Modeling improvements for leak detection in pipelines of LPG

Lizeth Torres and Cristina Verde

Abstract— The fluid behavior in a pipeline of LiquefiedPetroleum Gas (LPG)—according to the principles of conser-vation of mass and momentum—is governed by a set of PartialDifferential Equations (PDE). In this work, such equationsaremodified with the inclusion of an unsteady friction model andapproximated using the Finite Difference Method. As a resultof this approximation, a model is obtained and subsequentlyused for the design of a high gain observer to locate leaksin a pipeline. Some results based on real data are exposed todemonstrate the repercussions of the proposed modifications intasks of leak detection.

I. I NTRODUCTION

Risks in pipelines associated with accidental releases ofpetroleum products are still high. These risks have forced tothe scientific community to develop more efficient methodsfor Leak Detection and Isolation (LDI) in pipelines. Thepurpose of such LDI methods is to obtain information on thelocation of leaks in time, avoiding undesirable consequencessuch as economical losses, damages to the environment anddamages to the population, [1]. The two major difficultiesto locate leaks are (i) the reduced number of sensors alongthe extended pipelines and (ii ) the requirement of involvedmodels for complex networks.

The recent paper [2] presents a pretty well-documentedoverview on LDI techniques, distinguishingdirect transientmethods,inverse transientapproaches, andfrequency-basedanalysis. The first ones are conceived to give an interpretationof signal changes so as to detect leak effects in them (see e.g.[3], [4], [5], [6]); the second type of these approaches refersto the problem of recovering parameters of a time-domainmodel from a set of actual measurements, allowing then tocompare current measurements with a leak-free situation andmonitor leaks in this way (e.g. as in [7], [8], [9], [10], [11],[12], [13]); while the third family of techniques gathers allworks related to frequency response analysis with regard tothe effect of leaks (cf. [14], [15], [16], [17], [18]).

Most of the LDI algorithms are based on the assumption ofa constant friction coefficient, neglecting that this coefficientdepends on the dynamics of the flow. As consequence, thealgorithms with this assumption are only valid in a particularoperation point and any change in the operation conditionscan produce errors in the diagnostic. For this reason, severalunsteady friction models have been investigated, allowingto calculate this parameter in function of the flow. Forinstance in [19], a friction equation is formulated, whichrelates the wall shear stress in transient laminar pipe flow

The authors are with the Institute of Engineering-UNAM, Mexico City,Mexico. ftorreso@iingen.unam.mx, verde@unam.mx

to the instantaneous mean velocity and to the weighted pastvelocity changes.

In [20], two one-dimensional unsteady friction models areconsidered, an instantaneous acceleration-based model anda convolution-based model. The first one is shown to failfor certain transient event types and the convolution-basedmodel is successful for all transient event types. In [21], astudy of the effects of unsteadiness on flow characteristicsisperformed, showing that in fast transients, both local inertiaand friction forces must be evaluated in a specific manner.Since these two effects are correlated and seem to dependon the same quantities. A single expression in the form ofan addition to the steady friction term is proposed for both.In [22], a pipeline model that incorporates the Brunone’sunsteady friction model is proposed, but in contrast to thestandard treatment of the unsteady friction term as a sourceterm, the authors propose a nonconservative formulation.

Considering these studies, this article intends to show howsome improvements in the friction modeling can benefit toLDI tasks. Then, a high gain observer is designed to locatesingle leaks in LPG pipelines. The observer is based on amodel that includes the unsteady friction term presented in[21] and a flow-dependent friction coefficient. To computethe friction coefficient, theSwamee-Jainequation—an ap-proximation of the implicitColebrookequation– is used. Themodifications of the model have been effectuated in order toimprove the LDI observer presented in [23]. The idea ofimproving the observer emerged from the analysis carriedout in [24] that show the critical effects of friction in LDItasks.

Section II presents the PDE equations–with some im-provements in the computation of the friction–describingthe dynamic behavior of a pipeline of LPG, as well as anapproximation of such equations using the Finite DifferenceMethod. Section III exposes the conception of a nonlinearobserver based on the approximated pipeline model andconsidering the input dynamics. Section IV, on the one handpresents a comparison between the response of the improvedmodel and real data, and on the other hand expose someresults about the performance of the high gain observer.Finally, Section V provides some conclusions.

