order Upwind method for a repeat of 8000e10,500 iterates;and then a repeat of 10,500e15000, momentum equation dis-cretized and solvedwith the second ordermethod too. According toFig. 6 after performing each of this steps, it is observed that a leapoccurs in the amount of mass flow rate. Finally a repeat of15,000e16,000, the energy equation discretized by second orderUpwind method and it is observed that it has no effects on thesolution.

as sample for d ¼ 20 mm and p1 ¼ 4.5 bar).

Fig. 9. Variation of release flow rate as a function of the hole diameter for differentvalues of initial point pressure, (A) above-ground pipe, (B) buried pipe (as sample, forD ¼ 204.6 mm).

Fig. 10. Variation of release flow rate as a function of the initial point pressure fordifferent values of hole diameter, (A) above-ground pipe, (B) buried pipe (as sample,for D ¼ 204.6 mm).

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4. Results and discussion

4.1. Model validation

This section presents the validation of the numerical simulation.Due to the limited researches in the field of natural gas leakagefrom buried pipes and also lack of similar reference to currentstudy, for validate results and simulation method of this study, theresults have been compared with above-ground pipe state. Tovalidate the results of the simulation, numerical results of thisresearch have been compared with results of Montiel et al. (1998).

The model of Montiel et al. (1998). was a pipe with a length of1000m and an inner diameter of 163.6mm in distribution pipelineswhich has a hole in one end (the pipe is intended to be a closedreservoir) and methane with assumption of ideal gas was consid-ered as natural gas. Modeling of ref Montiel et al. (1998) wasassumed by a one-dimensional and the flow in the pipe adiabatic.In addition, the finite difference method (the Secant iterative al-gorithm) is used to solve the governing equations on the problem.

The model used for validation in the present study is dividedinto two cases. The first case contains a pipe with a length of1000 m that a hole is created at the end of it (it means that in firstcase, similar to ref. Montiel et al. (1998), the length of the pipe afterthe hole isn’t considered); the second case contains a pipe with alength of 1500 m that a hole is created at the distance of 1000 mfrom beginning point. In second case the hole divided the pipe intodownstream and upstream sections. Boundary conditions and so-lution assumptions for this simulation have been consideredsimilar to the reference Montiel et al. (1998). Then, numerical re-sults have been compared with Montiel’s results.

Fig. 7 compares the results of the normalized volumetricdischarge rate from hole between this research and the other re-searches. As it can be seen, there is a good agreement between theresults of ref Montiel et al. (1998) model and first case of currentmodel that shows the validity of the current modeling. The reasonof larger numbers in results of second case than Montiel research,can be considering due to the movement of the reversal flows fromdownstream section toward the hole. Also as it seen, by increasingthe hole diameter the flow reversal becomes more and the result

Fig. 11. Contour of the absolute pressure inside the pipe and also around the hole for above-ground pipe (as sample, for d ¼ 20 mm, D ¼ 204.6 mm and p1 ¼ 4.5 bar).

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196192

differences of the two researches gets more. This indicates thatignoring the pipe length after the hole caused some errors in theamount of leakage, compared to the actual state. Also the per-centage of relative difference between second case of present study(with L after the hole) and Montiel’s (Montiel et al., 1998) model isabout 7.11%, 3.8% and 5.53% for holes with 20 mm, 25 mm and30 mm diameter, respectively.

In Fig. 8, contour of Mach number is presented. As it may beobserved, in the investigated range of hole diameters, the averageof Mach number at the hole is approximately unity which indicatesthe existence of a sonic flow. This issue is another reason showingthe accuracy of this simulation and its related results.

