Experimental Study Exploring the Interaction of Structural and Leakage Dynamics

The laboratory investigation used the contaminant ingress into distribution systems (CID) facility at the University of Sheffield, which is a 141-m-long recirculating pipe loop. The facility consists of 50-mm nominal diameter 12 bar rated MDPE pipe, tensile yield stress of 15 MPa, with water fed from an upstream holding reservoir (volume of $0.95{\mathrm{m}}^3$) through a 3.5-kW Wilo (Burton Upon Trent, U.K.) MVIE variable speed pump. A 0.8-m removable section of pipe, 62 m downstream of the system pump, allows for the inclusion of different test sections housed within a $0.45{\mathrm{m}}^3$ capacity box containing a single side viewing window. The flow rate and pressure data are recorded using a single Arkon (Brno, Czech Republic) Flow System Mag-900 electromagnetic flow meter located immediately downstream from the system pump and a series of Gems (Plainville) 2200 pressure sensors. Data were acquired at 100 Hz using a National Instruments (NI) (Newbury, U.K.) USB-6009 data acquisition device (DAQ) and a Measurement Computing (Newbury, U.K.) PMD1820 DAQ for flow rate and pressure, respectively. Isolation of different sections of the pipe loop is achieved through the use of quarter-turn butterfly valves located at intervals along the pipe, including either side of the test section box. A schematic and image of the facility, relevant to the testing presented in this paper, is shown in Fig. 2.

Manufactured test sections containing longitudinal slits were produced as listed in Table 1, with all sections produced using the same specification pipe as the main pipe loop. The pipe dimensions, 50-mm internal diameter and 6.5-mm wall thickness, classify the pipe as thick-walled because the nondimensional diameter to wall thickness ratio is less than 20 ($d/s=9.69$). Pipe test sections were cut to 0.8-m length and compression fittings attached to allow for installation within the pipe loop. The use of compression fittings for installation and the resulting induced longitudinal stresses were assumed as having negligible influence on the structural behavior of the leak openings as concluded by Cassa and van Zyl (2011). Three repeat sections were manufactured for testing containing slits 60 mm long and 1 mm wide. A single pass of a 1-mm-thick circular slitting saw was conducted to minimize variation of the initial area negated the influence of the closure effect (compression) of the residual stresses upon removal of the material for each of the three repeat $60\times 1\mathrm{mm}$ test sections. The slit tips were then rounded using a 1-mm-diameter drill bit to prevent axial propagation of the slits under the applied loading conditions.

Unique to this study was the simultaneous measurement of leak area and axial strain. Quantification of the leak area was required for assessment of the structural behavior of the leak, and hence the dependence of the leak hydraulics on the time and pressure-dependent response (synchronous leak flow rate and associated discharge coefficient). A range of potential methods for measuring the leak area were assessed including Moire interferometry (Yen and Ratnam 2010) and laser scanning (Rabah et al. 2013); however, an image analysis technique was concluded to be the most effective and nonintrusive method providing high accuracy, efficiency, and simplicity. To accurately distinguish between the outside of the pipe and the leak opening, the pipe surface was painted white using an enamel spray paint. This provided a clear distinction between the white surface of the pipe and the black area of the slit opening. Images of the test section were recorded using a GigaView (Huntsville, Alabama) SVSI high-speed camera at 3 frames per second (fps), allowing for continuous image capture over a maximum 8-h time period. The section was illuminated by an array of IP65 (ANSI/IEC 2004) rated bright white strip light-emitting diode (LED) lights, which provided a consistent light source with minimal heating effect. A basic image-processing methodology was then employed to convert the red/green/blue (RGB) images to binary form using a defined constant threshold value as shown by way of example in Fig. 3. A pixel count of the black pixels, i.e., slit opening, was then completed to calculate the leak area with a maximum associated error of approximately $\pm 3.82{\mathrm{mm}}^2$ related to the image resolution and the coupled effect of the lighting and threshold value. A calibration image was used to quantify the physical area of each pixel prior to testing.

