Separation of pressure signals caused by waves traveling in opposite directions

Water distribution system infrastructure inherently deteriorates with age due to corrosion, weathering, malfeasance, or the build-up of sediments and microfilms inside the pipe (Colombo et al. 2009; El-Zahab & Zayed 2019; Duan et al. 2020). This is an important problem as extreme deterioration can have significant implications for the effective operation of such pipeline systems (e.g. reliability of water supply), the functioning of metropolitan environments (e.g. road or building closure due to pipe bursts) and society's health and well-being (i.e. the intrusion of pathogens into the networks through cracks and leaks) (Karim et al. 2003; Snider & McBean 2020). As such, it is important for water utilities to monitor the condition of their pipeline infrastructure and proactively maintain the health and integrity of their pipeline infrastructure.

A range of invasive and non-invasive methods for anomaly detection in pipelines exist (El-Zahab & Zayed 2019), but a promising method for the assessment of pipeline infrastructure condition is through the use of hydraulic transient analysis (Colombo et al. 2009; Duan et al. 2020). Hydraulic transient analysis couples the measurement of hydraulic signals (typically pressure) with either model-based (e.g. Zecchin et al. 2013; Gong et al. 2014; Capponi et al. 2017; Zhang et al. 2019) or statistical (e.g. Motazedi & S. 2018) signal processing approaches, with the aim of identifying condition faults within a pipeline (e.g. leakages, bursts, blockages, or corroded sections). Hydraulic transients are pressure and flow perturbations that propagate throughout a pipeline system at near the speed of sound and result either from rapid changes in boundary conditions (i.e. a valve maneuver) or can be induced in the system (e.g. through a pressure wave generator) for active hydraulic transient analysis methods (Colombo et al. 2009). Similar to radar, or sonar applications, hydraulic transients are useful as a diagnostic tool as they are very sensitive to the system conditions, resulting from the fluid structure interactions. For example, a pipeline burst will induce a negative pressure wave that can be measured at a great distance from the burst location. Similarly, a pressure wave induced into a pipeline system for the purposes of interrogation will reflect at leaks, blockages, or even corroded sections, enabling the reflected wave to be measured and analyzed (Covas & Ramos 1999).

One of the problems with hydraulic transient analysis has been that it is not always possible to directionally attribute a measured pressure perturbation to its direction of origin within the pipeline. That is, as the pressure response at a point is the superposition of pressure waves traveling from both the upstream and downstream directions, a single measurement of a perturbation will not enable the analyst to determine whether it came from an upstream or downstream location. This directional attribution is important, as it can greatly assist the analyst in locating the perturbation source unless simple systems with pressure measurements at pipe ends are considered.

To address this problem, attention has recently shifted to the use of signal-splitting techniques, which involve decomposing a measured pressure perturbation into its upstream and downstream traveling waves (Zecchin et al. 2014). That is, the signal-splitting methods are able to reconstruct the independent wave forms that are traveling from both the upstream and downstream directions. This is achieved through the use of two closely spaced sensors, and a signal separation algorithm, which is based on the physics of the system, that takes the two pressure measurements and determines the associated positive and negative traveling waves. This technique was first applied to a frictionless numerical system in Gong et al. (2012), where reflections from pipe wall corrosion points were effectively reconstructed. This was then extended to a system including friction by Zecchin et al. (2014) where it was shown to effectively determine the form of a reflected wave in a noisy pipe-reservoir-valve system. The method was then applied to a laboratory system in Shi et al. (2017), and in combination with an optical fiber pressure sensor in Shi et al. (2019), where wave reflections within a real system were effectively reconstructed.

An acknowledged potential issue with the signal-splitting approach is that, under some circumstances, the algorithm can become unstable and yield results with significant numerical artifacts. This is potentially problematic, as it reduces the reliability, and hence utility, of the proposed method. The focus of this paper is to study the signal-splitting approach and find solutions to mitigate the introduction of numerical artifacts into the separated wave forms. The approaches explored in this paper are alternative frequency- and time-domain formulations of the signal-splitting algorithm; the use of a differenced signal as opposed to the original signals; the use of low-pass filters, to pre-filter the signals; the effects of noise, distance between measurement sections and pipe material on the signal-splitting algorithm performance; and the wavelet transform analysis directly in the separated signals. The proposed approaches are tested on a range of numerical and laboratory systems, where it is found that the most successful approach is a time-domain algorithm coupled with pre-filtering, even when the assumptions about the pipe material rheology are not correct.

The delay operator, ⁠, operates the shift back in time of a function by the quantity ⁠.

