Frequency Domain Analysis for Detecting Pipeline Leaks

Frequency Domain Analysis for Detecting Pipeline Leaks Pedro J. Lee1; John P. Vítkovský2; Martin F. Lambert3; Angus R. Simpson4; and James A. Liggett5 Abstract: This paper introduces leak detection methods that involve the injection of a fluid transient into the pipeline, with the resultant transient trace analyzed in the frequency domain. Two methods of leak detection using the frequency response of the pipeline are proposed. The inverse resonance method involves matching the modeled frequency responses to those observed to determine the leak parameters. The peak-sequencing method determines the region in which the leak is located by comparing the relative sizes between peaks in the frequency response diagram. It was found that a unique pattern was induced on the peaks of the frequency response for each specific location of the leak within the pipeline. The leak location can be determined by matching the observed pattern to patterns generated numerically within a lookup table. The procedure for extracting the linear frequency response diagram, including the optimum measurement position, the effect of unsteady friction, and the way in which the technique can be extended into pipeline networks, are also discussed within the paper. DOI: 10.1061/~ASCE!0733-9429~2005!131:7~596! CE Database subject headings: Leakage; Water pipelines; Frequency response; Resonance; Transient flow; Linear systems. Introduction Fluid transients in pipes—water hammer waves—are affected by pipeline features, including leaks and blockages, thus leaving clues that can be used for the identification and location of such features. Analysis of transients can reveal a substantial amount of information concerning the integrity of the system. Numerous methods utilize transient behavior for the purpose of leak detection. These include the inverse transient method ~Liggett and Chen 1994; Nash and Karney 1999; Vítkovský et al. 1999; 2001!, the transient damping method ~Wang et al. 2002!, and timedomain reflectometry techniques ~Jönsson and Larson 1992; Brunone 1999; Covas and Ramos 1999!. This paper proposes the use of a generated input signal for leak detection and signal1 Postgraduate Student, Centre for Applied Modelling in Water Engineering, School of Civil and Environmental Engineering, The Univ. of Adelaide, Adelaide SA 5005, Australia. E-mail: [email protected] 2 Graduate Hydrologist, Water Assessment, Natural Resources and Mines, Indooroopilly QLD 4068, Australia. E-mail: [email protected] 3 Associate Professor, Centre for Applied Modelling in Water Engineering, School of Civil and Environmental Engineering, The Univ. of Adelaide, Adelaide SA 5005, Australia. E-mail: [email protected] 4 Associate Professor, Centre for Applied Modelling in Water Engineering, School of Civil and Environmental Engineering, The Univ. of Adelaide, Adelaide SA 5005, Australia. E-mail: [email protected] 5 Professor Emeritus, School of Civil and Environmental Engineering, Cornell Univ., Ithaca, NY 14853-3501. E-mail: [email protected] Note. Discussion open until December 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on October 9, 2002; approved on July 21, 2004. This paper is part of the Journal of Hydraulic Engineering, Vol. 131, No. 7, July 1, 2005. ©ASCE, ISSN 0733-9429/2005/7-596–604/$25.00. 596 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 processing techniques as a means of analyzing these signals in the frequency domain. The concepts of steady oscillatory flow and pipeline resonance are well established, and details can be found in Zielke et al. ~1969!, Zielke and Rösl ~1971!, Chaudhry ~1970, 1987! , and Wylie et al. ~1993!. While a pipeline reinforces and transmits input signals of a particular frequency ~for example, the fundamental frequency!, others are effectively absorbed within the system. Thus, pipeline systems are similar to frequency filters for weakly nonlinear unsteady systems, the characteristics of which are determined by system properties, such as boundary conditions, friction, and wave speed. The degree that each frequency component is absorbed or transmitted within the pipeline is defined by a frequency response diagram ~FRD!, also known as the transfer function for the system ~Lynn 1982!. This diagram relates both the magnitude and phase of the system output to the system input for different frequencies. A transfer function—describing the relationship between the frequency spectra of the input and the output—can be obtained using linear systems theory ~Lynn 1982; Liou 1998; De Salis and Oldham 1999, 2001!. A wide-band input signal fed into the system while measuring the output, generates the response at a wide range of frequencies