SIMULATION OF GAS

PIPELINES LEAKAGE USING

CHARACTERISTICS METHOD

Author: Ehsan Nourollahi

• Organization: NIGC (National Iranian Gas Company)

• Department of Mechanical Engineering , Ferdowsi University, Iran

Topics:

1. Introduction

2. Characteristics method

3. The numerical solution method & The implementation of the leakage effect

4. Results & Conclusions

Introduction

The pipe surface leakage or the pipe section dismissal can be created of some various reasons like as corrosion, earthquake or mechanical stroke which may be implemented in the pipe surface and also overload compressors.

Figure (1)

After the leakage creation, the flat expansion pressure waves are propagated in two converse sides

These waves have the sonic speed and after clashing to the upstream and downstream boundaries, return to the form of compression or expansion wave depending on the edge type

In the leak location, depending on ratio of pressure to ambient pressure be more or less than CPR quantity, the flow will be sonic and ultrasonic or subsonic respectively.

If the flow be sonic and ultra sonic, the sonic reporter wave don’t leak from out of the pipe to inter the pipe practically. Hence the changes of the flow field are accomplished due to the flat pressure waves and the real boundary conditions on the start and end of the pipe

mass flow outlet of the hole only depends on the stagnation pressure in the leak location and on area of the hole and is not related to the form of the orifice cross section

1

1 1

2

k

k

cr

out

kP

PCPR

Characteristics method

The continuity equation is:

The momentum equation is:

By extension of these equation, we have:

With attention to the definition of speed of the sound by:

For an ideal gas:

tx

u

)(

Dt

Du

x

P

kp

a 2

s

pa )(2

01

x

u

x

u

t

01

x

uu

t

u

x

Third condition of continuity for isentropic flow is:

or:

For isentropic flow are constant, then we have:

)1/(2)( k

Aref a

a

)1/(2)( kk

Aref a

a

p

p

t

a

akt

1

1

21

x

a

ak

k

x

p

p

1

1

21

refrefA pa ,,

By using of the relationship between the sonic speed and the pressure in an ideal gas, these equations are changed to the below forms after some steps of rewriting of the mass and momentum conservation equations:

These equations are set of quasi-linear hyperbolic partial differential equations.

0})({2

1})({

x

uau

t

uk

x

aau

t

a

0})({2

1})({

x

uau

t

uk

x

aau

t

a

Therefore a solution of the form: is

required. Except of special cases, there are no

analytical solution for these equations, then we should

study numerical solutions. In this paper we use

Characteristics method to achieve a numerical

solution.

Base of this method is transferring of two independent

equations as and to another group

as or

),(

),(

txuu

txaa

),( txaa ),( txuu

),( txcc ),( aucc

Figure 2. Graphical interpretation of the characteristics method

(a) Three-dimensional surface defining

(b) Projection of line on characteristics surface to plane at

The solution may be represented by the curved surface bounded by edges PQRS

0c

),( txcc

if in a special point on the surface of for a

reviewing special curve from that point, the slope of the

projected curve on the x-t plane be equal with quantity

of curve of that point, the passing direction of that point

is known as the characteristic direction. We have in

mathematical expression:

By using of this complete derivative definition, the sonic

speed and particle speed parameters are determined

with respect to the time of a characteristic length like as

the below :

),( txcc

Cdtdx char )/(

x

ac

t

a

dt

da

char

x

uc

t

u

dt

du

char

Therefore if defined as the form of:

In length of two characteristics, they are rewritten like as:

),( aucc auc

auc

2

1

02

1

02

1

22

11

cc

cc

dt

duk

dt

da

dt

duk

dt

da

The numerical solution method

At first the none-dimensional parameters of A and U are defined as below in the characteristics method:

In the above equation is sonic speed in the start point. Then Reimann non-dimensional characteristics are defined as following:

refref a

aA

a

uU ;

refa

Uk

AUk

A2

1;

2

1

An explicit equation between Reimann variables in

inner points of the solution field is presented below

which is for each step:

By distinction of the state equation in the boundary, a

mono-equation is created between Reimann variables.

So always in any boundary, one of these variables is

known and the other one is unknown then the

unknown Reimann variable can be calculated, so the

effect of the boundary transfers to the solution field is

obtained.

nini

ni

ni

ni

ni ab

x

t

1111

nini

ni

ni

ni

ni ab

x

t

1111

The implementation of the leakage effect

For implementation of the leakage effect on the flow field, the mesh is chosen in a way that the hole location would be stated between two nodes

When the hole is created in the pipe surface, as it’s said, the pressure ratio to the ambient pressure in below the hole which is inter the pipe, is more than the CPR in the later time steps.

Figure (3)

The leakage point in any time step act as a boundary

and two expansion waves depending to direction of

flow in the pipe, would reach to a and b points with a

little time difference and create the same change in

non-dimensional speed of U like the below form:

bb

bb

aa

aa

Um

QUU

Um

QUU

Therefore, the flow is checked in

the hole location and outflow of

the leak location, calculated by:

1

1

1

2...

k

k

llor k

kZRT

MPAQ

Then the unknown parameters and are

calculated like the below form:

Therefore, the state of two points in any time step with

considering to corrected leakage effect and hence by

notice to the equations that governed to the problem

are type of the hyperbolic equations, during the time of

the leakage effect is transferred permanently as a third

boundary addition to the upstream and downstream

boundaries to the solution field.

ba

bbb

aaa

Uk

Uk

)1(

)1(

Results and Conclusions

Consumptions:

Pipe Length : 250 meter

Hole Area : 1 cm

Number of grid system : 100 nodes

Initial gas pressure : 30 bar

Initial gas speed : 41 ft/s

Also temperature is constant and there are non viscose flow.

Boundary conditions:Upstream boundary condition is the reservoir with constant

pressure and the downstream boundary condition is stated

with three forms:

• The boundary with no changes with respect to the location

• The valve with constant coefficient of pressure drop

• close end

Figure 4. State (1) of the boundary conditions:4-a Pressure changes by increasing of the pipe length at primary

times4-b Changes of the exit mass flux by time

Figure 5. State (1) of the boundary conditions:5-a changes of the exit mass flux by increasing the hole area and

pipe length5-b changes of the exit mass flux by increasing the pipe pressure

and pipe length

Figure 6. State (2) of the boundary conditions:6-a. Pressure changes by increasing of the pipe length at primary

times6-b. Changes of the exit mass flux by time

Figure 7. State (2) of the boundary conditions:7-a Changes of the exit mass flux by increasing the hole area and

pipe length7-b Changes of the exit mass flux by increasing the pipe pressure

and pipe length

Figure 8. state (3) of the boundary conditions:8-a Pressure changes by increasing of the pipe length at primary

times8-b Changes of the exit mass flux by time

Figure 9. State (2) of the boundary conditions:9-a Changes of the exit mass flux by increasing the hole area and

pipe length9-b Changes of the exit mass flux by increasing the pipe pressure

and pipe length

With best wishes

of Iranian People