It is worth noticing that in the evaluation of G and in Equation (6), the assumption of a frictionless elastic system is not needed.

Two kinds of separation procedures can be used in the time domain, based on the hypothesis of frictionless (pure delay), or linearized friction, elastic pipe systems, without the need for the direct and inverse transform of the signals required by Equations (6) and (7).

For the pure delay case, the algebraic manipulation of the frequency-domain separation formulae leads to different recursive time-domain solutions.

The number of terms used in the expansion strongly affects the accuracy of the signal separation. Compared to the previously derived time-domain recursive formulae, Equation (13) only depends on the acquired pressure signal values at previous steps while Equations (9) and (11) require the use of both acquired and separated signal values at previous steps.

Once and are evaluated by Equation (14), Equation (5) provides the values of and ⁠.

Different procedures based on the derived formulae for the separation of the positive and negative direction components of pressure signals are used and compared in the following.

The separation procedure in the frequency domain based on Equations (6) and (7) has been applied already (e.g. Shi et al. 2017). If either pressure signals are numerically simulated or acquired through measurement, and are data vectors sampled on a regular time grid and their transformation in the frequency domain is obtained by the Fast Fourier Transform.

Before the evaluation of the separated components by Equation (6) can be undertaken, signals must be filtered to avoid spurious oscillations as much as possible in the results. The filtering is obtained by the Hadamard multiplication of the transformed signals with a filter vector. As an example, for a sigmoidal filter, the terms of the filter vector are evaluated at the same frequencies of the transformed signals by Equation (16) as shown in the following sub-section.

Once the data are filtered, the separated signals can be evaluated in the frequency domain by Equation (6) and then transformed back into the time domain by Equation (7) using the inverse fast Fourier transform.

Five different procedures can be used in the time domain.

The first two procedures are based on the pure delay hypothesis and Equations (9) and (11). In both cases, the vectors containing the separated signals in time are obtained by the linear combination of the measured or simulated signals at the previous time step. Since the separated signals at each time step depend on the same separated signals evaluated at previous time steps, the calculation must be recursive, cannot be parallelized, and may be numerically cumbersome.

The difference between the use of Equations (9) and (11) is only in the terms that are evaluated first, i.e. and or and ⁠. In both cases, Equations (5) allows the evaluation of the remaining terms.

Two further procedures are based on Equations (14) and (15), which are similar to those derived for the pure delay case but introducing the linearized friction term ⁠. The same considerations concerning the numerical efficiency and the equivalence of the results obtained by Equations (9) and (10), apply to these procedures also.

A fifth time-domain procedure is based on the pure delay hypothesis and Equation (12). In this case, the accuracy of the results strongly depends on the approximation introduced by the expansion order, which in turn corresponds to the number of the previous time steps needed for the evaluation of the separated signals. The increase in the number of the involved time steps also increases the computational burden.

As noted out in the literature (e.g. Shi et al. 2017) and confirmed by the results shown in the following sections, when Equations (6) and (7) are used, signals must be filtered to avoid spurious oscillations in the separated components.

When the signals are filtered, the products and are used instead of and ⁠, respectively. Equation (6) show that the filter could also be applied not only to and ⁠, as in the following, but to their component ⁠, ⁠, ⁠, and before the inverse transform.

The same applies to the other formulae and hence to obtain a voice of the wavelet transform of pressure waves propagating in positive and negative directions, the same Equations (6) and (7) can be used, provided that the corresponding voices of the wavelet transform of pressure acquired at the measurement sections are used.

With respect to the frequency-domain procedures, different values of must be introduced in Equations (20) and (21), corresponding to the application of several filters at different time scales. The information content of the different voices, even limited to a few components (Ferrante et al. 2022), allows an easier discernment of pressure waves traveling in different directions in noisy signals.

To assess the applicability of the wave separation technique, the time- and frequency-domain procedures are tested on numerical and laboratory signals. The distance between the measurement sections, l, is varied in the tests from about 1 to 50 m. The effects of other parameters, such as the acquisition frequency, the noise disturbance and those related to the rheology of the pipe material, are also considered.

As a first case, the separation procedure was tested considering the pressure signals at two nodes of a numerical model at a distance in a smooth pipe with m, and an initial steady-state flow of l/s. The signals were computed at a frequency of Hz using a model based on the method of characteristics with a steady-state piezometric head operating point of about 20 m (note that the signals presented in the following sections show only the transient perturbations about this operating point).

