Analyzing the role of consumer behavior in coping with intermittent supply in water distribution systems

Water distribution systems (WDS) are designed to deliver consumers adequate drinking water, meeting necessary quality criteria. Although WDS of the developing regions of the globe is designed initially to provide water continuously per specified standards, most operate inconsistently. Specifically, in South Asian countries, particularly India, most WDS provide water to consumers for less than 24 h a day (Charalambous & Laspidou 2017; Ghorpade et al. 2021). These systems are termed intermittent water supply (IWS) systems. Kumpel & Nelson (2016) estimated that an alarming (∼370 million) population in India relies on IWS systems to meet their water demands.

Several causes, such as population growth, urbanization, increased occurrence of extreme weather events, inadequate accessibility of freshwater, inadequate sizing of distribution network, and water losses, can be attributed to the IWS operation of WDS (Loubser et al. 2020). Ghorpade et al. (2021) reasoned that once any WDS falls into the ‘vicious’ trap of IWS operation, it might become challenging to climb up because of the intricate relationships between several drivers (design, operation, maintenance, institutional capacity, consumer satisfaction, revenue generation, etc.) that govern its performance.

From a detailed inspection, consumer behavior in IWS (irregular and unreliable) systems is distinguishable into Types A and B. Suppose the expected static pressures at the consumer (ferrule) points are less, or the supply is essentially uncertain, then the consumers tend to practice Type A behavior (Figure 1(a)) – store water within underground (sumps) storage tanks and later pump it into overhead (rooftop) tanks for use. On the contrary, if the static pressures at the consumer points are high enough to deliver the water to the rooftop tanks directly, the consumers practice Type B behavior (Figure 1(b)) – avoid sump storage and carry out successive pumping. Intriguingly, against the common assumption that the demand satisfaction in residential areas is minimal under IWS, practicing either of these two types of behavior is anticipated to allow the consumers to cope with the noncontinuous water availability and meet their demands more suitably (Andey & Kelkar 2009; Randeniya et al. 2022).

Several attempts toward modeling the hydraulics within IWS systems can be seen in the literature (Liou & Hunt 1996; Bourdarias et al. 2008; Marchis et al. 2010; Mohapatra et al. 2014; Leib et al. 2016; Campisano et al. 2018; Taylor et al. 2019; Mohan & Abhijith 2020; Randeniya et al. 2022; Suribabu et al. 2022; Weston et al. 2022; Sinha et al. 2023). However, consumer behavior in IWS systems has not been given the needed importance barring a few studies (Mohan & Abhijith 2020; Randeniya et al. 2022; Suribabu et al. 2022).

Mohan & Abhijith (2020) represented the outflows (water withdrawal) at the consumer points of highly (unreliable) intermittent WDS as uncontrolled orifice-based demand. They employed the uncontrolled-type head–discharge relationship suggested by Reddy & Elango (1989) and a noniterative time-stepping method using backward differencing to analyze the hydraulic characteristics of a draining IWS system. However, Mohan & Abhijith (2020) did not emphasize the water storage constraints (volume of sumps and/or rooftop tanks) and inadvertently associated the consumer demands with the leakage losses. Furthermore, their model also showed limitations in conserving mass and momentum and accurately estimating the draining time (Meyer & Ahadzadeh 2021).

Recently, Randeniya et al. (2022) and Suribabu et al. (2022) proposed distinct yet similar, straightforward methodologies to simulate the hydraulic behavior of the domestic storage tanks and analyzed the performance of IWS systems by applying the modeling environment of EPANET 2.2 (Rossman et al. 2020). On the basis of a comprehensive social survey conducted in Kakkapalliya, Sri Lanka, Randeniya et al. (2022) reported the attributes of domestic storage tanks and consumer experience in a typical IWS system. After attaching hydraulically equivalent domestic tanks to demand nodes, they constructed the hydraulic model of an intermittently operated WDS in EPANET 2.2 and analyzed the equity and performance under varying IWS operation schedules. However, the approach by Randeniya et al. (2022) was limited only to Type B consumer behavior. Such behavior by the IWS system consumers necessitates the availability of optimal or near-to-optimal pressure heads at the consumer points of the distribution network on a sustainable basis, which is rare in many real-world IWS systems (Cook et al. 2016; Strijdom et al. 2017; Ghorpade et al. 2021; Satpathy & Jha 2022). On the contrary, although the approach by Suribabu et al. (2022) was related to Type A consumer behavior, their study was limited to understanding the direct effects of filling the underground sump tanks (via uncontrolled withdrawal of water) on emptying the source tank. The impact of actual water consumption (meeting of demands) and subsequent emptying and filling (via pumping) of rooftop tanks on governing the water withdrawal at the consumer points and the overall behavior of a WDS during the IWS practice was not examined.