It has previously been surmised that material strain can be used as an indicator of the dynamic area of leaks in pressurized pipes (Ferrante 2012). In order to determine the relationship between the synchronous material strain and leak area and whether the strain may be used, a predictor of dynamic leak area, a selection of TML GFLA-3-50 (Tokyo, Japan) strain gauges were attached using CN (Tokyo, Japan) cyanoacrylate adhesive to the pipe in the axial pipe direction in discrete locations as listed in Table 1. Only axial strain measurements parallel to the slit length were selected for the experimental work because theoretically they presented the greatest potential relationship between localized material strain and leak area based on the mode of deformation, i.e., tensile strain along the length of the slit wall. This was confirmed in a preliminary phase of testing. A cylindrical coordinate system ($r,\theta ,z$) is used to describe the location of the gauges [approximate coordinate error $\pm (0\mathrm{mm},0.0048\mathrm{rad},1.5\mathrm{mm})$] as shown in Fig. 4, where (31.5, 0, 0) is the center of the slit area at the external radius of the pipe. Strain data were acquired at 100 Hz using NI 9944 quarter-bridge completion accessories connected by RJ50 leads (National Instruments, Newbury, U.K.) to a NI 9237 four-channel module housed within a NI CompactDAQ chassis. The strain gauges were water-proofed using flexible rubber mastic tape applied over the surface of the gauge, which was shown to have negligible effect on both the structural behavior of the leak and the measurements recorded by the strain gauges.

The experimental procedure aimed to effectively record the relationship between the structural and hydraulic behavior of a leak through the simultaneous measurement of four key parameters (pressure head, leak flow rate, leak area, and axial strain). The conservative but controlled system conditions were defined to produce data capturing the time-dependent viscoelastic behavior (creep and recovery characteristics) coupled with the leakage response. The procedure comprised two cyclic stages: (1) pressurization and resulting creep phase, and (2) depressurization and resulting recovery phase. Stage 1 provided data on the time-dependent leak behavior following the assumed instantaneous pressurization and ensuing quasi-steady-state conditions at a predefined initial pressure head, with Stage 2 providing data on the leak opening area behavior following the assumed instantaneous depressurization and material recovery. Three to five repeats were conducted for TS601a at predetermined initial pressures of 10, 20, and 25 m head, each of which included an 8-h pressurization phase and 16-h recovery phase. Fig. 5 summarizes the experimental procedure implemented for the pressurization and recovery phases. The lengths of time for the pressurization and depressurization stages were chosen due to the observed time periods of measurable structural creep and recovery during preliminary testing. The repeated test sections, TS601b and TS601c, were both tested at 20-m initial pressure head only, primarily for assessment of experimental repeatability. The pressures utilized within the experimental work represent relatively low pressures compared with operational water distribution system pressures, but were set due the physical constraints (e.g., size of overflow weir) of the laboratory facility.

After the installation of the individual test sections within the pipe loop, the system was left dormant for 2 days to allow equilibration of the material strain. A null offset for all the attached strain gauges was executed prior to testing, assuming the test sections were at rest (zero stress and strain). The pressurization phase was conducted by starting the pump at a predetermined speed to achieve the necessary initial system pressure head. A manual opening of the upstream test section box valve (item $e$) while the downstream test section box valve remained closed was completed resulting in a step pressure change in the test sections. The subsequent leakage flow rate was allowed to overflow the box into the collection tank before being returned to the main system holding tank by a separate automated submersible pump, thus maintaining a constant water level within the test section tank ($H_w=0.45\mathrm{m}$). After the defined 8-h creep period, the upstream valve was closed, resulting in a step change depressurization and isolation of the test section. The test sections were then left to complete the 16-h recovery phase before repeating the process. Tests were conducted between 3 and 5 days to allow for a detailed assessment of the effect of the loading time history on the material response.

