Dynamic effects of a regulating valve in the assessment of water leakages in single pipelines

These authors


Introduction
To reduce operation and maintenance costs, water utility companies must effectively manage water leakage in their supply systems (Ramos et al, 2023b;Conejos Fuertes et al, 2020).These leaks are often caused by excessive pressure or inefficient management by utility companies.Because all supply systems eventually suffer from leakages, efficient management is required to effectively identify the origin of leaks ( Ávila et al, 2021;Adedeji et al, 2017).Hence, water utility companies must establish pipe-leakage mitigation programmes aimed at improving the service efficiency and decreasing costs borne by water treatment plants, which is of the utmost important for establishing reliable digital twins in water distribution systems (Bonilla et al, 2022;Galdiero et al, 2015).According to the International Water Association (IWA) conventions (Lambert and Hirner, 2000;Lambert et al, 1999), water flow in the supply systems is divided into two types of flows: (i) the flow rate measured, which represents the authorised/billed consumption at households, industries, businesses and institutions and (ii) the uncontrolled flow rate, which is subdivided into apparent losses (customer flow meter inaccuracies, unmetered consumption, unauthorised consumption, water used for firefighting and water used for cleaning streets and public areas) and physical losses (leakages).Figure 1 shows the different levels used to categorise consumption types.
Leakage occurs in all water distribution systems (WDS) and depends on network features, the tightness obtained during the construction process, the operation practices of the corresponding water utility companies and the level of technology and expertise used for managing WDS.Water losses vary among different countries and different regions within countries; therefore, the strategy used for mitigating leakage depends largely on the understanding of each component of WDS (Germanopoulos and Jowitt, 1989).Because leakage is related to energy costs of WDS, effective leakage control is necessary to ensure social, economic, and environmental sustainability (Ramos et al, 2021;Bhaduri et al, 2015).If effective measures are not implemented to manage water losses due to leakages, drinking water supply will be insufficient to meet Billed by fixed quota users Fig. 1 Notation used for Water Balance Components the basic needs of people, thereby increasing costs exponentially.Therefore, studying water-leakage patterns in WDS is necessary for utility companies globally to prevent high distribution costs (Voogd et al, 2021).Every loss mitigation programme starts with an analysis of the water balance of the WDS being assessed (Figure 1).Next, implementing a mathematical model is necessary for studying the WDS.Herein, water leakage is simulated considering pressure-based consumption.Currently, water flow rates and node pressures in pipes are determined using simulation models for an extended period (quasi-static models), wherein the energy and continuity equations are solved simultaneously using the gradient method (Salgado et al, 1988).Excess water pressure can be reduced and managed by installing pressure reducing valves (PRVs) at specific points and replacing pipe sections to improve hydraulic behaviour, thus realising highly efficient water distribution management (Ramos et al, 2023a).PRVs help maintain an adequate service pressure throughout the day and specifically at night, reducing the water-leakage volume.Service pressure in WDS, which is indicated by the valve resistance coefficient, is directly related to PRV position.However, the continuous variations in PRV in WDS result in pressure oscillations that cannot be measured using the quasi-static model or extender period simulation (Almandoz et al, 2005) because this model does not consider the hydraulic system inertia.The accuracy of the results obtained using the quasi-static model decreases with increasing variations in PRV opening-closure.The occurrence of transient events helps in detecting leakages in a WDS (Brunone et al, 2022).Recently, Ayed et al (2023) proposed a transient flow model to assess the volume of water leaked from a pipe joint; however, this model failed to consider the water balance of a WDS and did not allow the measurement of water losses based on pressure-based consumption.This study proposes the development of a rigid inertial model (also known as Euler's equation or mass oscillation) aimed at assessing water leakage in a WDS.This model can be used to effectively measure pressure pulses produced owing to PRV openings-closures; no deformations are expected to occur in the pipe walls or the fluid owing to the service pressure in WDS (Coronado-Hernández et al, 2018).The proposed model affords new in the field of water-leakage mitigation in WDS because to the best of our knowledge, no existing study measures pressure variations related to PRV in short time periods.Moreover, the proposed model can provide a better understanding of water-leakage volumes and become a more reliable tool than the quasi-static model for utility companies to establish water loss management programmes.Because mitigating water leakage is of utmost importance for utility companies, the development of new mathematical models for leakage assessment is of great interest to the efficiency and sustainability of water companies.
Current mathematical models used for to assess water losses in WDS use quasistatic models to measure water loss volumes during a given period of time, usually 24 hours.These models neglect the system inertia generated owing to local and/or convective velocity.Nevertheless, water utility companies must constrain the service pressure at night to reduce water losses, which requires control-valve operations based on quasi-static models.When rapid variations occur in the operation of control valves, the system inertia can produce variations in the system pressure evolution and consequently in the water leakage.Pressure oscillations caused by variations in control-valve operations can be used to detect leakage and clogged pipes (Brunone et al, 2023;Covas et al, 2005).This study proposes algebraic differential equations using the local velocity term to assess water leaks based on the hydraulic system inertia.The proposed model can be used to evaluate water leakages considering opening and closure manoeuvres in single pipelines.