simultaneously. When a pipeline system is excited by such a signal, the frequency response function is related to the Fourier spectrum of the input and output signals by Hsvd = Ysvd Xsvd s1d where Xsvd and Ysvd5Fourier transforms of the input and output signal, respectively; Hsvd5frequency response function of the linear system; and v5angular frequency. Eq. ~1! is known as the system identification equation. Wylie et al. ~1993! and Ferrante et al. ~2001! used the impedance equations to generate the transfer function for single pipeline systems. The input and output were defined as the complex discharge and complex head at a point in the pipeline, respectively. Ferrante et al. ~2001! indicated that the location of a leak affects Fig. 1. Single pipeline used for proposed resonance leak detection method the relative magnitude of the resonance peaks in the transfer function, but the impact of a leak on the location of resonance frequencies is minimal, except in the case of large leaks ~contrasting with the effect of nondiscrete blockages, which do influence the location of resonance frequencies!. This finding leads to a proposed method that uses the relative sizes of resonance peaks of different harmonic frequencies as a means of leak location in a single pipeline as explained below. The techniques of extended blockage detection—proposed in Antonopoulous-Domis ~1980!, Qunli and Fricke ~1989, 1991!, and De Salis and Oldham ~1999, 2001! that use resonant peak shifts—are not directly applicable to leak detection. This paper uses the perturbation of flow generated by an inline valve that fluctuates in a specified pattern. An upstream reservoir and the fluctuating downstream valve discharging into a constant head reservoir bound the pipeline. An illustration of the pipeline configuration is shown in Fig. 1. Two methods of leak detection are proposed in this paper. An inverse technique in the frequency-domain forms the basis one of the method. In addition, a method for locating the section of leaking pipe within the system using a resonance peak-sequencing method is also introduced. Both methods require the accurate determination of the FRD. A discussion of the generalizations of the techniques forms the final part of the paper. Background The impedance method ~Fox 1989; Wylie et al. 1993; Ferrante et al. 2001! and the transfer matrix method ~Chaudhry 1970, 1987! make use of linearized forms of the momentum and continuity equations in the frequency domain. The linearized steady-oscillatory continuity and momentum equations are given in Chaudhry ~1987! as dq gAvi + 2 h=0 dx a S D vi dh + +R q=0 dx gA s2d s3d where R5frictional resistance term, equal to sfQ0d / sgDA2d for turbulent flows or s32nd / sgAD2d for laminar flows in which n5kinematic viscosity. The complex variables q and h represent the discharge and head oscillations about the steady-state ~pretransient! conditions in the frequency domain at the frequency v. The variables q and h are related to the discharge sQd and head sHd by Q = Q0 + Rhqeivtj and H = H0 + Rhheivtj, where RhPj denotes the real portion of the complex number P and subscript “0” refers to steady-state conditions. Eqs. ~2! and ~3! represent a set of ordinary differential equations that are solved using separation of variables. The solution is expressed in matrix form relating the frequency-domain discharge and head fluctuations in the upstream and downstream ends of the pipe segment, HJ 3 q h n+1 = cosh mk,k − − Zk sinh mk,k 1 sinh mk,k Zk cosh mk,k 4H J q h n s4d / where Zk = mka2k ivgAk ~the characteristic impedance for the kth pipe section!; and superscripts n and subscript k denote the node and link number within the system, respectively, in which the link k lies between the upstream node n and downstream node n + 1. The propagation function is m = a−1Î−v2 + igAvR; and ,k5the length of the uniform pipe section under consideration. The linearizations used to create Eqs. ~2! and ~3! are valid if abssqd ! uQ0u for Q0 Þ 0 , absshd ! uDHu, where DH is the difference in head at the extremities of the pipeline. abssPd = ÎPr2 + P2i denotes the absolute value of an arbitrary complex number P. Similar matrices can be derived for different link components within the pipeline. The problem considered herein considers the impact of a leak on the frequency response diagram for a single pipeline system bounded by an upstream reservoir and a downstream valve. Oscillating the valve opening provides the excitation to the system. The valve orifice equation, relating the head loss across the valve to the flow through the valve, is QV = CVtÎDHV s5d where QV5flow through the valve; DHV5difference in head across the valve; CV5valve coefficient; and t5dimensionless