Fig. 10 shows the evaluated discharge coefficients using the orifice equation ($\lambda =0.5$) and the recorded laboratory data (synchronous leak flow rate, leak area, and pressure) for the three test sections. The raw data were filtered to reduce the total number of data points resulting in a representative data sample equivalent to a sampling rate of 1 Hz. The mean discharge coefficient values were 0.642 [standard deviation $(\sigma )=0.036$], 0.5443 ($\sigma =0.078$), and 0.488 ($\sigma =0.079$) for TS601a, TS601b, and TS601c, respectively, at 20-m pressure head. The corresponding mean discharge for coefficient values for TS601a at 10- and 25-m pressure heads were 0.608 ($\sigma =0.0062$) and 0.642 ($\sigma =0.0085$), respectively. The distinctly reduced $C_d$ value for TS601c may be accounted for by the measured leak area error previously noted. The results presented in Fig. 10 confirm that for individual longitudinal slits in pressurized pipe, a constant discharge coefficient is applicable to describe the pressure and time-dependent discharge through the leak. This supports the appropriateness in the application of orifice theory within leakage modeling of dynamic leaks provided that the synchronous leak area can be accurately estimated.

The direct relationship between axial strain and leak area provides a means to predict the time-dependent leak area for individual test sections if the localized material strain could be known or modeled. The definition of a viscoelastic model for the pressure and time-dependent axial strain may, therefore, be useful as a predictor of the actual leak area and hence leakage flow rate. A rigorous calibration using the experimental data from TS601a was conducted in order to define a model for the dynamic axial strain and hence the dynamic leak area. The generalized Kelvin-Voigt mathematical representation of viscoelastic behavior was chosen for the calibration due to its efficiency and previous effective use in modeling viscoelastic pipes under transient loading conditions. A nonlinear least-squares methodology was employed using the Levenberg-Marquardt algorithm and a function tolerance of $1\times {10}^{-12}$ to fit the creep compliance terms ($J_n$) as given in Eq. (3), using the measured strain, pressure head data, and the convolution integral, Eq. (4). Calibration of the discrete daily leakage response and the 5-day response were evaluated. The results of the calibration program considering the full time history (5-day response) are summarized in Table 3. The retardation times (${\tau }_n$) were assigned prior to the calibration to capture the discrete time period material response. These predetermined values were used to represent the time periods in increasing orders of magnitude (multiples of 10 s) covered by the long-term response of the structural behavior captured within the experimental program. This methodology reflected the method used by Covas et al. (2005) when considering the calibration of the short-term viscoelastic response of pipes to transient propagation.

Employing the 11-component parameter model as detailed in Table 3, the strain data were converted to time-dependent modeled leak area [$A_L(t)$] using the calibration for TS601a given previously. The 11-component model was assessed to be the optimal model selection based on computational efficiency and modeling accuracy (standard error representative of approximately 11.3% of the experimental recorded standard deviation). The mean value for the evaluated discharge coefficient ($C_d=0.64$) based on the analysis presented in Fig. 10, the measured differential pressure head across the leak opening, and the modeled leak area were input into Eq. (5), a modified time-dependent form of the orifice equation, to evaluate the leakage through the longitudinal slit. Eq. (6) is the leak area model using the calibrated creep compliance model (where $J_{11}$ is the 11-component creep compliance model [Eq. (3)] using components listed in Table 3) and the linear strain area relationship for TS601a where $C_1=0.01765$ and $C_2=2.8\times {10}^{-5}$. Fig. 11 presents results from this procedure for TS601a at three discrete quasi-steady-state pressure heads (10, 20, and 25 m) alongside the measured leak flow rate from the experimental work showing extrapolation across pressure ranges is valid

Comparison of the net 3-day leakage volumes produced percentage errors of $-4.29$, 3.22, and 0.14% between the modeled and measured leakage volumes for the 10-, 20-, and 25-m pressure head tests, respectively, further confirming the validity of the developed explicit model.