Materials and Methods
This section shows the methodology used in this research, which is presented in Figure 2. The objective of this research is the development of a rigid water column model (RWCM) for computing water leakages in single pipelines, which could be extrapolated to other more complex water systems in future research.The proposal methodology is divided in five steps, as follows:

Step 1: Model case study
A model case study is developed, defining the parameters of the reservoir (i.e., head), pipes (i.e., roughness, length, and diameter) and valves (opening degree and dimensionless coefficient headloss).The development of the model enable the development of the different mathematical model (RWCM model) and topology model in the extended period simulation (EPS).The model is completed considered the inputs values, which are focused on the hourly consumption patterns and the base demand of the consumption joints.The input data are required for analysing leakages in water distribution systems, which can be characterized by water utility companies through the installation of flow meters.The definition of topological characteristics of a water installation should be defined (case study).

Step 2: Definition of the leakages coefficients
The leakages should be considered in mathematical models, therefore the different coefficients should be considered.Currently, Equation 1 can be used to evaluate the leakage in each element (i.e., line or tap).
where Q ul is the leakage flow rate (i.e., line or consumption point), p 2 /γ w is the pressure head (if the element is a line, the chosen pressure is the average pressure value of the line), N is the leakage exponent, and K f is the global emitter coefficient.
The leakage volume for a single pipeline is estimated by Equation 2 where T is the total time, ∆T is the interval time, and V ul is the leakage volume.
When the flow injected is equal to measured and uncontrolled flow, the calibration is considered correct, and the methodology enable the development of the steps 3 (Section 2.3) and 4 (Section 2.4).

Step 3: Extended Period Simulation (EPS)
The extended period simulation (EPS) has been used for different authors to compute water leakages in WDS (Almandoz et al, 2005;Ramos et al, 2023a;Giustolisi et al, 2008) in the last decades.Usually, EPANET software has been utilised considering the water balance proposed by the IWA (Lambert and Hirner, 2000), which is described in Figure 1.The Bernoulli equation is used to describe the water movement along an installation.The emitter coefficient (see Equation 1) must be calibrated.The calibration process finishes when the injected flow rate (measured by a water utility company) gives similar values compared to the results obtained with the mathematical model.
The used emitter coefficient in EPANET is: β varies between 10 −4 and 10 −7 as a function of the material (cement, steel or PVC) (Maskit and Ostfeld, 2014).It grows over time and this increase depends on the material.Different scenarios are defined in this case, which weighted the β value.Besides, N exponent is defined in the calibration model, estimating the value through of normalized valued, which oscillates between 0.5 and 2.5 according to published researches (Maskit and Ostfeld, 2014;Van Zyl and Clayton, 2007).
In this research, the extended period simulation is modelled for a case study to quantify the water leakages in a single pipeline.For the same case study, the proposed model (RWCM) is also applied (see Section 2.4) to observe the different when manoeuvres in a regulating valve is performed in a short time period.

Step 4: Rigid water column model
Currently, water leakage assessment models use the Bernoulli equation in extended periods because control-valve operations are performed in a controlled and prolonged manner.Herein, water leakage was determined using the rigid water column model, which is the main contribution of this research since a mathematical model is developed.
The mathematical model used to assess leakage in a simple pipe as presented in Figure 3 uses the water balance expressed as follows: where Q iny = injected flow rate (m 3 /s), Q m = flow rate measured (m 3 /s), and Considering the main components of the water balance proposed by the International Water Association: where All flow rate measured consumptions (Q m ) are considered independent of pressure head (p 2 /γ w ).These values can be evaluated using a nodal consumption where Q m can be simulated considering pressure-independent consumption.Downstream, pressure varies over time, so Q m is determined using Equation 6: where However, consumption Q ul is dependent on pressure (Almandoz et al, 2005;Giustolisi et al, 2008); that is, the higher the pressure, the higher the consumption and

Consumption Type Observation
Quae Water meters must be tested to determine their accuracy.
Quaq and Q uai Must be characterised by utility companies.

