Hydraulic modeling approach for evaluating the performance of flow-starved water transmission networks

CWS

continuous water supply

DDA

demand-driven analysis

EPS

extended period simulation

FDSR

flow-based demand satisfaction ratio

FSC

flow starvation condition

FSL

full supply level

GL

ground level

HGL

hydraulic grade line

IWS

intermittent water supply

LSL

low supply level

LPS

liter per second

MBR

master balancing reservoir

PDA

pressure-driven analysis

PDD

pressure-driven demand

PDO

pressure-driven outflow

PDA-FSC

PDA under FSC

SR

service reservoir

SWC

standing water column

VDSR

volume-based demand satisfaction ratio

WSS

water supply systems

WDN

water distribution network

WTN

water transmission network

H

height of FSL from the ground level of the MBR

L

total length of the immediate downstream pipe

FSL

full-service level of the MBR

L

total length of the immediate downstream pipe

LSL

low service level of the MBR

W1

withdrawal capacity of the networks at source head equal to FSL of the MBR

W2

withdrawal capacity of the networks at source head equal to LSL of the MBR

W3

withdrawal capacity of the networks at source head equal to GL of the MBR

h

height of LSL from ground level of MBR or length of the vertical portion of the immediate downstream pipe

L-h

length of the horizontal section of the immediate downstream pipe

x

distance of SWC from the LSL of the MBR at any arbitrary point in Stage 2

Despite intermittent water supply (IWS) not being ideal for supplying water to consumers, more than 1 billion people worldwide receive water intermittently (Charalambous & Laspidou 2017). Most of this population resides in developing countries (Charalambous & Laspidou 2017). Although IWS is discouraged due to contamination intrusion, consumer inconvenience, and the ill effect on infrastructure, its presence is a reality for a significant population (Charalambous & Laspidou 2017). Farmani et al. (2021) reaffirmed the presence of IWS in developing countries and assessed the resilience of IWS under COVID-19 conditions. Scarcity of water resources, unscientific network expansion, uncontrolled withdrawal, and high water loss are among the reasons for the prevalence of IWS (Simukonda et al. 2018; Kalbar & Gokhale 2019). Recently, Ghorpade et al. (2021) identified the drivers of the IWS as unsuitable design and analysis methods, lack of operation and maintenance, low consumer satisfaction, high non-revenue water, and weak institutional capacity.

Continuous water supply (CWS) provided to consumers with minimum threshold pressure, ensures equity as consumers are free to withdraw water as per demand (Randeniya et al. 2022). Achieving equity in water supply under IWS emerges as one of the challenges (Batish 2003; Randeniya et al. 2022). A precise operational control (pressure and demand) is needed to ensure that supply to consumers remains proportional to demand under IWS (Bhave & Gupta 2000; Batish 2003). Hence, a better prediction of flows in different parts of networks is vital for understanding the performance of IWS systems (Mohan & Abhijith 2020; Meyer & Ahadzadeh 2021). Recently, Sarisen et al. (2022) have examined and reviewed the challenges of modeling IWS systems and identified gaps in water demand modeling, leakage modeling, modeling of the water filling process, and the need for systematic data for real IWS systems.

The WDN is used to supply water to consumers within service areas. In contrast, the WTN transports bulk water from the water source to SRs (Figure 1). Hydraulic modeling of WDN and WTN is used to understand the performance of WSS. Despite the inherent difference in purpose and layout, the same modeling approach is adopted for both network types. Various modeling and simulation software packages have been developed to predict the performance of water networks (Cembrowicz & Ates, 1998; Rossman 2000; Conety Ravi et al. 2019; Bentley 2022). There are two prominent analysis approaches; demand-driven analysis (DDA) and pressure-driven analysis (PDA). In DDA, nodal demands are presumed to be completely satisfied irrespective of residual pressure available at these nodes (Gupta & Bhave 1996; Conety Ravi et al. 2019). In a situation of pressure deficiency, demand satisfaction at these nodes is impossible. Hence the DDA fails to predict the actual performance of WDN as nodal outflows are non-realistic in such pressure-deficient conditions. Bhave (1981) recognized this limitation and introduced the methodology of node-flow analysis, which later became the basis of PDA. Contrary to the DDA approach, the PDA approach is more realistic and uses the nodal head-discharge relationship for each node while analyzing the pressure-deficient WDNs.