II. PIPELINE DYNAMICS

Under some assumptions presented in [25], [26], thefluid behavior in thej-th section (∀j = 1, ..., N ) of aclosed pipeline is governed by the following momentum andcontinuity equations:

∂H(zj, t)

∂t+

b2

gAr

∂Q(zj , t)

∂zj= 0 (1)

1

Ar

∂Q(zj , t)

∂t+g

∂H(zj, t)

∂zj+J(Q(zj, t))+g sinαj = 0 (2)

where(zj , t) ∈ (Oj , lj−1)×(0,∞) are the time [s] and space[m] coordinates respectively, withOj as the origin of eachj-th section andlj as its length [m]. Hence, a whole pipelineis composed by differentj-sections with spatial subdomainszj forming the spatial domainz ∈ (0, L), i.e, z =

⋃

j zj,

whereL =∑N

j=1 lj is the length of the entire pipeline andN the total number ofj-sections. See Fig. 1.

H1 H3 Hj+1

l1

H2 Hj HN

HN

Q1 Q2 Qj

QN

lj lN

Fig. 1. Pipeline with sectionsj = 1, ...,N

H(zj , t) is the head pressure [m], Q(zj , t) the flowrate [m3/s], b the wave speed in the fluid [m/s], g thegravitational acceleration [m/s2], Ar the cross-sectional area[m2] assumed to be constant along the entire pipeline,αj theelevation angle of thej-th section, andJ(Q(zj, t)) representste friction losses affecting the fluid dynamics of the pipe.

Classically,J(Q(zj , t)) is expressed in thesteadyformgiven by

J(Q(zj , t)) = JS(Q(zz, t)) =fQ(zj , t)|Q(zj, t)|

2DA2r

(3)

wheref is theDarcy-Weisbachfriction coefficient. In most ofmodel-based LDI approaches, this coefficient is consideredto be constant, even if it is sometimes updated when a leakis detected, [27], [28]. But this coefficient actually dependson the so-called Reynolds number (Re) and the roughnesscoefficient of the pipe (e). This dependence is expressed bythe Colebrookequation, but this is given in an implicit way.Optionally, an approximated explicit formulation off is givenby theSwamee-Jainequation expressed as

f = 1.325

{

ln

[

0.27

(

e

φ

)

+ 5.74

(

1

Re

)0.9]}−2

(4)

where the Reynolds number can be calculated with:

Re =ρV φ

µ=

4ρQ(zj, t)

πφµ(5)

This equation is valid for10−8 < e/φ < 0.01 and5000 <Re < 108. Whereµ is the viscosity of the fluid.

Notice thatf is calculated from the flow in thej-th sectionaccording to Eq. (4), such that it can be defined asf |zj = fj.However, to keep the possibility of assuming this coefficientto be constant along the pipeline, the variablef is kept in thenotation.

In order to obtain a more complete friction modeling,someunsteadyfriction lossesJU (Q(zj , t)) can also be added

(namelyJ(Q(zj , t)) = JSQ(zj , t)+ JUQ(zj, t)) in Eq. (1)-(2), according to the model below:

JU (Q(zj , t)) =k

2Ar

(

∂Q(zj, t)

∂t+ bΦA

∣

∣

∣

∣

∂Q(zj, t)

∂zj

∣

∣

∣

∣

)

(6)

where ΦA = sgn(Q(zj , t)) and k denotes theBrunonecoefficient given byk =

√0.0476/2 for laminar flow and

by

k =

(√

7.41

Relog (14.3/Re0.05)

)

/2

for turbulent flow.A closed-form solution of Eq. (1)-(2) is not available. In

the present paper, the Finite Difference Method is chosenbecause of its simplicity and because the structure of theresulting models—roughly triangular—is considered to besuitable for the conception of high gain observers.