valve opening5CdAV / sCdAVdRef, where the subscript “Ref” refers to a reference valve opening size at steady state; and AV5area of the valve aperture. Substitution of t = t0 + RsDteivtd into Eq. ~5! followed by linearization gives HJ 3 q h n+1 1 0 2DHV0 = 1 − QV0 4H J 3 q h n 0 + 2DHV0Dt t0 4 s6d where QV0 , DHV05steady-state flow through the valve and the steady-state head loss across the valve, respectively; Dt5complex variable describing the magnitude of the valve oscillation; and t05dimensionless valve opening at steady state. Eq. ~6! is first-order accurate for small perturbations given that abssDtd ! t0. As Dt is defined as real, abssDtd is replaced by Dt for the duration of this paper. The discharge from the leak is expressed by the orifice equation: JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 / 597 Table 1. System Parameters for Pipeline Example in Fig. 1 Fig. 2. Percentage error at v / vth = 1.0 as a function of the magnitude of valve perturbation QL = CdALÎ2gsHL − zLd s7d where CdAL5lumped discharge coefficient of the leak orifice; HL5head at the leak orifice; zL5elevation of the pipe at the leak; and QL5discharge out of the leak orifice. Linearizing Eq. ~7! gives HJ 3 q h n+1 = 1 − 0 QL0 2sHL0 − zLd 1 4H J q h Parameter Value L1 L2 D1 D2 H1 H2 a1 a2 f1 f2 Dt QL Q0 CdAL / Apipe vth 1,400 m 600 m 300 mm 300 mm 50.0 m 20.0 m 1 , 200 m / s 1 , 200 m / s 0.020 0.022 0.1 4.373 10−3 m3 / s 0.0153 m3 / s 0.002 0.15 Hz n s8d where QL0 , HL05steady-state flow out of the leak and steadystate head at the leak, respectively. The variables q and h represent the discharge and head oscillations about the steady-state ~pretransient! conditions. Notice that each transfer matrix determines the head and discharge oscillation downstream by drawing upon upstream information. These transfer matrices can be combined by multiplication to form an overall transfer matrix for the entire pipeline system. Multiplication starts from downstream and progresses upstream. The system is solved subject to boundary conditions ~Chaudhry 1987!. Due to the linearized nature of the transfer matrix equations, however, care must be taken so that the magnitude of the sinusoidal driving function does not exceed the linearization approximations. Fig. 2 illustrates the percentage linearization error as a function of the valve perturbation using the nonlinear method of characteristics model. The error is quantified as the deviation of the signal spectrum from the case when the system is behaving linearly at the input frequency. The error grows exponentially and reaches a value of 1% at a Dt / t0 of 0.15. Note that the results of Fig. 2 were derived for the first-harmonic frequency of the system and should not be used as a predictor of the errors at all other frequencies. Properties of the Frequency Response Diagram Two important parameters that can affect the FRD are the measurement position and the presence of a leak within the pipeline. The system in Fig. 1 is used to illustrate the impact of these parameters on the FRD of a pipeline system. The system parameters are shown in Table 1. In the case of a linearized pipeline system with asymmetric boundary conditions, the FRD consists of a series of evenly spaced peaks and troughs with the peaks occurring at the odd multiples of vth and the troughs occurring at the even multiples of vth, where vth is the fundamental angular frequency of the pipeline, defined as 598 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 vth = ap 2L s9d where L5length of the pipeline. While the location of the resonance ~peaks! and antiresonance ~troughs! points on the frequency axis is fixed by the fundamental frequency, vth, the response magnitudes at these frequencies are affected by the position of the measuring point ~Muto and Kanei 1980!. Fig. 3 shows the magnitude of various peaks in the FRD as a function of the measurement position along the pipe for the system shown in Fig. 1. Each series in Fig. 3 describes the magnitude of response from a particular harmonic frequency ~vr = v / vth = 1.0, 3.0, 5.0…!, and the response changes significantly with a change in the measuring position. They are also known as the “mode shapes” of the system. The shape of the FRD at any position in the pipeline can be derived from the relative magnitude of the harmonics at that position. The FRD at 742 and 2,000 m along the pipe are shown as insets in Fig. 3. The peaks of all harmonics converge at the extremities of the pipeline. The FRD produced from the downstream end of the pipeline ~at 2,000 m, at the excitation valve in Fig. 1! is particularly important as it displays equal magnitude peaks as a result of this convergence. From Fig. 3, the frequency responses for all harmonic