Most PDA approaches implement demand modification based on outlet boundary condition, i.e. relating outlet pressure to flow (Bhave 1981; Wagner et al. 1988; Fujiwara & Li 1998), which is termed as PDA based on pressure-dependent demand (PDD). A few studies directly used outlet properties to determine the flow/outflow, termed as PDA based on pressure-dependent outflow (PDO) (Reddy & Elango 1989; Chandapillai 1991). Recently, Conety Ravi et al. (2019) reviewed and summarized developments in PDA studies. Other approaches, such as volume-driven analysis (Taylor et al. 2019; Sivakumar et al. 2020) are also based on these PDA modeling approaches and the similar water withdrawal mechanism.

Although PDAs have more realistic demand modeling of outlet boundary than DDA in pressure-deficient conditions (Gupta & Bhave 1996), most PDA studies model tanks as reservoirs, assuming infinite water sources, ignoring the inlet boundary condition. Whenever the upstream network fails to provide sufficient flow to match the flow prevalent in the downstream network, the downstream network operates at a lower flow; this phenomenon is termed herein as flow starvation. PDA studies predict a safer picture than the actual condition in such cases. Traditionally PDA models have focused solely on outlet boundary conditions, flow starvation requires more explicit modeling of inlet boundary condition. This limitation also arises due to modeling a single component (WDN or WTN without an upstream network), usually adopted in PDA, which conceals the effect of one component from the other. As a WSS is a highly integrated system, combined modeling of different components is necessary. A modified PDA approach is needed to account for flow starvation (a common phenomenon in IWS) to estimate flow and pressure accurately through the water network (Kalbar et al. 2014).

Various researchers have developed PDA models to simulate the IWS regime (Tabesh et al. 2001; Batish 2003; Ingeduld et al. 2007). Since conventional modeling approaches do not simulate partial empty pipe networks caused by pressure deficiency under flow starvation (Mohan & Abhijith 2020), the network's performance under FSC cannot be simulated. PDA models found in the literature do not consider the FSC prevalent in IWS (Mahmoud et al. 2017; Shirzad 2020). Modeling studies focused on IWS also fails to capture FSC (Mohapatra et al. 2014; Taylor et al. 2019; Randeniya et al. 2022). Mohan & Abhijith (2020) presented a novel modeling approach for partial flows under FSC; however, it is reported that mass-conservation is not maintained in their method (Meyer & Ahadzadeh 2021).

Additionally, the method's applicability is limited by two assumptions. Firstly, the model assumes fill-draw operation (drawing of water occurs subsequently after filling the tank). However, WDN with limited operational hours (2–3 h) can satisfy the fill-draw operation condition where the filling of the tank and supply from the tank does not overlap, but the same cannot be replicated in the WTN due to longer operational hours (16–24 h). Secondly, the model assumes the possibility of partial flow in any pipe within a WDN. This assumption is not valid in the WTN, where water is drawn from the MBR to the elevated SRs through vertically erected inlets, and hence the partial flow is not physically possible. Such an inlet can only carry flow when fully pressurized; therefore, it is not realistic to think of partial flow into the elevated SR. From this understanding, it is physically possible for water from the MBR, which is at a higher elevation, to fill a SR at a lower elevation as long as the water column has a high enough head; otherwise, no flow occurs into the SR.

Current modeling approaches do not account for the inlet boundary condition of WTN during flow starvation. Hence, the present study is focused on developing a hydraulic modeling approach for flow-starved WTN. The proposed approach is based on modeling WTN and presents a more realistic representation of the network and deployed control mechanisms. The present study associates partial flow in non-pressurized pipe in the downstream vicinity of balancing reservoirs with the inability of the upstream network to provide sufficient inflow and modify inlet boundary conditions accordingly.