In order to discretize in space Eq. (1)-(2), initial conditionsare required expressing the profiles ofQ(zj , t) andH(zj , t)at the initial instantt = 0, in this work they are defined asfollows:

Q(zj , 0) := Q0j(zj), H(zj , 0) := H0

j (zj)

Moreover, temporal profiles ofQ(zj , t) andH(zj , t) at theboundaries of the spatial subdomains of each sectionj arerequired, i.e., atzj = Oj and zj = lj respectively. Theseboundary conditions can be functions denoted as follows:

H(zj = Oj , t) := Hinj ; H(zj = lj , t) := Hout

j (7)

Q(zj = Oj , t) := Qinj ; Q(zj = lj , t) := Qout

j (8)

Additionally, if the set ofN pairs of equations (1)-(2) fora whole pipeline has to be solved, boundary conditions arealso required, then let denote

H(z = 0, t) := Hin; H(z = L, t) := Hout (9)

Q(z = 0, t) := Qin; Q(z = L, t) := Qout (10)

Taking as boundary conditions,Hinj andHout

j , the finite-dimensional model for thej-th section of a pipeline is givenby the following equations:

Qi=β

[

a1

Hi−Hi+1

∆zji+µ0Qi|Qi|+ µj+µ

∣

∣

∣

∣

Qi −Qji+1

∆zji

∣

∣

∣

∣

]

(11)

Hi+1 = a2

(

Qi −Qi+1

∆zji

)

; ∀i = 1, . . . , ns (12)

where i is the index of the spatial discretization andns isthe total number of spatial discretization segments of thej-thsection.

The physical parameters associated to model (11)-(12) arecondensed in

β =2

2 + k, a1 = gAr; a2 =

b2

gAr;µ0 = − f

2φAr

andµj = −gAr sinαj , µ =

bΦAk

2

while the boundary are set as follows:

H1 = Hinj , Hns

= Houtj

Note that in order to simplify the notation, the time isomitted as an argument of the functionsQi andHi.

If a single leak occurs in thej-th section, this section willhave a new boundary conditionQf corresponding to the flowrate of the leak at the pointzj = zf , such that the section willbe partitioned into two segments with a boundary conditionin between (see Fig. 2). As a consequence, the pressure inthe leak point is given by

Hf = a2

(

Qi −Qi+1 −Qf

∆zji

)

(13)

III. O BSERVER DESIGN

In [23], an algorithm based on structured residuals hasbeen presented to identify a section of the pipeline witha leak, as well as a nonlinear observer to locate the leakposition accurately—once the region was identified.

Here, a high gain observer based on the model (11)-(13)is proposed considering (i) a friction coefficient as a functionof the flow, and (ii ) an unsteady friction term. The proposedobserver is conceived to be employed after the identificationof a sectionj with a leak, using for instance the approachproposed in [23].

In order to design the observer, let consider system (11)-(12) with ns = 2 and the Eq. (13). The choice ofns isdue to the requirement of representing a flow rate before theleak point and a flow rate after it, i.e., a model with at leasttwo discretization segments is needed. Hence, the minimalmodel to represent aj-th section with a leak is expressed asfollows:

Q1 = β

[

a1

H1 −Hf

∆zj1+ µ0Q1|Q1|+ µj + µ

∣

∣

∣

∣

Q1 −Q2

∆z1

∣

∣

∣

∣

]

Hf = a2

(

Q1 −Q2 −Qf

∆zj1

)

(14)

Q2 = β

[

a1

Hf −H3

lj −∆zj1+ µ0Q2|Q2|+ µj + µ

∣

∣

∣

∣

Q2

lj −∆zj1

∣

∣

∣

∣

]

∆zj1 = 0

Qf = 0

where the adjoint states∆zj1 andQf represent the positionof the leak zf and its respective flow rate, both to beestimated. Notice that∆zj2 = lj − ∆zj1, where lj is thelength of thej-th section with a leak. See Fig. 2.

lj

zf = Leak position

Dzj1 Dzj2 = lj-Dzj1

Fig. 2. Segmentation of thej-th section as a consequence of a leak

Then, the state vector of system (14) is:

x(t) = [ Q1 Hf Q2 ∆zj1 Qf ]T (15)

wherex is composed by the elementsxi with i = 1, ..., 5.