4.2. The effect of pipe diameter, hole diameter and pressure onleakage

In this section, the effects of various parameters on natural gasrelease for both above-ground and buried pipe states are presented.Effective parameters on the amount of leakage for low-pressure gaspipelines can be identified as: pressure of initial section (p1), pipelength (L), pipe diameter (D), hole diameter (d) and mean tem-perature of the pipeline (Tf). The effect of pipe length appears in thepressure drop along the pipe and as it motioned in previous sec-tions, the current model is developed for urban gas distributionpipelines operating at low pressure with relatively small pipe

Fig. 12. Contour of the absolute pressure around the hole for buried p

diameter; therefore due to these, the pressure drop along the pipeis negligible. Also, due to adiabatic flow assumption temperaturechanges along the pipe is very low. Based on the foregoing, effectiveparameters of this study are as p1, d and D.

The model of this investigation is a polyethylene pipe (piperoughness 1.5 mm) (Handbook of Polyethylene (2007)) with alength of 500 m which is located at the urban gas distributionsystem. A hole 300 m from the initial point of the pipe is created onthe lateral surface of it and the flow has a developed state beforereaching the hole.

The inlet temperature is 288 K and the problem is investigatedwith the different amounts of beginning point pressure in the rangeof 3 bare5 bar absolute. Hole diameter is investigated in the rangeof 5 mme80 mm and pipe diameter investigated for three mostcommon pipe sizes (with nominal diameters: 114.6 mm, 163.6 mmand 204.6 mm) with used in urban gas distribution systems. In thecase of buried pipes, the soil above the pipe is considered 1.5 mhigh.

Analysis of the results indicates that, per a specified amount forinput pressure, the larger amount of the hole diameter, causesincreasing in the pressure difference between the initial point andhole (p1�p3) which results in increasing the volumetric flow rateoutput.

Fig. 9 provides volumetric flow changes in terms of the holediameter in different amounts of inlet pressure for gas pipeline

ipe (as sample, for d ¼ 20 mm, D ¼ 204.6 mm and p1 ¼ 4.5 bar).

Fig. 13. Contour of the velocity around the hole, (A) above-ground pipe, (B) buried pipe (as sample, for d ¼ 20 mm, D ¼ 204.6 mm and p1 ¼ 4.5 bar).

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196 193

states including above-ground and buried pipes (because theoverall behavior of charts for different values of the pipe diameter isthe same, this chart is given as a sample for a case withD ¼ 204.6 mm). The result of curve fitting for leakage flow changesdepending on hole diameter, showed that the leakage flow withhole diameter (d) and ratio of the diameters (d/D) varies as a secondorder and fourth order function, respectively. Also, by comparingFig. 9(A) and (B), as expected, the volumetric flow rate of leakage inthe buried pipe state is less than the one in the above-ground state,because of the resistance of the soil column on top of the hole.

In Fig. 10, the volumetric flow rate change is shown according tothe inlet pressure for different diameters of the holes. As shown inthis figure, the volumetric flow rate has an approximate linearfunction with inlet pressure. Also by increasing the initial pointpressure, the pressure difference of initial point and the hole getsmore and as a result the leaked gas flow from the hole increases. Bycomparing Fig.10(A) and (B) it can see that, the volumetric flow rateof released gas from the hole in the buried pipe is less than above-ground pipe, which is due to soil resistance.

Due to compressibility of investigated fluid in this study, all thequantities have various values in different points of the hole.Therefor in order to quantify at the hole, mass-weighted averagehave be taken into account.

Figs. 11 and 12 respectively show the absolute pressure contourin the pipe and around the damaged section for the above-groundand buried pipelines. As seen in the Fig. 11(A), the pressure dropalong the pipe is very small but sudden pressure changes is

significant around the damaged section, due to the sudden changein the sectional area in damaged area (small hole diameter in ratioto the pipe diameter). In other words, when the hole is too small, itcauses the pressure at the damaged section to be greater than theambient pressure. Also, a survey of results shows that by increasingthe hole diameter, the pressure difference between the ambientpressure and the damaged section decreases.

The achievement of noted cases, creating an expansion waveinto the pipe that makes the gas flow moves toward the hole. Asshown in Fig. 13(A), the consequent of the velocity at the damagedsection of above-ground pipe equals to the sound speed.

