Transient damping method for narrowing down leak location in pressurized pipelines

: Numerous leak detection methods have been developed for pipeline systems because of the shortage of water resources, increased water demand, and leak accidents. These methods have their advantages and disadvantages in terms of cost, labor, and accuracy; therefore, it is important to narrow down the location of a leak as easily, rapidly, and accurately as possible. This study applies the technologies based on the execution of a transient event (transient test-based technologies (TTBTs)), and a model is presented for representing the relation between the leak location and the damping of the pressure transient due to the leakage. The model is verified with laboratory experiments in which the leak location can be narrowed down to be less than 10% to 30% of the total pipe length. The model is found to be more effective if the leak location is nearer to the upstream end. In addition, the leak location found by the damping model varies with an approximate absolute error of 2% to 5% of the pipe length. It is suggested that the damping model is suitable for narrowing down and not for finding the leak location, and should be used in combination with other leak detection methods.


INTRODUCTION
Water leakage in pipelines occurs in all water distribution systems due to age, corrosion, or a third party (Zhang et al., 2015), and causes considerable economic loss, such as that related to shortages of drinking and irrigation water. The amount of water leakage in water distribution systems varies widely between different countries, regions, and systems (Puust et al., 2010). Especially in Asia, home of 53% of the world's urban population, the estimated annual volume of leakage in urban water utilities is approximately 29 billion cubic meters; hence, water utilities are losing nearly 9 billion US dollars per year (Asian Development Bank, 2010). Leakage is not just an economic issue, it also has environmental, health, and safety implications (Puust et al., 2010). For example, leakage in pipelines causes ground Correspondence to: Masaomi Kimura, Graduate School of Agricultural and Life Sciences, The University of Tokyo, 1-1-1 Yayoi,  subsidence, contamination, and sinkholes, which results in damage to the infrastructure (Ali and Choi, 2019). Moreover, leakage possibly influences water quality by introducing contamination into water distribution networks through leaks in low-pressure conditions (Colombo and Karney, 2002). Hence, detecting the existence and exact locations of leaks as quickly and accurately as possible is of utmost importance.
Although various leak detection methods have been developed (such as ground-penetrating radar, acoustic leakdetection, and infrared spectroscopy), no single method has been able to satisfactorily meet the operational needs from the perspectives of cost and labor. Hence, a simple, cheap, and reliable method for leak detection would be of great economic value (Pudar and Liggett, 1992). Leak detection by measurement of the pressure in a pipe can be employed in the daily maintenance of pipelines, because manometers are less expensive than flowmeters and can be easily installed at air valves on pipelines. However, a pressure change caused by leakage is too small to be detected, even in a steady flow. Additionally, in cases of low water pressure, it is difficult to detect leaks by capturing the force caused by the pressure change (Chatzigeorgiou et al., 2015).
As a recent solution to the aforementioned problems, transient test-based technologies (TTBTs), in which a transient hydraulic event (such as that caused by a water hammer) is used for leak detection, are attracting interest. TTBTs are expected to offer leak detection methods at a lower cost and with less labor compared to other methods (Meniconi et al., 2011). In a transient hydraulic event, a pressure wave sufficiently strong to be detected can be generated, and only a pressure measurement taking a few minutes (or even less) is required for detecting leaks. After a transient hydraulic event occurs in a pipe, a pressure wave travels repeatedly between both ends of the pipe stretch. The movement of this pressure wave is observed as cyclical pressure transients at an arbitrary point in the pipe stretch. Leakage in a pipeline system will result in an increased damping rate and the creation of new leak-reflected signals within the pressure transients. Most TTBTs have been developed and applied to water pipeline systems using information contained within these two effects (Duan et al., 2010). Brunone (1999) and Brunone and Ferrante (2001) investigated the effect of leak-reflected signals in the pres-sure transients, and demonstrated how leaks can be detected by leak-reflected signals in both laboratory and field experimental pipes. While the method adopted was simple and easy, it could not be applied in the case of a slight leakage and if noise due to the pipe structure is present (Asada et al., 2019). Inverse transient analysis is a more powerful TTBT method, using both leak-induced damping and leak-reflected signals (Covas and Ramos, 2010;Shamloo and Haghighi, 2010;Vítkovský et al., 2007) , and it is theoretically applicable to pipes of any structure or characteristic. However, the computational complexity is generally enormous, and it is important to reduce it, for example, by narrowing down leak location with other methods in advance. Leak-induced damping is thought to be minimally affected by the noise compared to leak-reflected signals, thus it is effective for the leak detection in pipeline systems with a complicated structure (Asada et al., 2019). In addition, it was revealed that the damping rate of the pressure transients is faster because of an increased energy dissipation from the leak, because the leak location is nearer to the downstream end in the case of rapid and complete closing of the valve (Asada et al., 2019). In this study, the leak-induced damping is theoretically modelled for leak location by considering energy dissipation from the leak and friction in a pipeline. The effectiveness of the damping model for narrowing down leak location is demonstrated based on the experimental results.