The flow rates provided by the system (14) are used here astheir outputs, i.e.,y(t) = [Q1 Q2]

T . Such that, from theseoutputs and the flow measurements at the extremes of thewhole pipelineym(t) = [Qin Qout]

T , the observation errorcan be calculated:

e(t) = [Qin Qout]T − [Q1 Q2]

T (16)

Additionally, the boundary conditionsH1 and H3—pressure measurements corresponding to thej-th sectionwith a leak—are considered as inputs, i.e.:

u(t) = [H1 H3]T = [Hin

j Houtj ]T (17)

Hence, the whole model takes the form of a nonlinearstate-space representation as follows:

x(t) = f(x(t)) + g1(x(t))u(t) + g2(y(t)) (18)

y(t) = [h1(x), h2(x)]T = h(x(t))

for functions f, g1, g2, h1, h2, h resulting from (14). Thevectorg2(y(t)) gathers the steady friction terms.

With the following notations:

Φ : x 7→{

ξ1 = [h2,Lf+g1h2]T

ξ2 = [h1,Lf+g1h1,L2f+g1

h1]T (19)

whereL denotes de Lie derivative, andξ1, ξ2 are composedby the elementsξ1i and ξ2j respectively, withi = 1, 2 andj = 1, 2, 3, the system inξ coordinates becomes:

ξ11 =ξ12

ξ12 =ϕ12(ξ1, ξ2, u, u)

y1 =ξ11

ξ21 =ξ22

ξ22 =ξ23 + ϕ22(ξ1, ξ2, u, u)

ξ23 =ϕ23(ξ, u)

y2 =ξ21

(20)

ξ11 =ξ12 = Ψ1(Q1)

ξ12 =ϕ12(ξ1, ξ2, u, u) = Ψ2(Q1, Hf , Q2,∆zj1, Qf)

y1 =ξ11

ξ21 =ξ22 = Ψ3(Q2)

ξ22 =ξ23 + ϕ22(ξ1, ξ2, u, u) = Ψ4(Q1, Hf , Q2,∆zj1, Qf)

ξ23 =ϕ23(ξ, u) = Ψ5(Q1, Hf , Q2,∆zj1, Qf )

y2 =ξ21

This system is under the form of auniformly observablesystem, for which some high gain observer can be designed,[29].

By considering two subsystems of the form:

ξi = Aiξi + ϕi(ξ, u, u), yi = Ciξ

i,

for i = 1, 2 and appropriateAi, Ci, ϕi resulting from (20),the observer design can for instance be done on the basis oftwo separate single-output designs as follows:

˙ξi = Aiξ

i+ϕi(ξ, u, u)−S−1i CT

i (Ci ξi−yi), i = 1, 2 (21)

where Si = [Si(l, k)]1≤l,k≤niis classically calculated as

follows:

Si(l, k) =(−1)l+kCk−1

l+k−2

θl+k−1i

(22)

where the binomial coefficient is defined:Ckn = n!

(n−k)!k!andθ1, θ2 > 0.

Notice that the principal difference between the observerproposed here and the observer presented in [23] is theassumption of constant inputs in the last one. Thus, toconsider the dynamic of the inputs, observer (21) has beenimplemented with the new coordinates, while the observerof [23] has been implemented with the original states.

IV. EXPERIMENTAL RESULTS

In this section, some results related to the suggested ob-server are presented. Real data–provided by sensors installedin a real gas pipeline–were used to validate its performanceexperimentally.

The whole pipeline—with lengthL = 60003 [m]—hasgas stations at:zA = 0 [m], zB = 10326 [m], zC = 28886[m], zD = 43866 [m], zE = 47412 [m] and zF = 60003[m]. The elevation data corresponding to each of thesecoordinates arezA = 2247 [m], zB = 2245 [m], zC = 2252[m], zD = 2334 [m], zE = 2500 [m] and zF = 2208[m]. The pipeline is schematized in Fig. 3, whereas thecorresponding physical parameters are given in Table I:

0 10,000 20,000 30,000 40,000 50,000 60,0002200

2250

2300

2350

2400

2450

2500

2550

length [m]

elev

atio

n [

m]

A B C

D

E

ExtractionF

Fig. 3. LPG pipeline under study

TABLE I

PHYSICAL PARAMETERS

Parameter Value UnitsAr 0.1939 [m2]ρ 530 [kg/m3]b 830 [m/s2]f 0.01255 -k 1.0909 -e 14.8e− 3 [mm]µ 1e− 6 [N.s/m2]

The constant friction coefficient given in Table I has beencalculated using the Darcy-Weisbach equation [26] to be usedby the observer presented in [23].