WSSs in India are designed to operate as CWS systems as mandated by CPHEEO (1999); however, most systems operate as IWS. In IWS, most consumers do not receive the entire demand, and supply is uncertain (Farmani et al. 2021). Typically consumers store water in storage tanks to cope with intermittency (Agrawal et al. 2007) and start to draw water as soon as the supply is resumed (Ghorpade et al. 2021). The poor institutional capacity of water utilities cannot enforce demand-side management such as effective metering, appropriate tariff structure, action on non-payment (Burt & Ray 2014). Additionally, there are no proper valve controls on the water network to limit withdrawal except for complete shutoff, and zoning is done to limit withdrawal (Ghorpade et al. 2021). These operation practices result in the emptying of SRs due to over withdrawal. The empty SRs lead to unequal water distribution in downstream service areas and more withdrawal on upstream (Sashikumar et al. 2003; Kalbar et al. 2014).

The gap between the design and operation of WSSs in India is responsible for forming SWC, which is primarily caused by the inability to provide sufficient inflow, control withdrawal, or provide adequate balancing storage at any water tank. WSSs in India are designed with a design life of 30 years. The population is forecasted for the following three phases: (a) present phase: planning period; (b) intermediate phase: the period after 15 years from commissioning; and (c) ultimate phase: The period after 30 years from commissioning (CPHEEO 1999). Most IWS systems have the low capacity in terms of providing pressure as they fail to supply water with sufficient pressure due to simultaneous demand (Sashikumar et al. 2003; Kalbar et al. 2014). However, studies (Bhave & Gupta 2000; Tabesh et al. 2001) have shown that IWS leads to more water withdrawal than estimated demand. Field studies on IWS have reported an increase in peak factor indicating enhanced withdrawal (Sashikumar et al. 2003; Kalbar et al. 2014). This excess water withdrawal capacity is problematic. The WTN is designed for the ultimate phase, and the pump station is usually designed for the intermediate phase, which leads to higher withdrawal capacity of the WTN with respect to the upstream network (CPHEEO 1999).

Additionally, the provision of design norms such as the minimum diameter of the pipeline leads to higher design diameters than the actual requirement. This also increases the withdrawal capacity of WTNs significantly (Ghorpade et al. 2021). The WTNs have excess capacity for initial phase of design life (CPHEEO 1999). Hence, design reports recommend that the design flow be realized in the field with the help of valve throttling, head-dissipating devices (Bhave & Gupta 2000), and hydraulic isolating structures (Kalbar & Gokhale 2019). However, in practice, flow control is attempted with only sluice valves, resulting in poor performance (Ghorpade et al. 2021). Eventually, there is no valve operation over the period, and the WTNs operate with limited control in the poorly managed system. The tank configuration of balancing reservoirs, uncontrolled operations, and significant physical losses in the systems contribute to the SWC formation.

The performance of a WSS under FSC is very different from that in CWS. Considering the tank configuration used in India, starting with an initially full tank, three different operation stages occur in the IWS system. These supply stages are described in the following section in the sequence of their occurrence (Figure 2).

In intermittent operation in India, inflow to the MBR is lower than outflow due to the WTN's higher withdrawal capacity. The volume of water in the MBR depletes at the rate of the difference between outflow and inflow. In Stage 1, inflow remains constant, whereas outflow decreases with a decrease in the water level. The system's performance in this stage is illustrated in Figure 2(a). The performance at this stage can be simulated in common software that allows PDA modeling using a tank element as a frictional resistance-free hydraulic element (Rossman 2000; Bentley 2022).

After Stage 1, transition starts, where vertical partial flow exists in the upper portion of the vertical pipe and the lower portion remains pressurized. During this transition, the water level in the MBR is below LSL (suppose at ‘x’ distance from the LSL of the MBR); hence the pipe section immediately downstream to the MBR does not get fully pressurized (Figure 2(b). During Stage 2, the WTN's withdrawal capacity changes with the pressurized part of the WTN. The decrease in water level leads to a reduction in the head (increase in x) and a reduction in the frictional resistance (reduction in the portion of pressurized pipe network). The immediate downstream pipe is vertical in most cases. Since the vertical pipeline is minimal in length with the least frictional resistant part of the WTN due to its higher diameter, the effect of reduced head in determining flow is more drastic than the reduced frictional resistance. As a result, the piped WTN's withdrawal capacity decreases, lowering the outflow.