Analysis was conducted to evaluate the average maximum change of area for all the test sections during the first 8 h of the recovery phases: $-6.38\times {10}^{-05}{\mathrm{m}}^2$ ($\sigma =2.76\times {10}^{-06}{\mathrm{m}}^2$), $-6.10\times {10}^{-05}{\mathrm{m}}^2$ ($\sigma =1.50\times {10}^{-06}{\mathrm{m}}^2$), and $-5.99\times {10}^{-05}{\mathrm{m}}^2$ ($\sigma =1.79\times {10}^{-06}{\mathrm{m}}^2$) for Days 1, 2, and 3, respectively. The low standard deviations for the repeated test sections suggest that the susceptibility and magnitude of longitudinal slits to deformation is dependent primarily on the slit length not the width, which varied across the length of each manufactured slit resulting in different initial areas. This corresponds to the findings from numerical simulations conducted by Cassa and van Zyl (2011) investigating the structural behavior of equivalent leaks in linear-elastic materials. The discrepancy in the relative magnitude of strain for each test section corresponds to the distance of the strain gauge from the slit edge. In other words, the material strain is a function of the proximity to the slit. Further analysis of the axial strain distribution parallel and perpendicular to the leak length through physical observations and numerical simulations would advance the understanding of the mechanism of opening, i.e., whether deformations are due to localized buckling or bulging of the material. An evaluation of the significance of the manufacturing process on the inherent stresses within the extruded pipe may also provide a greater level of understanding of the observed phenomena.

The validated relationship between the measured leak area and the localized material axial strain allows for evaluation of the synchronous dynamic leak area if the strain is known or can be modeled. The simple fitted linear equations provide an alternative means to define the time-dependent leak area if the leak is not visible, i.e., buried. The influence of the external ground conditions on the leakage behavior of dynamic leaks (e.g., transferred loading, soil hydraulics, additional flow resistance, and ground temperature) remains a comparatively unexplored area of research. This unique data set and the calibration between leak area and strain, therefore, provides an opportunity to develop a methodology to explore the performance of buried leaking pipes, assessing the fluid-structure interaction and the associated structural dynamics and leak hydraulics, addressing the limited current knowledge on this topic.

The temperature range recorded during testing means that the magnitude of the observed structural deformations may be considered as relatively more extreme than for pipes in situ due to the relationship between temperature and creep compliance, i.e., increasing temperature reduces the stiffness of the material. The typical seasonal soil temperature variations in the United Kingdom for pipe burial depths between 750 and 1,350 mm (Water Regulation No. 1148 1999) lie between 4 and 21°C (Banks 2012).

It is generally agreed that the dynamic leak area is the dominant influence on the observed sensitivity of leaks to pressure (Clayton and van Zyl 2007; Cassa and van Zyl 2011; Ferrante 2012). This interpretation is qualitatively supported by the discernible linear correlation between the measured leak area and flow rate from the experimental results. Studies have previously used an effective leak area when modeling leakage due to the uncertainty of the changing leak area and the potential dependence of the associated leak discharge coefficient. Using the synchronous measurements of the leak flow rate, pressure head, and leak area, the time-dependent discharge coefficients for each test section were evaluated and found to remain constant over the full range of testing (five discrete loading phases) despite a maximum change in leak area of greater than 250% for TS601a at 20-m pressure head, for example. Two additional test sections with longitudinal slits of $20\times 1$ and $40\times 1\mathrm{mm}$ were subsequently produced to confirm this finding, with calculated mean discharge coefficients of 0.608 ($\sigma =0.0091$) and 0.610 ($\sigma =0.0062$), respectively. It may, therefore, be assumed that the theoretical discharge coefficient is independent of the leak deformation but dependent on the orifice type. This provides confirmation that the structural behavior, namely, the change of leak area, is the most critical determining factor of the dynamic leakage behavior. Ultimately this validates the inference made that accurate leakage models based on the orifice equation may be utilized for longitudinal slits in thick-walled viscoelastic pipes under fully turbulent flow conditions. This is achievable by using a discrete constant discharge coefficient, knowledge of the applied time-series pressure, and the synchronous leak area quantified from physical data or structural modeling. In reality, the actual leak area may not be measurable. However, knowledge that the discharge coefficient is independent from the dynamic structural leak behavior may, therefore, allow an approximation of the leak type and area to be made based on the observed pressure and leakage relationship. Further work to confirm this observation (constant discharge coefficient) for other leak types in different pipe materials is still required but remains a commonly adopted assumption (e.g., Cassa et al. 2010).