The time-domain recursive procedure based on Equation (12) does not yield comparable results unless a large number of terms in the expansion are used (e.g. more than 500), with a resulting large increase in computational burden with respect to the other two procedures. Based on these findings, only results of the application of Equation (9) for the pure delay and (14) for the linearized friction are shown in the following, as they are representative of the time-domain procedures.

As to be demonstrated, the use of filters reduces the amplitudes of the oscillations shown in Figure 2 and improves the separation of the numerical signals in the frequency domain by Equations (6) and (7). By way of an example, Figure 8 shows the results of the same procedure used for Figure 2 when a sigmoidal filter with ⁠, ⁠, and is applied to the signals. These values are a trade-off between the need of avoiding spurious oscillations and the need of preserving the shape of the signals and were obtained by a preliminary parametric study. In fact, the filters modify the pressure wavefront shape reducing the slopes and introducing a bias in the separated signals between the arrival times of the pressure waves.

The effect of the signal sampling frequency, ⁠, was analyzed in the range between 2 and 2 Hz. The oscillation amplitude of the downsampled unfiltered signals was comparable and the filter effects were the same provided that the pass-band and stop-band edge frequencies were both scaled by ⁠.

The effect of the noise was analyzed by adding a white noise to the signals and of Figure 2 with a standard deviation of of the maximum of the simulated values, which is assumed to be representative of the total error band of a pressure transducer. The noise affects the results when both frequency- and time-domain separation procedures are used (Figures 9(a) and 9(b)).

The results of the time-domain formulations are less affected by noise on shorter time scales, but due to the recursive procedure, the standard deviation of the separated signals increases in time.

The weight term in Equations (9) and (11) can be used to reduce both the drift and the noise effect increase on the long-duration behavior for the time-domain separation procedures. As an example, Figure 10 shows the results for the same case of Figure 9(b) with ⁠. Even such a small variation with respect to unity eliminates the drift and the increase of the noise in time.

The use of the linearized friction can be considered similar to the use of the weights and applied to the terms in the recursive formulation. For this reason, it is not surprising that the separation procedure with corresponds to a drift and noise effect reduction comparable to that of Figure 10.

As expected, the noise, being singular everywhere, has a great impact on the numerical differences of the signals.

As a further step in the investigation, pressure signals at the two measurement sections are simulated at a distance of apart using the same numerical model used in Figure 9 with added noise as specified in the previous sub-section. The separation procedures in the frequency domain show that noise and distance between sections induce artifacts in the separated signals that can hide the information content (Figure 12(a)), while the time-domain separation procedures are very effective in the separation of the simulated noisy signals (Figure 12(b)).

Interestingly, the use of a sigmoidal filter (results not shown) in this case neither reduces the noise effect nor improves the results.

The separation procedure in the frequency domain does not require any particular assumptions concerning the pipe material, provided that its rheology can be modeled by Equation (4). By way of contrast, the separation procedures in the time domain are based on the pure delay or linearized friction assumptions, which apply only to an elastic pipe material. For this reason, the separation procedures in the frequency domain should perform better than their counterparts in the time domain when a viscoelastic pipe is considered.

To explore this issue, a viscoelastic pipe system (i.e. ⁠) was simulated using the method of characteristics with a sensor spacing of about 50 m. The pipe material rheology is fully described by the series composition of a spring with a Kelvin–Voigt element, which is made by a damper with viscosity in parallel with spring with Young modulus E. In this case, where ⁠, is the pipe constraint coefficient, is the water density, and e is the pipe wall thickness. For the simulations shown ⁠, Pas, and Pa. A white noise was added to the signals as in the previous cases, and the sigmoidal filter was used.

In the following, the separation procedures are tested on experimental signals acquired during transient tests.

Two polymeric pipes were used. The oriented polyvinyl chloride (PVC-O) DN110 PN16 upstream pipe, with the total length 99.18 m, an internal diameter of 103.0 mm, and a wall thickness 2.7 mm, was joined to the high-density polyethylene (HDPE) DN110 PN10 pipe, with a total length 92.79 m, 96.8 mm, and 6.6 mm.

An electromagnetic flow meter (FM) was used to measure the discharge during the initial steady-state conditions, with an accuracy of 0.2% of the measured value. Four piezoresistive pressure transducers, with a full scale (f.s.) of 6 bar and an absolute and accuracy of 0.25% f.s., were connected to the HDPE pipe at P_E1 and P_E2, and to the PVC-O pipe at P_O1 and P_O2. The acquisition frequency was 1 Hz for the FM and 2,048 Hz for the pressure transducers.