This article presents a pressure-dependent volume-driven approach and a novel simulation tool (named EPyT-IWS) to overcome the aforementioned limitations of the existing state of the art. The proposed tool allows us to simulate consumer behavior in IWS systems realistically and comprehend whether this enables the consumers to cope with the restricted water availability. The suitability of EPyT-IWS for conducting the extended-period analysis was demonstrated by applying it to a typical WDS, delivering water to 2,650 consumers (530 households) initially proposed by Suribabu et al. (2022), with the essential characteristics of a representative IWS system. The results corroborate the ability of the proposed tool to mimic the domestic consumer response to intermittency closest to reality and to perceive the knowledge of how far away the WDS is from an equitable operation.

The artificial string comprises a tank, three nodes, three pipes, and a flow control valve (FCV), and the imaginary string constitutes a tank, a node, and a connecting pipe. Specific nodes and pipes of the artificial strings are dummies. Dummy nodes have zero base demand and the same elevation as the consumer node, except artificial node-3, which has the same elevation as the household. All dummy pipes have a negligible length. The constituents of the artificial and imaginary strings for Type A and Type B consumer behavior are detailed in Table 1.

In typical (irregular and unreliable) IWS systems, the house connection constitutes a minimum of three 90° Elbows, a stop valve, and a pipe that discharges freely into a storage tank, all connected in series (Suribabu et al. 2022). The minor head losses within the house connection would be significant, and the minor loss coefficient (⁠⁠) would add up to ∼5.90 (Bhave 1991). Therefore, the flow rate through the house connection pipe at every consumer node would be directed by the static pressure head at the ferrule point and the friction losses within the house connection arrangement.

At each time step, the EPyT-IWS algorithm computes the value for every consumer node j and updates the flow setting of the artificial FCV within the artificial string. The flow rate through the artificial pipe 1 and artificial pipe 2 will equal the FCV setting if the artificial tank level is less than its depth. Else, the flow rates would be zero.

The value would govern the inflow to the artificial tank connected to every consumer node. For Type B consumer behavior, the outflow from the artificial tank would be administered entirely by the time-series demand value of artificial node 3. However, for Type A consumer behavior, the imaginary string components need to be accounted for while estimating the artificial tank outflow.

Typically, under Type A consumer behavior in IWS systems, pumps are operated to fill the rooftop tanks and to meet water demands. EPyT-IWS avoids physically adding the pumps to the hydraulic model but indirectly simulates the pump operation using the in-built hydraulic solver. The hydraulic solver takes an adequate pumping rate (⁠⁠) for every consumer node to simulate the imaginary tank filling and artificial tank emptying. The EPyT-IWS algorithm assumes that the demands at consumer nodes are met at any time step t if the water level within the imaginary tank is above a certain value (minimum required water depth, in m). Similarly, the algorithm assumes no pumping occurs until the water level within the imaginary tank is below a specific depth (threshold depth for pump activation, in m). The algorithm considers that even when the imaginary tank level is below the value for pump activation, the pump cannot operate until the water level within the artificial tank is above a critical value (critical water depth, in m). Once the pump switches on, the artificial tank is emptied, and the imaginary tank filling would continue until either of the two conditions occurs – the artificial tank level dropping below value or the imaginary tank reaching full.

For performing the extended-period hydraulic simulation using EPyT-IWS, appropriate hydraulic time step (⁠ in s) and simulation time duration (⁠ in days) values must be specified as input. Special care needs to be given while specifying and values. Typically, 1,800 or 3,600 s is selected as the value for hydraulic analysis using EPANET 2.2. Nonetheless, the value of this scale might make it numerically challenging and time consuming for the EPANET solver to derive numerical solutions due to the necessity of simulating many tanks additionally incorporated into the hydraulic model by the EPyT-IWS algorithm (Randeniya et al. 2022). For this reason, a relatively shorter time step in the scale of 60 s is deemed adequate for analysis with EPyT-IWS. Concerning ⁠, a longer duration (in the range of weeks or preferably months) should be selected to fully overcome the effects of the initial conditions (levels of the underground sump tank and/or overhead rooftop tank) on the EPyT-IWS outcomes. Besides and ⁠, the initial state of the components of the artificial strings and imaginary strings attached to the system nodes of the WDS needs to be indicated. By default, EPyT-IWS takes the initial water level within the artificial and imaginary tanks as 0 and 0.5 m, respectively. Also, the default value at every consumer node j is selected as five times its base demand value. The values for ⁠, ⁠, and are set as 0.1, 0.2, and 0.5 m by default.