peaks are also at a maximum at the downstream boundary, resulting in stronger signals and a subsequent higher signal to noise ratio ~Lee et al. 2002a!. The importance of this measurement position becomes apparent when the impact of a leak on the FRD is considered. The impact of a leak on the FRD measured at the end of the pipe is shown in Fig. 4. The figure indicates that a leak causes a nonuniform pattern in the resonance peaks in the FRD. This finding has also been observed in Ferrante et al. ~2001! and Lee et al. ~2002a, 2003!. When the response is measured at the excitation boundary of the pipeline, the FRD of an intact pipeline ~no leak! consists of a series of equally spaced peaks of equal magnitude. The presence of a leak results in a deviation of the FRD from this known pattern and clearly indicates the presence of a leak prior to the application of any leak analysis technique. Fig. 3. Response of major harmonic frequencies as measurement position is changed along the nonleaking pipeline of Fig. 1, insets indicate the frequency response diagram obtained at two different measurement positions Inverse Resonance Technique As indicated in Fig. 4, a single leak in a pipeline can lead to a change in the shape of the FRD, caused by the frequencydependent damping of components in the transient signal. Inverse fitting minimizes the sum of the difference squared between measured and modeled frequency response functions by varying the value of the leak size sCdALd and leak position sxLd within the transfer matrix model. The leak size is changed within the transfer matrix of the leak @Eq. ~8!# and the leak position is changed by the length of the pipe sections between the leak and system boundaries @Eq. ~4!#. This method is similar to the inverse transient method of Liggett and Chen ~1994!. The objective function is given by the least-squares criterion M E= fhmj − h jg2 o j=1 s10d where E5objective function value; hmj and h j5measured and calculated frequency domain amplitude responses at the jth frequency, respectively; and M5number of measurement points. The minimization algorithm used for the results in this paper is the shuffled complex evolution ~SCE! algorithm ~Duan et al. 1993!. The SCE algorithm performs a global search based on the simplex method and does not require the use of local gradient information. The settings for the SCE algorithm are as follows: the number of complexes=number of fitted parameters=2, error Fig. 4. Frequency response from the leaking and intact pipeline of Fig. 1 measured at the inline valve at downstream boundary tolerance=1 3 10−12, the bounds for the leak size sCdALd are from 0.0 to 0.005, and the bounds for the leak position are from 0.0 to 2,000 m. Results for an Example Problem The results from the application of the inverse resonance method for a FRD extracted from the numerical model for a point 800 m from the upstream boundary is given in Fig. 5. The frequency response was extracted from the system shown in Fig. 1. The inverse resonance method was performed using data generated by the transfer matrix equations. In total, 100 data points from the FRD were used in the inverse resonance fitting. After 1,140 iterations, the inverse resonance method was able to determine the exact leak size and position of the leak ~CdAL = 0.0014 m2 and leak position xL = 1 , 400 m!. The relative error between the numerical data and the inverse resonance result is at the machine precision s1 3 10−12d. The advantage of the inverse resonance technique in comparison to the inverse transient method ~time domain! is that the model is not discretized in space, and a leak can be located at any point in the pipeline. The modeling of frequency-dependent effects, such as unsteady friction also has speed advantages when calculated in the frequency domain. The application of the inverse resonance technique requires the accurate modeling of the pipeline system to determine pipe integrity information. The following section presents an alternative method of locating a leak in the pipeline by analyzing the shape of the FRD alone. Fig. 5. Inverse resonance fitting for the frequency response extracted for a location of 800 m from the upstream boundary JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 / 599 Fig. 6. Impact of changing leak size and position on the frequency response diagram extracted at the inline valve at downstream boundary: ~a! leak at 700 m CdAL = 0.00014 m2; ~b! leak at 700 m CdAL = 0.00028 m2; ~c! leak at 1,400 m CdAL = 0.00014 m2; and ~d! no leak Resonance Peak-Sequencing Method for Leak Location Fig. 4 shows that a FRD extracted from numerically generated pressure variations at the excitation boundary of a pipeline system with no leaks yields