Q uah
Pressure-based consumption.
Q uaf Depending on frequency.
vice versa.Equation 7demonstrates that water loss is highest at night.For assessment purposes, the emitter exponent was considered to be 0.5, which has been widely adopted by many authors (Almandoz et al, 2005;Ramos et al, 2023a).
where K f = emitter coefficient, p 2 /γ w = pressure at node 2 and γ w = water specific weight.
The remaining the water consumption variables are presented in Table 1.Substituting Equations 6 and 7 in Equation 5and disregarding the consumptions from Table 1, Equation 8 is obtained: The continuity and momentum equations for a simple pipe are given as follows: where f = friction factor, g = gravity acceleration, l = length of pipe (m), A = cross-sectional area of pipe, d 0 = internal diameter of the pipe (m), Σk m = global coefficient of minor losses, H = hydraulic grade line, and R v = valve resistance coefficient.
Considering that water leakages tend to reduce the increased pressure in the pipe created owing to automatic variations in PRV operations, the pipe is not expected to suffer deformations and the water column is not expected to be compressed during these variations.Therefore, a rigid pipe is considered, which implies that the wave speed tends to infinity (a → ∞).Hence, (∂Q iny /∂x = 0), thereby demonstrating that the continuity equation is reduced to Equation 11: Upon dividing gA in the momentum equation, Equation 12is obtained: When integrating a generic t value along a pipe with an area of A between two points with a length of l, then: (13) After integrating and rearranging the terms: Finally, when assessing energies at points 1 and 2, thus: where z = corresponds to an elevation of the pipe profile.This equation simulates water behaviour based on the rigid water column method (Coronado-Hernández et al, 2018).By replacing the terms and solving Equation 15: The friction factor can be determined using the Swamee-Jain formula.f = 0.25 log( ks 3.7d0 ) + 5.74 Re 0.9 ) where k s = absolute roughness of a pipe and Re = Reynolds number.The Reynolds number is defined by relationship vwd0 ν , where v w = water velocity (injected flow rate) and ν = kinematic viscosity.

Step 5: Comparison and discussion results
This step considers the assessment of water leakages using different typical behaviours and applications of the EPS and RWCM.The analysis shows the discrepancies on the estimation of volume leakages for a case study considering domestic and industrial consumption patterns.
The next section presents the application of the used methodology to a case study.