Damping model
The damping of pressure transients in the pipeline with leakage is represented by means of an exponential law, according to the results obtained by Ramos et al., (2004), assuming the damping by friction loss and leakage are mutually independent (Wang et al., 2002), as follows.
where ΔH is the change in the piezometric head generated by the water hammer (m), ΔH 0 is the initial change in the piezometric head (m), t is the time (s), t* is the time nondimensionalized by the wave propagation period T (t* = t/T), R is the friction-induced damping coefficient, and R L is the leak-induced damping coefficient. To calculate the value of R, a numerical simulation needs to be used because the damping by the unsteady state friction has to be considered in the complicated model proposed by two conceptually different approaches. In the first approach, the so-called weighting function-based, the unsteady state friction is given by a weighted integral of past fluid accelerations (Trikha, 1975;Vardy and Brown, 1995;Zielke, 1968.) In the second approach, the so-called instantaneous acceleration-based, it is assumed as a function of the instantaneous local and convective accelerations (Brunone et al., 1991;. In this study, the value of R is measured by the experimental pressure transient in case of no leakage without resorting to a numerical simulation. Additionally, Meniconi et al. (2014) reported that there is a biunique correspondence between the damping of the pressure transient at any pipe section and the energy dissipation of the entire pipeline system. Therefore, the value of R L related to leak location is derived from the energy dissipated from the leak.
In a single pipeline with total length L (m) and pipe cross-sectional area A (m 2 ), water is supposed to flow at a flow rate Q (m 3 s -1 ) from an upstream reservoir to a downstream end valve, in which the continuity equation is established as follows: where Q up is the flow rate upstream from the leak, Q down is the flow rate downstream from the leak, and Q leak is the rate of leakage volume in a steady flow. A leak is assumed to exist at a point x L *L from the upstream reservoir (0 ≤ x L * ≤ 1), where x L * is the distance to the leak nondimensionalized by the total length L. A wave with the value of ΔH generated by rapidly closing the valve propagates through the pipeline at wave speed c (m/s), which decreases because of the leakage during the transient event. The period is the time taken to propagate for two round trips through the pipe in the case of a reservoir-pipeline-valve system (T = 4L/c). The interpretation of transient conditions is simplified in almost the same manner as the models of spring oscillations by measuring displacements with respect to the spring's equilibrium position (Karney, 1990). Thus, the energy dissipation under transient conditions can be considered using the change ΔH with respect to the basis of the steady-flow condition. For the case in which ΔH is zero at the leak, the energy in the pipeline is preserved because the change in kinetic and elastic energies is balanced, as in the case of no leakage. Therefore, the change in the energy from the leak in the pipeline is derived from the case in which the change in the piezometric head at the leak is ΔH, as shown in Figure S1.