equal magnitude and evenly spaced harmonic peaks. The impact of a single leak on this diagram is frequency dependent and results in an uneven damping of the harmonic peaks across the frequency axis. This frequency-dependent impact was also shown in Wang et al. ~2002! and forms the basis of the transient damping leak detection method. The leak-induced damping pattern on the FRD provides valuable information concerning the leak, and provides a leak location method that is based on a rank sequencing of resonant peaks. This method does not require the inverse resonance method’s matching of system responses, but allows the location of a leak within a region of the pipeline through the matching of rank sequences to entries in a lookup table. To understand the operation of this method, the impact of a leak on the FRD of the system is analyzed. Fig. 6 contains the FRD of the pipeline in Fig. 1 for varying leak parameters. The FRD is extracted at the downstream valve boundary. Four situations are presented: A nonleaking case, a leak at the 1,400 m point with a CdAL of 0.00014 m2, a leak at the 700 Fig. 7. Impact of leak size on the frequency response of the first three harmonic peaks ~measured at the downstream boundary at 2,000 m with a leak at 1,400 m from upstream boundary! 600 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 m point with a CdAL of 0.00028 m2, and a leak at the 700 m point with a CdAL 0f 0.00014 m2. Ranking the sizes of the resonant peaks in order of magnitude can summarize the shape of the frequency diagram. Fig. 6 shows that the order of the resonant peaks ranked in terms of the magnitude of the peaks has changed from @2,5,3,1,4# when the leak is positioned 1,400 m from the upper boundary, to @3,1,4,5,2# when the leak is shifted to 700 m from the upper boundary. The numbers in the square brackets indicate the “nth” peak in the FRD. An increase in leak size from 0.00014 m2 to 0.00028 m2 resulted in lower magnitudes in all resonant peaks, but the overall shape of the FRD, in particular the relative sizes of the peaks, remains unchanged. The impact of leak parameters on the resonant peak pattern can be seen in Figs. 7 and 8. These figures show the variation of the response magnitude of the first three resonant peaks ~vr = v / vth = 1.0,3.0,5.0!, measured at the end of the pipeline in Fig. 1, as a function of the leak size and leak location, respectively. Fig. 7 assumes that the leak is fixed at the 1,400 m position, while Fig. 8 is plotted with a constant leak CdAL of 0.00014 m2. The Fig. 8. Impact of leak position on the frequency response of the first three harmonic peaks for a fixed leak size of CdAL of 0.00014 m2 Table 2. Peak-Ranking Sequence and Corresponsing Leak Position Leak location range ~% of pipe length from upper boundary! Peak ranking * * * hvr1 . hvr3 . hvr5 ~Region * * * hvr1 . shvr3 = hvr5 d * * * hvr1 . hvr5 . hvr3 ~Region * * * shvr5 = hvr1d . hvr3 * * * hvr5 . hvr1 . hvr3 ~Region * * * hvr5 = hvr1 = hvr3 * * * hvr3 . hvr1 . hvr5 ~Region * * * hvr3 . shvr5 = hvr1d * * * hvr3 . hvr5 . hvr1 ~Region * * * shvr5 = hvr3d . hvr1 * * * hvr5 . hvr3 . hvr1 ~Region A! B! C! D! E! F! 0–25% 25% 25–33% 33% 33–50% 50% ~or 0 or 100%! 50–66% 66% 66–75% 75% 75–100% plot of resonant peak response versus leak size in Fig. 7 indicates that the magnitude of response at each harmonic component decreases as the size of the leak increases. While the rate of decrease in the response varies slightly between resonant peaks, the order of the peaks, ranked in terms of magnitude, remains unaffected by the leak size. In contrast, Fig. 8 shows that the relative sizes of the resonant peaks change significantly with the location of the leak. The resonant peak responses are seen to intersect at five leak positions along the pipe, dividing the system into six unequal sections. The intersection of any two resonant peak response curves in Fig. 8 indicates an exact leak location. Between each intersection point is a region where the resonant peaks, when ranked in order of magnitude, are arranged in a particular sequence. These sequences remain constant within each region and represent a corresponding shape in the FRD. The shapes of the FRD measured at the downstream valve for leaks located at two different positions in the pipe are inset into Fig. 8. When a leak is present in a particular region of the pipe, the shape of the frequency response diagram follows the sequence coding for that particular region. The shape of a measured FRD can, therefore, be used as a means of locating a leak within a single-pipeline system. A summary of these sequences for the first three harmonic peaks are tabulated in Table 2 and can be used as a means of locating a leak within a particular