In case of buried pipes, the damaged section is a common sec-tion between the pipe and the soil. According to the Fig. 13(B), inthe case of buried pipe, we also have a sonic flow; but just in theproximity of the hole and inside the soil (immediately after theentry of gas flow into the soil), we see a subsonic flow. The reasonfor this matter is the resistance of the soil that reduces the gas flowspeed when entering the soil. In addition, the pressure caused bythe soil above the damaged section makes the input and outputpressures in the damaged section different.

Entering natural gas flow into the soil and its hit with soil par-ticles and also passing air through the soil particles causes thenatural gas emission into the soil and creates two vortices on theright and left sides of the gas flow in the soil. Fig. 14 shows thevelocity vectors inside the pipe and soil around the hole. Bymagnifying and increasing the size of the velocity vectors, thecreated vortices are observed in the Fig. 14(C).

Fig. 14. Velocity vectors around the hole for buried pipe.

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196194

(mm)

(Nm³/hr)

10 20 30 400

1000

2000

3000

4000

5000

6000

7000p =3 bar [Eq. (22)]p =3 bar [Numerical Simulation]p =3.5bar [Eq. (22)]p =3.5 bar [Numerical Simulation]p =4 bar [Eq. (22)]p =4 bar [Numerical Simulation]p =4.5 bar [Eq. (22)]p =4.5 bar [Numerical Simulation]p =5 bar [Eq. (22)]p =5 bar [Numerical Simulation]

p =3 bara [Eq. (22)]p =3 bar [Numerical Simum lation]p =3.5bar [Eq. (22)]p =3.5 bar [Numerical Simulation]p =4 bara [Eq. (22)]p =4 bar [Numerical Simum lation]p =4.5 bar [Eq. (22)]p =4.5 bar [Numerical Simulation]p =5 bara [Eq. (22)]p =5 bar [Numerical Simum lation]

Q

d

1

1

1

1

1

1

1

1

1

1

Fig. 16. Comparison of normalized release flow rate from hole between numericalsimulation and Equation (22).

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196 195

By analysis of the results of this simulation and using curvefittingmethod, it is observed that volumetric flow rate of leaked gashas a second, fourth and linear relation with the hole diameter,ratio of the hole diameter to the pipe diameter and initial pointpressure respectively. According to this, normalized volumetricflow rate of natural gas leakage from urban distribution gas pipes,can be respectively calculated by Eqs. (22) and (23) for above-ground and buried pipes.

Q ¼ 0:748�1þ b4

�d2p1 (22)

Q ¼ 0:44�1þ b4

�d2p1 (23)

As it can be seen, these equations have an agreement with Eq.(2). In Eq. (2) the amount of leaked gas calculated from an orificewhich installed at the end of a vertical pipe. The orifice located atthe outlet of a pipe and between two flanges. This equation can onlybe used for an approximate estimation of discharged gas from anorifice into the atmosphere. In the other word, this case is differentwith the real accident of gas leakage and gives approximate andunreal results. Therefore based on the results of this research, Eq.(2) modified into the Eq. (22).

Comparison between Montiel’s model (Montiel et al., 1998) andresults of Eq. (22) are presented in Fig. 15. As it could be seen,general behavior of both charts is similar. As it mentioned before,the leakage value obtained by Eq. (22) is higher than Montiel’smodel (due to reversal flow from downstream side toward hole). Asignificant increase in outflow from the hole can be seen for largerhole diameters. Also by increasing hole diameter, difference of thehole pressure and downstream pressure get more and we could seemore reversal flow from downstream side toward the hole.Therefore, considering the length of pipe after the hole have animpact on large diameter holes. In other words, disregarding lengthof downstream in previous studies causes unrealistic prediction forleakage. According to the figure, the percentage of relative differ-ence between the results is 2%, 4.5%, 11%, 12.43% and 12.3% for10 mm, 15 mm, 20 mm, 25 mm and 30 mm holes, respectively.