A generalized Kelvin-Voigt creep compliance formulation [Eq. (4)] was employed within the viscoelastic calibration due to the effectiveness of this mathematical representation in capturing both the instantaneous elastic and time-dependent creep and recovery material responses over specified time periods. As may be expected, the results of the calibration show that the greater the number of creep compliance components the better the fit to the experimental data, highlighting the importance of employing both the short- and long-term components to define a model that considers the full loading-history and the time-dependent creep and recovery responses. Smaller separate models ($<11$ total components) are capable of accurately predicting the daily strain response in isolation without considering the full loading history. Application of an 11-component model was determined to be the minimum requirement to effectively replicate the observed structural behavior because five discrete retardation time periods were identified from the time-dependent structural response. This, therefore, represents an accurate and parsimonious model essential in the development of computationally efficient tools for use within both academic and industrial applications. Separation of the daily standard errors indicated that the calibration goodness of fit was weakest for the first day. This highlights a potential limitation of mathematical representations in accurately predicting the total physical response of viscoelastic structures to applied loading, emphasizing that such models are only ever approximations of the true behavior. This modeling error is inconsequential with regards to the modeling error for application in real systems because it is not anticipated that in reality a leak would either exist in newly laid pipeline that is pressurized for the first time or be present in a pipe that has been fully depressurized for a time greater than the total material recovery time. The investigation highlights the need to consider the entire loading history, or alternatively a time period greater than the time required for the material to reach a quasi-steady state. Fig. 11 confirms the effectiveness of Eq. (5), a time and pressure-dependent form of the orifice equation, as a means of modeling the leakage behavior of discrete longitudinal slits in MDPE pipe considering the loading history and the interdependence of the structural behavior and the leak hydraulics.

Leakage exponent-based analyses are often used as a means to assess the sensitivity of leaks to pressure. The results presented in this paper question the validity of such an approach when considering viscoelastic materials that not only display pressure-dependence but also time-dependence. The impact of this is in reducing the effectiveness and benefits of leakage assessment and control techniques using the FAVAD (or similar) leakage model for predominantly viscoelastic material-based systems. In order to understand the benefits of pressure management in reducing the total losses in a system comprised of polyethylene pipe, an appreciation of the complex dynamic nature of the leaks in this material must be considered. The explicit leakage modeling technique developed within this paper allows for accurate calculations of time-dependent leakage based on pressure head data and a leak area model calibrated from recorded axial strain data. A leakage model for longitudinal slits in viscoelastic pipe based on the characterization and calibration within this paper considering all parameters including geometry, material rheology, and loading conditions will provide a means to develop a generalized leakage model considering all the significant influencing parameters. The methodology presented for characterizing the leakage behavior may be developed for assessing the equivalent short-term response of leaks subject to dynamic pressures, e.g., pressure transients due to valve closures, pump shutdown, or changes in demand (Collins et al. 2012). Focus on the short-term response has significance for the assessment of contaminant ingress risk associated with the existence of low or negative pressures in water distribution pipes. Likewise, active leakage control techniques such as leak detection and localization based on inverse transient analysis may be greatly enhanced by the inclusion of the pressure and time-dependent orifice equation.