Analysis of Results
In order to assess changes in water leakage, the following characteristics of a simple pipe has been considered: l = 1300 m, d 0 = 300 mm, k s = 0.0015 mm, ∆z = z 1 − z 2 = 45 m, Σk m = 5, and R v = 210 ms 2 /m 6 .For this analysis, a kinematic viscosity (υ) of 1x10 −6 m 2 /s was considered.To perform the water balance, the following data were obtained: Q iny = 140.0l/s and Q r = 91.0l/s (with Q md = 75.5 l/s and Q mi = 15.5 l/s).Figure 4 shows the modulation coefficients for domestic and industrial consumption.For the domestic consumption, the peak coefficient is found from 9:00 to 10:00 h, while the minimum coefficient of 0.1 is detected between 4:00 to 6:00 h.For the industrial consumption, the maximum and minimum values are 1.6 (12:00 to 13:00) and 0.4 (0:00 to 1:00), respectively.1.8 2.0 -12] [12-13] [13-14]  [14-15] [15-16] [16-17] [17-18] [18-19] [19-20] [20-21] [21-22] [22-23] [23-24]   C md (-) The water balance has been addressed considering a domestic and industrial consumptions and the water leakage.In this sense, Equation 8 can be reduced as: The Extended Period Simulation (EPS) was run to know the water behaviour over 1 day.The input parameters were determined by solving Equations.8, 16, and 17 considering a zero local velocity (dQ iny /dt=0).An emitter coefficient (K f ) of 9.29 l/s/wmc 0.5 was previously calibrated.The water flow in the entrance of the single pipeline was at t = 0 h for 79 l/s, which were considered as the initial conditions for the mathematical model.Figure 5 shows the results obtained from applying the EPS.The maximum pressure is 39.5 m , occurring from 4:00 to 6:00 a.m.For this period, the highest water leakage of 59.1 l/s is reported.The highest flow in the water treatment plant is 197.3 l/s (from 9 to 10 am) and the minimum is 79 l/s (from 4 to 6 am).The EPS represents leaked flow patterns effectively.Since water utilities need to control downstream pressure head, then a value of 15 m was imposed in the pressure reducing valve (PRV).The EPS was run using a PRV, which is named as EPS-PRV.If p2 γw = 15 m, then the resistance coefficient (R v ) can be computed using Equation 19: The rigid water column model (RWCM) was simulated using formulations 8, 16, and 17.A regulating valve is manipulated for two scenarios: (i) considering an opening manoeuvre (RWCM-Opening) with a minimum R v value of 9000 ms 2 /m 6 (at 30 s); and (ii) a closure manoeuvre (RWCM-Closure) with a maximum value of 210 ms 2 /m 6 (at 30 s). Figure 6 presents the simulation during the first 180 s for the EPS, EPS-PRV, RWCM-Opening, and RWCM-Closure.Figure 6a shows the behaviour of pressure head for the different simulations, where the EPS trends to a constant value of 39.4 m for the analysed period.Similarly, the EPS-PRV provides a constant value of 15 m, as expected for the function imposed by the valve.The RWCM-Opening starts at 39.7 m (t = 0 s),but after the opening of the regulating valve, the pressure head reaches a constant value of 38.68 m at t =38.5 s.For the RWCM-Closure, the pressure head attains its minimum value after the closure manoeuvre (30 s) with a value of 14.26 m.The injected flow has also different patterns considering the analysed simulations(Figure 6b).The total flow presents a constant value of 79.6 l/s for the first 180 s (EPS), while the EPS-PRV shows a reduction of 22.3 l/s in comparison to the EPS, since a constant value of 57.3 l/s is detected by the model.Considering the results of the RWCM-opening, the minimum injected flow is reached at 5 s with a flow of 57.1 l/s.After 38.5 s, the injected flow trends to a constant value of 79.8 l/s.The injected flow attains a minimum value of 56.4 l/s (at t=35.5 s) for the RWCM-Closure.The behaviour of water leakage (Q ul ) is similar to the behaviour of injected flow (Q iny ) since domestic and industrial consumption are not varying for this period, as shown in Figure 6c. Figure 6d presents the variation of the resistance coefficients (R v ) analysed for the current simulations.Table 2 presents the leakage volumes computed for periods of 30, 60, and 180 s.As expected, the EPS gives the greatest leakage water volume compared to the other simulations.Considering the first 30 s (see Figure 6) the leakage volume computed for

Conclusions
In this study, a mathematical model to simulate the behaviour of water-leakage flow considering the rigid water column model (RWCM) was developed.The proposed model is more effective compared to the extended period simulation (EPS) -since it is suitable to represent rapid manoeuvres in regulating valves.The proposed RWCM model was applied to a case study where domestic and industrial consumptions were reported.The initial condition of the mathematical model is the one obtained from the extended period simulation.For the assessment, the emitter coefficient was calculated using the results of the quasi-static model.The numerical resolution of the algebraic-differential system provides a satisfactory solution to the water-leakage flow problem because the resolution satisfies the condition that states the water-leakage flow increases with pressure and vice versa.The proposed model can represent pressure head and leakages flow rate pulses for short and large time periods, which could be applied in more complex water systems.This study confirmed that the system inertial will have substantial influence on the water-leakage flow, thus yielding results that can be substantially different from those obtained using the extended period simulation.The implementation of the proposed model can be considered for developing digital twins approaches in water improving the sustainability management of the water systems.

Fig. 2
Fig. 2 Methodology flow rate due to apparent losses, Q uae = undetected flow rate due to metering errors, Q uaq = flow rate billed by fixed quota users, Q uaq = flow rate by illegal use, Q uah = flow rate used for fire hydrants, Q uaq = flow rate used for system flushing, and Q ul = flow rate due to physical leakages.

Fig. 5
Fig. 5 Analysis of the water balance under an EPS

Table 1
Other Consumption Variables