The separation procedures are tested on the pressure signals measured at two pairs of measurement sections, i.e. P_O1 and P_O2 with a spacing distance of 50.02 m along the PVC-O pipe and P_E1 and P_E2 with a spacing of 48.90 m along the HDPE pipe. The viscoelastic parameters used to evaluate were calibrated in Ferrante & Capponi (2017).

Figure 15(a) shows both the signals acquired at the pressure transducers P_O1 and P_O2 and the separated signals obtained by the frequency-domain procedure. The results seem to be even better than those obtained by the same separation procedure applied to numerical signals, with reduced oscillations and influence of noise. This is most probably to the signal smoothing resulting from the polymeric pipe material rheology.

As further validation, the separation procedure in time- and frequency domains are applied to the pressure signals acquired at the measurement sections P_E1 and P_E2 during the same transient as that in Figures 15 and 16.

These results appear to confirm that the viscoelastic effects generally improve the results for both time- and frequency-domain separation procedures since the HDPE material can be considered to have a greater viscoelastic effect than PVC-O. A measure of the viscoelasticity of the two materials is provided in Ferrante & Capponi (2017) where the use of the fractional derivatives allows for this kind of evaluation.

Techniques for separating the effects of pressure waves traveling in opposite directions have the interesting feature of helping diagnose parts of complex systems. The information carried by waves about their causes and encountered singularities can be associated with a location within the system without the need to model the system in total, boundary conditions included.

As demonstrated within this paper, the separation of pressure signals acquired at two measurement sections into their respective positive and negative traveling wave components can be obtained by different procedures, and under different conditions. Procedures available in the frequency domain have been revised, and other procedures entirely in the time-domain are proposed (without the need for transformation to and from the frequency domain). The performance of all these procedures has been tested using numerical and experimental signals, considering different distances between sections and different pipe materials.

The procedure obtained by the Fourier transform of the signals, their algebraic manipulations in the frequency domain, and then by the inverse transform, has been seen to be significantly affected by spurious oscillations that can hide the information content of the separated signals. The oscillations do not depend on the sampling frequency of the signals and are also present in noise-free signals. These oscillations can, however, be partially removed by a low-pass filter but their tuning for specific applications can be onerous. The choice of the filter parameter values requires a trade-off between the reduction of the oscillations and the ability to retain the sharpness of step wave fronts in the pressure signals.

The wavelet transform of the separated pressure signals can be obtained if a wavelet function is used instead of a filter. This accomplishes two tasks, i.e. detecting pressure waves propagating in the two opposite directions and avoiding the need to filter out oscillations introduced by the separation procedure or noise.

The procedures obtained by the time-domain recursive formulations were also tested. Two recursive-based procedures requiring the values of the separated signals at previous time lags yield near identical results. A procedure requiring only historic values of the original pressure signals was also tested; however, its efficiency and reliability were found to be limited by the large number of terms needed (results not shown). The time-domain procedures are affected by two main issues that become more relevant over long time durations. The first issue is the drift of the separated signals, which is an increase in the mean absolute values of the two wave components. The second issue is the noise error compounding in time caused by the terms involving the previous time steps, which propagate and add noise from one time step to the next.

The use of a weight factor (like a relaxation coefficient) applied to the historic terms in the time-domain formulations can reduce the drift and the noise aggregation observed in the separated signals. The use of a linearized friction term to account for frictional losses can be considered equivalent to that of the weight factor, with the difference being that the former is applied to all terms of the recursive formulation and, most important, being physically based it depends on parameters of the system that can be measured or estimated (such as the geometry, the reference flow and the friction factor). A filter applied to the acquired signals can also be used if needed to reduce the impact of noise.

The distance of the measurement sections appears not to be a limiting factor. Separation procedures can provide similar results for sections 1 or 50 m apart, as long as the pipe section between the measurements is uniform. The separation procedures are also effective for pipes of a viscoelastic material. Those signal-splitting procedures derived in the frequency-domain yield signals less affected by oscillations, while those derived in the time-domain perform well, even despite the fact that they are based on the assumption of an elastic pipe material.

In conclusion, separation procedures are a promising tool application in transient-based diagnosis. Their use in subsections of a complex system is one of the most interesting features, as the methods do not require a full model of the pipeline system to interpret the measured signals. Findings show that time-domain procedures appear to be more accurate and reliable when numerical or experimental signals are used, both in elastic and viscoelastic pipes, especially when friction effects are modeled. In general, the results of the proposed investigation can be considered a necessary preliminary step in assessing these techniques. Field tests in functioning systems are the natural and necessary continuation of this research activity.

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The authors declare there is no conflict.