After Stage 2, outflow decreases to match inflow, and equilibrium of water level is achieved in the pipe immediately downstream to the MBR. The equilibrium water level marks the beginning of Stage 3 and the end of Stage 2. At this stage, a unique combination of the head available to the WTN and its resistance is attained so that outflow of MBR equals inflow. The water level at this stage is depicted in Figure 2(b) when the operating water column becomes stable. This equilibrium stage is the SWC condition (Kalbar & Gokhale 2019).

After supply duration is over and the WTN is closed, the MBR is refilled in the remaining duration of the day so that the next day's supply can resume from Stage 1.

This study develops a novel modeling approach predicting flows and pressure during FSC. This study proposes withdrawal capacity as an indicator of network capacity. The withdrawal capacity of the pipe system (pipe and control equipment) is defined with reference to the source water level. It is assumed that withdrawal capacity equals total outflow from an infinite volume water source at the constant head to a piped system under pressure-driven flow. In simpler terms, it is the total outflow expected through a piped system if a reservoir with infinite volume is connected to it under pressure-driven flow. Flow starvation occurs whenever inflow at any reservoir (with limited storage capacity) is lower than the piped network's withdrawal capacity and flow mismatch leads to a water level below the LSL of the reservoir. Hence, we propose a novel PDA under FSC (PDA-FSC) approach to model this condition.

In a flow-starved network, the pipe downstream to the storage reservoir behaves like storage reservoir and pipe. Although these storage reservoirs, i.e. MBR/SR are made of real/physical material having some hydraulic resistance, the reservoir has a larger diameter than the pipe network and negligible hydraulic resistance. Hence, the MBR/SR is modeled using a tank element with zero hydraulic resistance. The same approach can be extended to the vertical section of the immediate downstream pipe to the reservoir by considering it as part of the reservoir. Its resistance can also be neglected compared to the rest of the water network. For a more realistic representation of reservoirs under FSC, an irregular-shaped water tank is used instead of the conventional circular tank. The following assumptions are made and they provide the basis for adopting the proposed modeling approaches for the WTN under FSC.

Assumption I: MBR, along with the vertical section of the immediate downstream pipe modeled as a single tank element, can exhibit the same performance as physical MBR under FSC.

Assumption II: Physical WTN can be effectively represented in the hydraulic model WTN by excluding the vertical portion of the immediate downstream pipe to the MBR.

Assumption III: During operation, all changes are considered to occur under quasi-steady flow condition. Hence, smooth transition between steady-state simulation is able to depict the performance adequately.

Assumption IV: During entire supply period, valves to all SRs remain open as assumed in design stage, and all SRs try to draw water simultaneously irrespective of water received previously. This assumption allows to model extreme of poorly managed WTN.

The network upstream of the MBR (Sump–Pump–Pumping Main–MBR) is used to determine inflow to the MBR. Part of Figure 1 illustrates this network schematically. The physical pure water sump is represented as a reservoir element to replicate the constant water level in a pure water sump. The water discharges through the inlet of MBR at a constant level, as shown in Figure 1. The MBR is also represented as a reservoir element with the same elevation as the FSL of MBR. A pump curve derived from a single duty point (data obtained from pump manufacturer) is used to simulate the pump's performance. After obtaining a complete upstream network to the MBR (refer to Supplementary Material data SI-2/Badapokharan_MBR_Pumped network inflow.wtg), steady-state simulation is used to find the MBR inflow.

After obtaining inflow and withdrawal capacities (W1, W2, W3), comparison is made to check the existence of FSC. If the inflow is greater than W1 or W2, conventional PDA based on PDO should be enough to model the performance of WTN. If the inflow is between W2 and W3, FSC exists, and SWC formation is possible between GL and LSL of the MBR. If the inflow is lower than W3, this method is not applicable, as SWC formation takes place below GL of the MBR.