section of the pipe. The rank sequences for intersection points are also shown in Table 2 and correspond to exact leak locations in the pipeline. To illustrate how sequence coding can be used to locate a leak within a reservoir-pipe-valve system, assume a FRD measured at the valve has the sequence of fhvr3sPeak 2d . hvr5sPeak 3d . hvr1sPeak 1dg when the first three resonance peaks are ranked in order of magnitude. From Table 2, the ranking indicates that a leak is located in the range of 66–75% of the total pipe length from the upper boundary. Similar rank-order tables can be generated for a higher number of peak harmonics, which would result in a finer discretization of the pipeline, and higher location accuracy. For example, the use of the first six harmonic peaks will result in 32 divisions of the pipe. The generation of such coding can be performed automatically. The response of each harmonic component as a function of leak position can be approximated by a sinusoidal function, neglecting the small slope effect of the hydraulic grade line on the response function ~Schroeder 1967! S hvrn = cos s2n − 1dpx L D s11d where, hvrn5response of the nth harmonic peak component; x5leak location along the pipe; and L5total length of the pipe. The intersection of different harmonic response curves and hence the relative magnitudes of the peaks at each intersection point can be determined from Eq. ~11!. This leads to the formation of a code for each leak position or region in the pipeline. The resonance peak-sequencing method is best used in conjunction with the inverse resonance method to limit the extent of the search space used in the inverse calibration process. The combination of these two techniques provides a fast and efficient way of detecting leaks in a single pipeline. Obtaining the Frequency Response Diagram The application of the inverse resonance method and the resonance peak-sequencing method hinges upon an accurate evaluation of the FRD. This section illustrates the way in which the FRD can be generated quickly under physical constraints, and also how this technique can be extended to complex pipeline behavior, such as unsteady friction and pipes in network configurations. Frequency Response Extraction Process Chaudhry ~1987! proposed a frequency sweeping technique where the system is sequentially excited by a sinusoidal perturbation of the valve at different frequencies. The response for each input frequency is then plotted as a function of the input to output power ratio. The problem with this approach is the long time required to sweep through each frequency, as each frequency requires that the system settles down to steady oscillatory conditions. In addition, the excitation of a pipeline at the resonance frequencies has the potential to inflict substantial damage to the system ~Ogawa et al. 1994!. Devices capable of generating a smooth and controlled variation in pressure ~e.g., pumps, turbine, motorized valves! must also be installed in the pipeline for this extraction procedure. A more attractive alternative is to consider every injected transient in a pipeline as a combination of individual frequencies and apply Eq. ~1! for the derivation of the frequency response function. This approach allows the entire frequency response function to be efficiently extracted from a single transient test of any shape ~given sufficient input bandwidth!. This approach increases the flexibility in the extraction process. Extension to Multiple Pipe Networks The proposed extraction of the FRD for a pipeline section and the subsequent leak detection methodology relies on the pipeline lying between well-defined boundary conditions, with the excitation valve at one end, and a reservoir at the other. However, few systems exist within this narrow definition; they often contain multiple pipe sections or exist as complex networks. The extension of the techniques presented in this paper to such networks is, therefore, important from a practical standpoint. The inverse resonance technique can be directly applied in a network configuration. The model calculates the expected response from the network using the transfer matrix equations and an implicit solution technique presented in Ogawa et al. ~1994!. JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 / 601 Fig. 9. Illustration of network analysis using the proposed procedure The system integrity can be determined using Eq. ~10! and a network steady oscillatory transient model. The measured FRD would be obtained as described previously for the single-pipeline case. The peak-sequencing technique, however, will not lend itself to this type of network-wide analysis as the FRD of a network will not be uniform, and the location of the