where ΔE is the change in energy per unit time (N m s -1 ), ρ is the fluid density (kg m -3 ); g is the gravitational acceleration (m s -2 ); and ΔQ leak is the change in Q leak when the wave with the value of ΔH exists at the leak. The rate of the leakage volume is a function of the pressure head at the leak and the size of the leak, and Q leak + ΔQ leak is expressed by the orifice equation as follows: where a is the product of the discharge coefficient and the cross-sectional area of the leak hole (m 2 ), H L is the steady state piezometric head at the leak (m) and z L is the elevation at the leak (m). Substituting Equation (4) to Equation (3), (5), it can be simplified as follows: where h L is the steady state pressure head at the leak (m). The value of ΔH at the leak is influenced by the line packing via the influence of friction, by which ΔH continues to rise to the maximum head after the wave passes until it is reflected back (Duan et al., 2012;Liou, 2016), while ΔH reaches the full Joukowsky head immediately in case of no friction. The full Joukowsky head is the change in the piezometric head converted from the flow rate by closing the valve, and can be expressed as follows (Joukowsky, 1904): where ΔH J is the full Joukowsky head (m). Liou (2016) presented an analytical formula for ΔH in a downstream valve during a half period (0 ≤ t* ≤ 0.5). However, this underestimates an actual result for ΔH because it neglects the line packing by the unsteady state friction, which cannot be derived analytically.
Thus, considering the line packing by the steady and unsteady state friction and smooth variation of ΔH, a formula is presented here for ΔH at the downstream valve, by simply assuming power variations, as follows: where α is the rate at which the initial head change increases to the maximum in the downstream valve (0 ≤ α ≤ 1), and β is the ratio of the maximum head change in the downstream end to the ΔH J value (β ≥ 1). The variation in ΔH at the leak during a period (0 ≤ t* ≤ 1) is formulated based on Equation (8). In a transient event, the wave with ΔH reflects in antiphase at the upstream reservoir, and it reflects in phase at the downstream valve. For the case of x L * ≥ 0.5, the reflective wave from the downstream valve reaches the leak when ΔH at the leak is varying to zero; whereas, for the case of x L * < 0.5 it reaches the leak after ΔH at the leak has varied to zero as shown in Figure S2. Thus, ΔH (0 ≤ t* ≤ 1) can be classified into two types, according to whether x L * ≥ 0.5 or x L * < 0.5. For x L * < 0.5, ΔH is formulated for the duration of a half period (0 ≤ t* ≤ 0.5) as follows: For x L * ≥ 0.5, ΔH is formulated for the duration of a half period (0 ≤ t* ≤ 0.5) as follows: The value of ΔH (0.5 ≤ t* ≤ 1) is equal in absolute value and opposite in sign to the value (0 ≤ t* ≤ 0.5). Figure 1 shows the variation in ΔH at the leak during a period, calculated using Equations (9) and (10), for the case of ΔH J = 5 m, α = 0.1, and β = 1.2 and using x L * = 0.2 for x L * < 0.5 and x L * = 0.8 for x L * ≥ 0.5. The total energy E in a pipe, on the left side of Equation (6), is expressed as the elastic energy form (Karney, 1990;Meniconi et al., 2014) and can be derived from the work of water and pipe by ΔH as shown in the shaded area in Figure 2.
where E p , E w are the Young's moduli of the pipe material and water (N m -2 ), respectively, D is the pipe diameter (m), b is the pipe wall thickness (m), and ε p , ε w are the strain rate of the pipe and water, respectively. Substituting Equation (11) to Equation (6), As can be seen clearly from Figure 1, the first and third terms in Equation (12) become zero and the second term only needs to be integrated for t* ranging from 0 to 0.5 and then doubled.
By solving Equation (13) for ΔH J using Equations (9) and (10), the formula for the damping of ΔH due to the leakage is expressed as follows:

Collection of experimental data
An experimental test was conducted in a pipeline to collect pressure transient datasets with simulated leakage to evaluate the effectiveness of the damping model at the Institute for Rural Engineering, Tsukuba, Japan. The pipeline has a spiral structure with bends at 25 m intervals, is composed of stainless steel, and has a 24.2 mm inner diameter, a thickness of 1.5 mm, and a length of 900 m (Figure 3). The pipeline includes a pump and a pressurized tank (maximum pressure 3.0 kg cm -2 ) at the upstream end, and manual and ball valves at the downstream end. The manual valve is used to control the velocity of flow downstream from the leak, and the ball valve is used to generate a transient event via rapid and complete valve closure. A manometer (with a gauge pressure range of 0 MPa to 0.1 MPa and an accuracy of 0.5% of full range) is set just upstream from the ball valve to collect the pressure transient data. The experimental test cases are shown in the left side of Table I. The pressure transient data measured in each case is represented as the time variation of ΔH in Figure S3. The wave speed in the pipeline was calculated by the time for a period of the measured pressure transient. Simulated leaks were established at three points: 150 m (upstream leakage (UL)), 450 m (middle leakage (ML)), and 750 m (downstream leakage (DL)) from the upstream end, corresponding to x L * values of 0.167, 0.500, and 0.833, respectively. The study assumes that the leak detection method uses the knowledge of the relative leak size to the pipe cross-sectional area a/A in advance using a water leakage test. Thus, the relative leak size was derived from the rate of leakage volume measured under two different hydrostatic conditions using the following equations, which are based on the orifice equation: where H 1 and H 2 are a pair of static piezometric heads, and  a z L is set as zero in this study b ε L = (x LO * -x L *) × 100 Q leak1 and Q leak2 are the rates of leakage volume measured for the cases of static piezometric heads H 1 and H 2 , respectively.