The pipeline is instrumented with pressure sensors ateach station (A to F) and flow rate sensors at the ends(A and F). In this pipeline is possible to extract LPG at390 [m] downstream the station D. Hence, taking advantageof this situation, the known extraction is considered as aleak to validate the proposed observer. The data used forthe validation correspond to 8 hours of LPG extractionapproximately.

Fig. 4 exposes a comparison between real data and theresponse of the model given by the Eq. (11)-(12) withns =5. The good agreement between them can be corroboratedin Fig. 5, where percentage errors are plotted.

0 200 400 600 800 1000 1200 1400700

750

800

850Pressure head at station B

time [min]

[m]

0 200 400 600 800 1000 1200 1400650

700

750

800Pressure head at station C

time [min]

[m]

0 200 400 600 800 1000 1200 1400500

550

600

650

700Pressure head at station D

time [min]

[m]

0 200 400 600 800 1000 1200 14000.26

0.28

0.3

0.32

0.34

0.36Flow rate at station A

time [min]

[m3 /s

]

Fig. 4. Comparison between real data and model given by Eq. (12)

0 500 10000

0.5

1

1.5

2Pressure head error at the station B

time [min]

[%]

0 500 10000

0.5

1

1.5

2

2.5Pressure head error at the station C

time [min]

[%]

0 500 10000

2

4

6Pressure head error at the station D

time [min]

[%]

0 500 10000

5

10

15Flow rate error at the station A

time [min]

[%]

Fig. 5. Modeling error

Now, to validate the proposed observer, it is supposed thatthe sectionj with a leak has already been identified usingstructured residuals as in [23]. This section is obviouslywhere the known extraction is located, delimited by thestation D and E with a lengthlj = 3546 [m].

For the observer implementation, the inputs were thepressure measurements at the ends of the section with theleak, as indicated in Eq. (17), i.e. the pressure heads at

the stations D and E. Whereas, the observation error wascalculated using Eq. (16).

Figure 6 exhibits the position estimation performed by theobserver (14), where is noticeable that the estimation is veryclose to the real value. This figure shows too the estimationrealized by the observer proposed in [23]. In Fig. 7, theestimation errors of both observers are plotted.

Both observers do a good estimation untilt = 240[min], but after this time, the dynamics of the pressures andflows rates change remarkably (see Fig. 4). Consequently,the estimation performance of both observers is affected.The dynamic changes did not affect so much the goodestimation performed by the observer (21), thanks to theinclusion of the unsteady friction model and the frictioncoefficient calculated from the flow rates. However, theperformance of the observer proposed in [23] is not so goodwith respect to the observer (21), its disadvantage is the useof a constant friction coefficient which does not considerdrastic variations in the pipeline dynamics. Since the constantfriction coefficient—presented in Table I—is calculated fromthe pressure heads and flows in an operation point, this isnot more valid when there are significant variations in thedynamics.

50 100 150 200 250 300 350 400 450 500360

380

400

420

440

460

480

time [min]

Len

gth

[m

]

Real leak positionEstimated leak position by the observer proposed in this workEstimated leak position by the observer proposed in (Torres et al. 2012)

Fig. 6. Position Estimation

50 100 150 200 250 300 350 400 450 500−40

−30

−20

−10

0

10

20

time [min]

[%]

Estimation error (Observer proposed in (Torres et al.))Estimation error (Observer proposed here)

Fig. 7. Error Estimation

V. CONCLUSION

In this article was presented a nonlinear observer basedon an approximated model of the momentum and continuityequations governing the behavior of a pipeline. The principal

contribution of this work is the modification of the dynamicalmodel—for leak detection and isolation purposes—with aunsteady friction term and a friction coefficient calculatedfrom the flow. The improved model was used to design anonlinear observer—tested with real data—showing a goodperformance with respect to another observer already pre-sented in the literature.

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