resonance peaks— along with the impact of a leak—is unclear and network specific. For this case, the network can be subdivided into pipe segments and the FRD determined for each individual pipe. This is made possible by the fact that water distribution networks are usually comprised of pipes that can be isolated from the main network by stop valves located at the end of each section for the ease of pipe maintenance. With this in mind, every complex pipeline system can be broken down into individual single pipes where the FRD of each pipe can be extracted. While this can be easily achieved by the closure of valves at both ends of the pipe, hence completely isolating the pipe from the network, a more attractive procedure is where only one valve is closed and both flow and pressure are maintained during the test. Consider the section of a pipeline network shown in Fig. 9. The network is a combination of individual single pipelines that can be partially separated from the main network through the use of a single stop valve at one boundary of the pipe section. The closure of that valve introduces a definable system boundary for the pipeline labeled “A,” and a perturbation injected adjacent to the closed valve generates the transient signal needed for the analysis. The transient can be generated by the controlled and measured perturbation of a side discharge valve located adjacent to the closed valve. The transient trace consists of the initial input pressure pulsation, followed by reflections from within the pipe up to a time of 2L / a, and finally reflections from the rest of the network. Note that any leak within this pipe section will create a reflected signal that will return to the measurement station within a time of 2L / a and the transient trace, up to this point in time, will contain all the integrity information for this reach of pipe. This initial length of data ~t = 0 to t = 2L / a! may be extrapolated to produce a synthetic signal that oscillates at the fundamental period of the pipe section by assuming that a reservoir exists at the far boundary as shown in Fig. 9. As the period of a pipe section for asymmetric boundary conditions is 4L / a, the original 2L / a section of data needs to be first extended to cover the complete period of oscillation. An approximate construction of the second half of the period for a reservoir boundary reflection is given by inverting the first 2L / a section of data and attaching it to the end of the original section to produce the full period. The 602 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JULY 2005 synthetic signal, consisting of a series of this constructed period, is then analyzed using the procedures set out in the previous section and Eq. ~1! to produce the FRD for this pipeline section. Impact of Unsteady Friction Unsteady friction is a frequency-dependent phenomenon that affects the behavior of transient signals, and it is a direct result of the losses associated with rapidly changing velocity profiles in unsteady flow. Such a frequency-dependent effect would be expected to induce changes in the shape of the frequency response diagram. Vítkovský et al. ~2003! have shown that unsteady friction induces a damping trend in the peak magnitudes of the frequency response diagram, causing greater attenuation at higher frequencies. This frequency-dependent damping creates a smooth trend in the peak magnitudes and can be easily distinguished from the more complex leak-induced pattern within the frequency response diagram. For example, while a leak causes the peak magnitudes to oscillate, the effect of unsteady friction is a gradual attenuation imposed onto the peaks. The impact of unsteady friction can be incorporated in both leak detection procedures to increase their applicability in real pipelines. For the inverse resonance method, unsteady friction can be included in the forward calculation of the FRD using the unsteady friction model derived in Vítkovský et al. ~2003!, which follows the frequency-domain representation of Zielke ~1968, 1969! and Vardy and Brown ~1995!. In the peak-sequencing technique, the removal of the unsteady frictional distortion of the peaks can be achieved by an array of scaling factors derived numerically for a leak-free case between steady friction and unsteady friction results. The scaling factors are ratios between the numerical steady and unsteady FRD results at the frequencies of interest for the intact system, SFsvd = hssvd hU+Ssvd s12d where SF5scaling factor; hS5response under steady friction; and hU+S5response under combined steady and unsteady friction. These SFs remove the trend imposed by unsteady friction on the peaks of the FRD allowing the ranking of the peaks to be carried