Method of narrowing down leak location by damping model
The procedure for calculating R L in the experimental cases is as follows: (1) The change in the piezometric head ΔH at the downstream end of the pipe is measured with and without the leakage for cases 1 to 6; (2) The total damping coefficient R + R L and the frictioninduced damping coefficient R for cases 1 to 6 are derived from the exponential variation in the two values, which are calculated by averaging each of the absolute values of ΔH for t* ranging from 0 to 1 and for t* ranging from 1 to 2, with and without the leakage; The values of the leak-induced damping coefficient R L for cases 1 to 6 can be found by subtracting R from R + R L . The value of R L in Equation (14) is calculated by changing α ranging from 0 to 1 by increments of 0.01, β ranging from 1 to 2 by increments of 0.01, and x L * ranging from 0 to 1 by increments of 0.001. The absolute error of R L from Equation (14) and the experimental pressure transient is calculated for cases 1 to 6, and the leak locations are searched in order that the absolute error of R L is almost negligible. The objective of the present study is to narrow down leak location using the damping model. However, the leak location cannot be completely narrowed down by using only the information of the leak-induced coefficient R L, as shown in Figure S4. Such a behavior was highlighted by Meniconi et al., (2014) in different pressure transients by numerical experiments. In fact, a given damping of pressure transients is not exclusive of a unique pressure transient, and provides multiple couples of solutions (i.e. the values of α and β) if no other information is available. Therefore, the time variation in ΔH is used to further nar-row leak location. The value of ΔH (0 ≤ t* ≤ 0.5) in Equation (8) is calculated by changing α within a range of 0 to 1 by increments of 0.01, and β within a range of 1 to 2 by increments of 0.01. The root mean squared error (RMSE) of ΔH (0 ≤ t* ≤ 0.5) from Equation (8) and the experimental pressure transient is calculated for cases 1 to 6. Therefore, leak location searches return almost negligible absolute error of R L and RMSE of ΔH.