After confirming the existence of FSC, the entire network (Sump–Pump–Pumping Main–MBR–WTN–SRs) is used for obtaining the performance of WTN which is depicted in Figure 5(b) (Refer to Supplementary Material data SI-2/EPS_Badapokharan_Operation as per design_base_model.wtg). Here, MBR and its immediate vertical downstream pipe are represented by the tank element (irregular-shaped tank), as depicted in Figure 3. Representation of SRs, in this case, remains the same as that of the model used for withdrawal. The MBR is a tank that fills from top, i.e. a tank inlet that discharges above the water surface. It can be modeled as a tank's inlet consisting of a pressure sustaining valve followed by a short length of large diameter pipe in EPANET. A detailed procedure for the same has been described by Rossman (2000). In this study, WaterGEMS is used and an equivalent EPANET model is included in Supplementary data SI-2 (Refer to Supplementary Material data SI-2/EPS_Badapokharan_Operation as per design_base_model.net). WaterGEMS can model the top-filling behavior as it has a built-in functionality in the tank element that provides a separate inlet at any tank level. For simulating the performance of the WTN in all stages of supply (i.e. normal, transition, equilibrium), extended period simulation (EPS) is done. To ensure that the simulation covers performance over all supply stages, the FSL is set as the starting boundary condition of operation, i.e. the starting water level in the MBR. The time step of the simulation is kept small so that the performance of the WTN during Stage 2 is captured. Multiple time steps are tried to check the convergence and stability of the results. The time step value for the initial trial is selected based on the withdrawal capacity of W2 and the volume of the vertical portion of the immediate downstream pipe. The initial time step is selected to be lower than the time taken to empty the vertical section of the immediate downstream pipe, considering the constant withdrawal capacity W2. This process serves as a rule of thumb for selecting the time step. Once results converge, the stability of the result for Stage 1 and Stage 3 is established by observing the water level in the MBR and SWC's height with a lower time step.

After PDA-FSC simulation, the design demand for each SR is obtained based on the downstream population, per-capita demand, and the WTN operating hours (CPHEEO 1999). As all flows should match estimated or design demand for equity in practice, the Flow-based Demand Satisfaction Ratio (FDSR) at SRs is defined as the ratio of delivered flow (LPS) to required demand (LPS) at any instant. Similarly, volume-based DSR (VDSR) at SRs is defined as the ratio of delivered volume (L) to required volumetric demand (L) during a time period. FDSR values greater than one imply the SR receiving greater flow than designated demand. Similarly, a FDSR value of less than one is an indication that the SR is receiving less flow. The values of VDSR can be interpreted similarly for volumetric water demand. In this context, ‘Equality’ in WTN means each SR receives same flow or volume of water, whereas ‘Equity’ implies that each SR receive water proportional to their designated demand. FDSR and VDSR can serve as suitable indicators of equity for particular stage or period, respectively. The goal of operation should be minimizing the difference among SRs' satisfaction indicators, thereby improving equity. These indicators can be used to understand the real-life implication of WTN under FSC. Consequently, FDSR or VDSR is used for the performance evaluation of WTN in terms of equitable water distribution.

Since the performance of the flow-starved WTN is highly dependent on the MBR inflow, the impact of the MBR inflow variations on the performance of the WTN is examined. Through this examination, the robustness of the modeling approach is also tested. Drivers for the increase and decrease in inflow are investigated to cover the entire spectrum, and their estimates were found. An increase in inflow can be due to various reasons, such as larger pumps for future-proofing (+20%) and flow increase due to increased water level in the sump (+7%). The major reason for the decrease in inflow (up to −20%) can be a leakage in the main or lower water availability at the source during summertime. Hence ±20% is considered as a possible range of inflow variation, and five scenarios were created within a 20% deviation of base inflow simulation. The five scenarios are

• 1. Scenario 1: Inflow = base inflow

• 2. Scenario 2: Inflow = 0.8 times base inflow

• 3. Scenario 3: Inflow = 0.9 times base inflow

• 4. Scenario 4: Inflow = 1.1 times base inflow

• 5. Scenario 5: Inflow = 1.2 times base inflow

A WTN from Badapokharan, Maharashtra, India, is selected to demonstrate the proposed PDA-FSC modeling approach. The overview and layout of the network are presented in Supplementary Material, Table S1-1 and Figure S1-2, respectively. The node and pipe details are provided in Supplementary Material, Table S1-3 and Table S1-4, respectively (Supplementary data SI-2). The system represents a WTN with MBR as source and SRs as demand points. Figure 5(a) shows the network model configuration used to determine the WTN withdrawal. Figure 5(b) shows the network's layout used for EPS performance evaluation during the three supply stages of the WTN operation.