Fig. 16 compares numerical values and results of Eq. (22). Dataanalysis shows that the minimum percentage of relative error

(mm)

(Nm³/hr)

10 15 20 25 30

500

1000

1500

2000

2500

3000

3500

Montiel' s model [21]Eq. (22)

Q

d

Fig. 15. Comparison of normalized release flow rate from hole between Montiel’s(Montiel et al., 1998) model and Equation (22).

between these results is about 1.276% (which is for a case withd ¼ 5 mm and p1 ¼ 5 bar) and the maximum of it is about 6.625%(which is for a case with d ¼ 30 mm and p1 ¼ 3 bar).

5. Conclusion

In this investigation, a two-dimensional, turbulent andcompressible gas flowmodel has been built to predict leakage fromabove and underground natural gas pipeline. Based on the nu-merical results, the following points could be made:

� By increasing the hole diameter, the pressure of damaged sec-tion is reduced and consequently the pressure difference in-creases between the damaged section and initial section, whichresults in the increase in the leaked flow. In this investigation,small diameter holes are examined and it makes the pressuredifference between the damaged section and the ambientpressure to be lowand have a sonic flow in the damaged section.

� For larger hole diameters, taking into account the length of pipeafter the hole will have a significant impact on results due to theincrease of the reversal flow from the downstream toward thehole. In the other word, considering the length of pipe after thehole show an impact on large diameter holes. The reversal flowfrom downstream part toward the hole causes higher amountsin prediction of the gas leakage. For example in a case withD ¼ 163.6 mm and p1 ¼ 5 bar, considering the pipe length afterthe hole causes 2e15% increasing in the amounts of leakage forvarious hole diameters.

� In case of buried pipes, the damaged section is a common sec-tion between the pipe and the soil and similar to the above-ground pipe state in this case, we also have a sonic flow, butjust in the proximity of the hole and inside the soil, we see asubsonic flow. The reason for this matter is the resistance of thesoil that reduces the gas flow speed when entering the soil. Inaddition, after entering natural gas flow into the soil and hittingwith soil particles and also passing air through the soil particles,a pair of the vortices created on both sides of the gas jet whichpenetrated into the soil.

� By analysis of the results of this simulation paid to provide asimple and useful correlation to calculate the amount of naturalgas leakage from urban distribution pipelines for both above-

A. Ebrahimi-Moghadam et al. / Journal of Natural Gas Science and Engineering 34 (2016) 185e196196

ground and buried gas pipes. According to this correlations, thevolumetric flow rate of natural gas leakage, changes asascending functions approximately into the second, fourth andfirst orders with hole diameter, ratio of the hole diameter to thepipe diameter and initial point pressure, respectively. Also inmost of the investigated cases, the percentage of relative errorbetween simulation results and correlation results is below 5%which implies high accuracy of the presented correlation.

� This model is therefore a step forward in the effort to developmore precise and powerful calculation tools to foresee the ef-fects and consequences of potential accidents caused by the lossof containment of hazardous materials.

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Nomenclature

Cg: correction factor of the gas specific weightCPR: critical pressure ratiod: hole diameter (mm)D: pipe diameter (mm)E: total energy (J)F: external body forces (N/m3)Ftf: correction factor of the gas temperatureg: gravitational acceleration (m/s2)G: special weightk: conductivity factor (W/mK)l1: upstream length of the pipe (m)l2: downstream length of the pipe (m)Ma: Mach numberp: pressure (bar)Q: volumetric of rate of leaked gas (Nm3/hr)R: gas constant (J/kgK)T: temperature (K)u: velocity (m/s)v: specific volume (m3/kg)x: displacement (m)Z: compressibility factor of gas

Subscript

1: initial point of the pipea: atmospherecr: criticaleff: effectivef: fluid passing through porous mediai, j, l: directionss: solid material of the porous mediat: turbulent

Greek letters

a: permeability of the porous material (m2/s)b ¼ d/D: orifice ratiod: Kronecker deltaε: turbulence dissipation rate (J/kgs)g ¼ cp/cv: heat capacity ratioY: porosity of the porous mediak: turbulence kinetic energy (J/kg)m: viscosity (kg/ms)r: density (kg/m3)t: stress tensor (Pa)