RESULTS AND DISCUSSION
First, the value of ΔH in Equation (8) is fitted to the measurement value of ΔH (0 ≤ t* ≤ 0.5) for cases 1 to 6 to investigate the accuracy and the efficiency of this equation. In all cases, Equation (8) reproduces ΔH (0 ≤ t* ≤ 0.5) with the minimum RMSE of approximately 0.2 m ( Figure S5). The reason that the RMSE is considered is that the measurement value of ΔH includes high frequency noise due to the spiral structure of the pipeline, and the model in Equation (8) neglects the variation of ΔH during the relative short time until the complete closure of the valve (t* = 0-0.03). In this study, the RMSE of ΔH is set as ε h < maxε h = 0.5 m to narrow down the leak location for cases 1 to 6, where ε h is the RMSE of ΔH (m) and maxε h is the maximum value of ε h . The leak-induced coefficient R L largely influences ΔH after a period from the initial state. The absolute error of R L is set as ε d < maxε d = 5.0 × 10 -5 so that the error of the ΔH is negligible using Equation (14), where ε d is the absolute error of R L and maxε d is the maximum value of ε d The right side of Table Ⅰ presents the results of the narrowing down of leak location for cases 1 to 6. Figure 4 presents error plots for the dimensionless leak location x L * under the conditions and shows they are dense around the true leak location x L * for cases 1 to 6. The vertical axis is the dimensionless hybrid error ε* (0 ≤ ε* ≤ 1) of the RMSE of ΔH and the absolute error of R L . The dimensionless error ε* is represented so that the value of ε h and ε d can be evalu- ated as a combined single function, as in the weighted sum method treating multi-objective function (Marler and Arora, 2004), as follows: As presented in Table Ⅰ and Figure 4, narrowing down the leak location can be successfully performed by the damping model in all cases, because the true leak location x L * exists within the narrowed down leak locations x LN *. The narrowing down rate of leak location for the total pipe length is of 30% level in DL (case 1, case 4), 20% level in ML (case 2, case 5), and less than 10% in UL (case 3, case 6); thus, it is larger as the true leak location x L * is nearer to the downstream end. This is because the variation range of the leak location for a value of R L is larger as the leak location is nearer to the downstream end, when the parameters α, β are varied, as shown in Figure 5. In addition, the leak location x LO * is optimized when the dimensionless error ε* is minimum for cases 1 to 6. The optimized results in Table Ⅰ show that the error ε L of the leak location for the total pipe length varies largely from approximately -5% to 2% for cases 1 to 6. The accuracy of the damping model is the same (or lower) compared to that of the previous leak detection methods using damping (Asada et al., 2019;Wang et al., 2002). Therefore, optimizing the leak location using only the damping model results in an inaccurate estimation.
It is important to narrow down the leak location by the damping model with suitable conditions, and find the leak location using a combination of other methods, such as inverse analysis. Compared with DL and ML cases, in the UL case, the true leak location x L * is farther from the optimized leak location x LO * of the minimum dimensionless error ε* in Table Ⅰ and Figure 4. These differences result from neglecting the variation of ΔH during the valve closure, which gives the larger errors of α and β to small energy dissipation from the leak in case of UL. Therefore, by improving the damping model considering the variation of ΔH by closing the valve, it is possible to narrow down the leak location further under severe conditions. Additionally, the pressure transient is more affected and inhibited by viscosity diffusion in the pipe as the pipe length L is larger and the wave speed c and the pipe diameter D are smaller (Duan et al., 2012;Wahba, 2008); thus, the damping model cannot be used in this case. In further studies, the validation of the damping model needs to be investigated for different types of pipes and multiple leaks, so that the damping model can be widely applied to field pipes.

CONCLUSION
This paper presented a method for narrowing down leak location using the damping model of pressure transients in a pressurized pipe. The leak location was narrowed down from approximately 30% to less than 10%, and it was revealed that the effectiveness of the damping model increases as the leak location is closer to the upstream end. In this method, we assumed that 1/16(ΔH/h L ) 3 << 1, which can be realized by restricting the valve in advance and suppressing the flow rate in field pipes. Under this condition, the application of the proposed method to field pipes can have considerable benefits, because the operation of rapidly closing the valve will be relatively easy given that the valve opening and the load on the pipe due to the pressure change are small. Therefore, the proposed damping model has a possibility of narrowing down the leak location simply and rapidly in field pipes by investigating the effectiveness of the model in different types of pipes in a detail.

ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI Grant Number JP19J10410. Figure S1. Head and flow rate profiles for the case in which the change in the piezometric head at the leak is ΔH Figure S2. Diagram of the wave propagation through the pipeline and the pressure transient at the leak for the cases of (a) x L * ≥ 0.5 and (b) x L * < 0.5 Figure S3. Time variation of the change in piezometric head in (a) case 1, (b) case 2, (c) case 3, (d) case 4, (e) case 5, and (f) case 6 Figure S4. Narrowing down results of the leak location using only the information of R L in the condition: ε d < 5.0 × 10 -5 , where ε d is the absolute error of R L between the calculated and measured value in (a) case 1, (b) case 2, (c) case 3, (d) case 4, (e) case 5, and (f) case 6 Figure S5. The fitting curve of ΔH calculated by Equation (8) in (a) case 1, (b) case 2, (c) case 3, (d) case 4, (e) case 5, and (f) case 6