The present study aims to demonstrate the performance of WTN under poor institutional capacity of a water utility with ineffective demand-side management and operational controls. For such a situation, it is assumed that if consumers receive excess water, they continue to withdraw due to uncertainty of supply, which is one extreme of most cases in India (Burt & Ray 2014; Ghorpade et al. 2021). This leads to the withdrawal of total water received at each SR by consumers. On the supply side of SR, all SRs are considered to withdraw water from the MBR simultaneously, as contemplated in the design. This operational regime is regarded as the suitable scenario to demonstrate the performance of WTN under three supply stages (normal, transitional, and equilibrium) as the downstream system valve setting remains the same and hence the effects of water level variations are easily captured. Although the present study includes only one operational regime, other more realistic operational regimes such as each SR getting closure after receiving total daily demand, can be modeled if the correct control mechanism is implemented.

Mohan & Abhijith (2020) have suggested using the mass balance approach to validate simulation results of flow and pressure-starved conditions. However, such an approach is unnecessary for the PDA-FSC as it is implicitly taken care by the tank element in the modeling. Deployment of irregular shape tank in the model eliminates the need for modeling partial flow present in the vertical portion of the immediate downstream pipe to MBR. Assumptions are made in the conceptual framework of PDA-FSC to improve the ease of modeling. Assumptions I and II are based on the insignificant hydraulic resistance of the vertical portion of the immediate downstream pipe to MBR. The applicability of the PDA-FSC approach is verified by comparing the instantaneous value of withdrawal capacity and the approximation used in modeling. As described in the explanation of Stage 2, only the pressurized section of the immediate downstream pipe (Length equal to L-x in Figure 2) offers hydraulic resistance. However, conventional software cannot modify network length at each time step during simulation. Hence, the present study uses an approximation for pressurized section and considers constant (L-h) length to be pressurized throughout the simulation.

Three estimates are obtained from simulation for the withdrawal capacity of WTN. Withdrawal capacities of WTN at different piezometric heads, represented by HGLs (from GL to FSL of MBR at 0.5-m interval) are found in two extreme scenarios of pressurization (at the extreme value of x, i.e. h and 0).

• 1. Approximation I: where L-x = L-h

• 2. Approximation II: where L-x = L

Figure 6 shows that the estimates of withdrawal capacity (Approximation II) are lower or equal to the withdrawal capacity (instantaneous value), and withdrawal capacity (Approximation I) are consistently higher or equal to the withdrawal capacity (instantaneous value). The difference between the three values of withdrawal capacities is abysmally low at all HGLs. Maximum errors in the estimation of instantaneous withdrawal capacity are +0.85 and −0.17% of instantaneous withdrawal capacity in Approximation I and Approximation II, respectively. Additionally, the instantaneous withdrawal capacity always lies within two approximations. Significant uncertainty of flow and pressure in the water system exist due to uncertainty associated with correct network parameters and demand loading conditions (Bargiela & Hainsworth 1989). Hence, error arising from these approximations are acceptable. The hydraulic model could not be verified experimentally due to inadequate monitoring of operating conditions (flow and pressure) and unpredictable operation schedule at case study location. An additional validation of the PDA-FSC approach for two-reservoir network (represented in Supplementary Material, Figure S1-5) is included in Supplementary data SI-1.

To determine the FDSR, the demand of SRs and flow toward SR are compared. Table 1 presents flows obtained from simulation under various supply stages and design demands. Observations for Stage 2 have not been included because it lasts only for a few minutes. Observations based on FDSR during different supply stages reveal that the FDSR varied widely across the SRs and in various supply stages. As shown in Table 1, several SRs (R13, R23, R20, R18, R17, R14), typically with lower FSL (Supplementary Material, Figure S1-2), receive more flow than their design demand, and few SRs do not receive water (R22, R21, R19, R16, R15). These observations imply that flow distribution between SRs is not equitable. FDSR worsens after reaching Stage 3 as the number of SRs having zero inflow increases from 5 to 7, with a decrease in demand satisfaction for most of the other SRs. VDSR results reveal cumulative effect of entire supply period where several SRs (R13, R23, R20, R18, R17, R14) are overserved whereas the remaining SRs are underserved. The flow control devices are recommended to achieve equitable water distribution at pipe branch leading to overserved SRs. The goal of the WTN is to match supply and demand at the whole network and each individual SR. It is evident from Table 1 that the total WTN withdrawal under PDA is many folds higher than the design demand. Even under normal operation which have higher withdrawal, this increase in withdrawal is not distributed equally among SRs. This inequity demonstrates performance failure and need for the supply-side control to SRs.