out as described previously to determine the position of the leak within the pipe. Fig. 10 shows the FRD for the leaking pipeline of Fig. 1, with the properties of Table 1 affected by unsteady frictional damping. The SFs calculated from Eq. ~12! are used to modify this FRD and the result is compared with the FRD of the leaking pipe under Acknowledgments The material presented in this paper is made possible by the Australian Postgraduate Award, Adelaide Researcher Scholarship ~National!, and by funding from the Australian Research Council. Notation Fig. 10. Results of corrected unsteady friction frequency response diagram ~FRD! with steady friction FRD ~note that the scaled unsteady results and the steady results almost overlap perfectly! steady friction conditions. There is a good match between the scaled FRD and the FRD with no unsteady friction. This procedure can be used to allow the peak ranking technique to operate in systems affected by frequency-dependent damping. Impact of Noise and Background Fluctuations The presence of fluctuations in water demands, operating hydraulic elements, and other sources of externally induced noise in operating pipelines can contaminant transient measurements and impacts the accuracy of the leak detection method. Previous publications, such as Liou ~1998!, indicate that the extraction procedure for the FRD will result in the removal of noise of a different structure to that of the input signal. This technique is commonly known as match filtering and is widely used in radar applications. The increased noise tolerance as a result of this match filtering is illustrated in Li et al. ~1994!, Dallabetta ~1996!, and Liou ~1998!. Conclusion The presence of a leak within a single pipeline induces changes in the shape of the frequency response function ~FRD! and can be used as a means of identifying the position of the leak within the system. Two methods of leak detection in a pipeline system have been presented in this paper including the inverse resonance method and resonance peak-sequencing method. The former draws upon existing inverse transient techniques and carries out a parameter estimation process by fitting the FRD from a numerical model to a measured FRD. The resonance peak-sequencing method involves a comparison between the shape of the FRD and known shapes generated by leaks at various positions in the pipe. Summarizing the shape of the FRD in a sequence of harmonic peaks, ranked in order of magnitude, can provide the means through which this comparison is carried out. Both methods provide accurate information concerning the leak. The resonance peak-sequencing method is a fast and efficient method of locating a leak within a region of the pipeline and should be used in conjunction with the inverse resonance method to limit the size of the search space. The use of a controlled transient injection and the system identification equation can lead to the fast estimation of the frequency response curve, making the use of frequency response techniques in leak detection in field situations a distinct possibility. The application of these methods to complex pipe systems and to frequency-dependent friction within the system is also illustrated within the paper. The following symbols are used in this paper: A 5 area of pipeline; AL 5 area of leak orifice; AV 5 area of valve orifice; a 5 wave speed; Cd 5 coefficient of discharge for leak orifice; CV 5 coefficient of discharge for valve orifice; D 5 diameter of pipeline; E 5 objective function; f 5 Darcy–Weisbach friction factor; g 5 gravitational acceleration; H 5 hydraulic grade line or transfer function; H0 5 steady-state hydraulic grade line, center of perturbation; HL 5 head at the leak orifice; HL0 5 steady-state head at the leak; h 5 complex hydraulic grade line perturbation; i 5 imaginary unit, Î−1; L 5 total length of pipeline; L1 , L25 lengths of pipe subdivided by the leak; n 5 peak number; Q 5 discharge; Q0 5 steady-state discharge, center of perturbation; QL 5 discharge out of the leak orifice; QL0 5 steady-state flow out of the leak; QV0 5 steady-state flow through the valve; q 5 complex discharge perturbation; R 5 frictional resistance term or correlation function; t 5 time; x 5 distance along pipe; Z 5 characteristic impedance; z 5 elevation of the pipe at the leak; DHV0 5 steady-state head loss across the valve; Dt 5 magnitude of the dimensionless valve aperture perturbation; m 5 propagation constant; n 5 kinematic viscosity; t 5 dimensionless valve aperture size; t0 5 mean dimensionless valve aperture size, center of perturbation; v 5 frequency; vr 5 harmonic number5v / vth; and vth 5 fundamental frequency of system. 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