The PDA-FSC approach provides an opportunity to assess the effect of inflow variation on the WTN performance. Since insufficient inflow is a causal factor in flow starvation and leads to reduced flow at the downstream SRs, the reduced inflow can further enhance the formation of SWCs in SRs. The SWC formations in SRs often leads to incomplete demand satisfaction at the consumer end.

One of the significant problems identified through the PDA-FSC approach is FSC in the WTN. Three options exist for eliminating FSC (1) increasing MBR capacity, (2) increasing MBR inflow to match withdrawal and (3) reducing SRs' withdrawal. It is evident from the PDA-FSC results (Figure 7) that increasing the MBR capacity can elongate the duration of normal operation. Since normal operation lasts for only 8.64% (1.38 h out of 16 h) of the total duration, a multi-fold increase in capacity is needed to eliminate the SWC, which may not be the best economic option. From sensitivity analysis, increasing inflow by 20% results in a delay in the formation of the SWC; theoretically, increasing inflow can eliminate the appearance of the SWC; however, it may not be economically feasible. Out of the three management options, reducing excessive SR withdrawal seems to be most economically viable option as it involves minor infrastructural changes such as control valves and equipment as opposed to major rehabilitations of infrastructure needed for other options. The option of controlling withdrawal also addresses the problem on the downstream side of MBR and creates a better opportunity for equitable distribution.

The developed PDA-FSC approach helps to understand the implications of poor inlet pressure and demand management during operation. The results of the PDA-FSC approach also help in attributing the cross-dependence of different WSS components (pumping main, MBR, WTN). Generally, the additional capacity of the WTN is considered an advantage for WSS operation. However, the present study demonstrates the downside of unnecessary additional capacity of the WTN and how it can create inequitable water distribution in poorly controlled WTN.

The PDA-FSC approach can incorporate several prevailing operational practices of the IWS. The modeling approach provide an opportunity to reflect on the control system of the WTN and create an avenue for designing the operation schedule according to the topography and demand. Several control equipments (flow control valve, pressure reducing valve, orifice, and manifold) have been advocated for effective pressure management, i.e. to match supply and demand at both network and individual SR levels. This modeling approach can assess the control equipment's effectiveness in achieving the desired goal. The PDA-FSC approach can be used as a management tool to verify or devise operation schedules for different inflow scenarios based on prevailing operation circumstances such as operator-based operation and valve throttling.

The study demonstrates the importance of operational control based on hydraulics in this system. It may thus lead to a policy focus on operation instead of having bigger pipes through public investment in gravity-fed WTN. The present study shows that achieving equitable supply is related to planning as well as the operation of the network. The PDA-FSC modeling approach used in the study is easy to replicate with a low learning curve. This is vital for developing countries where comprehensive modeling is seldom used for designing operation schedules.

The present study highlights the limitations of the existing hydraulic modeling approaches for flow-starved conditions in water networks, a common phenomenon in the IWS system prevailing in most developing countries. The modeling approach developed in this study provides a better and easier way to simulate flow-starved conditions in the WTN. The methodology developed has been implemented on a WTN in India. The results from this modeling approach show that redundant withdrawal capacity in an inadequately controlled WTN can lead to inequitable water distribution indicating performance failure. The study further demonstrates the impact of a flow-starved network and its inability to fulfill the downstream service reservoir's demands. The results from the developed modeling approach can be used to derive better operational schedule for IWS.

Authors acknowledge the funding received under the sponsored research project from the Department of Science and Technology (DST), India – Water Technology Initiative (WTI) – 2017 DST/TM/WTI/2K17/39. The authors acknowledge all the cooperation and support provided by Palghar Zilla Parishad, Palghar, Maharashtra, India. The authors also acknowledge editor and two anonymous reviewers for insightful and comprehensive suggestions.

All relevant data are available from an online repository or repositories (https://dx.doi.org/10.5281/zenodo.7162154).

The authors declare there is no conflict.