The elements of the test network include a reservoir (a proxy for a water treatment plant), a pump, an elevated storage tank, 22 nodes, and 22 pipes. The reservoir was assumed to have an elevation of 95 m and was located far from the storage tank. The pumping operation to fill the storage tank was assumed to occur at a fixed pumping rate of 150 litres per minute (LPM). The storage tank has an elevation of 107.5 m, and all consumer nodes have 100 m elevation values. The Hazen–Williams roughness coefficient value for all 22 pipes was assumed as 130. The storage tank outlet was always assumed to be open, causing the tank filling due to pump operation and its draining due to withdrawal at consumer points to happen concurrently. The details regarding the number of house supply connections corresponding to each consumer node, its average daily demand, and the length and diameter of pipes are specified in Table 2.

Different test cases (Table 3) were selected by varying the pump operating schedule to simulate the consumer behavior in IWS systems. For all test cases, the time window was kept at 120 days to evaluate the ability of consumers to cope with the restricted and uncertain water supply practices over the long run of an IWS system. For 9 (out of 11) cases, except Cases 5a and 7a (Table 3), the consumer behavior in every system node was assumed as Type A. Cases 1–3 correspond to consistent operation of pumps for only 1 (from 6 AM to 7 AM), 3 (from 6 AM to 9 AM), and 6 h (from 6 AM to 12 PM) every day, respectively. Cases 4–7 correspond to unreliable intermittency. For all these four cases, the water supply practice for the first 60 (out of 120) days was kept alike: 3 h pumping (from 6 AM to 9 AM) for the first 30 days; no pumping and no water supply during 31–45 days; and restarting of pumping on the 46th day and only 1 h pumping (from 6 AM to 7 AM) for the next 15 days. For the subsequent 60 days, 3 h pumping was adopted for Cases 4–6 by varying the clock time of pumping (Table 3). For Case 7, 13 h pumping (from 11 AM to 12 AM) was adopted. Cases 5a and 7a correspond to the case with the consumers of the most upstream node (Node 1) adopting Type B consumer behavior, and the remaining consumers aggregated in the rest of the nodes continuing with Type A behavior. The diurnal variations in the household water consumption at every node under all the cases analyzed were represented using a demand pattern (Table 4) that was evolved from scrutinizing the time-series water demand data generated by Jethoo & Poonia (2011) for an Indian city (Jaipur) based on the socioeconomic data. The consequence of alternate water sources such as groundwater, tankers, bottled water, and community water filters was entirely ignored for modeling purposes.

For every consumer node j, a desirable RW value (⁠⁠) can be evolved by associating the number of house supply connections aggregated at the node j with the total number of house supply connections served by the WDS. For example, for Node 1, the desirable RW value would be 0.047 (i.e., 25/530). If the value for every consumer node is equal to ⁠, then the IWS practice adopted can be deemed equitable over time T.

The extended-period hydraulic analysis of the test network under varying cases (Table 3) was conducted using EPyT-IWS by selecting the as 60 s and as 120 days. The default values were selected for initial water depth within the artificial and imaginary tanks and for ⁠. The other input parameter values selected are detailed in Table 5. The simulations were run using a desktop computer with Intel^® Core™ i7-6700 CPU @ 3.40 GHz processor, 16.0 GB installed memory, and a 64-bit operating system. It may be noted that due to data paucity, the design of the consumer system (dimensions of the tanks, characteristics of the house connection network, domestic pumping characteristics) was arrived at depending on the restricted field knowledge of the consumer behavior in IWS systems. Several assumptions have also been made to oversimplify the components of the artificial and imaginary strings.

Although all six nodes were at the same elevation, the filling and emptying dynamics of the underground sump tanks were found to be entirely distinct for them under the three cases, apparently due to the differences in their specific positions relative to the storage tank. Under Case 1, EPyT-IWS predicted that the water level within the underground sump tank in Node 1 would reach a maximum within ∼50 min. However, during the same time, the water levels within the sump tank in Node 3 were predicted to be only 0.63 m, and that in Node 6 were only 0.04 m. The water level profiles under Case 1 predicted for Nodes 1 and 3 differed distinctly from that for Nodes 6, 13, 16, and 22 (Figure 6(a)). The tank water level was found to attain a pseudo-equilibrium around the value for these four nodes. This is because the EPyT-IWS algorithm envisages the consumers to start the water pumping as soon as the tank level rises above the limit to overcome the water shortage caused by the emptying of rooftop tanks (Figure 7). Intriguingly, the underground sump tank level in Node 1 was never predicted to reach the value, while the same in Node 3 oscillated between 0.5 and 0.6 m during the pumping hours (Figure 6(a)).

By increasing the pumping hours from 1 to 3 h, the minimum depth to which the water level drops within the sump tank in Node 1 increased from 0.96 to 1.22 m, and the highest level reached within Node 3 rose from 0.88 to 1.55 m. An entirely contracting sump tank level profile compared to Case 1 with a mean value of 1.01 m was predicted for Node 3. For the other four nodes, an escalation of the slopes of the initial rising part of the profile curves was virtually evident (Figure 6(b)). For instance, concerning the sump tank in Node 22, the time required to attain the pseudo-equilibrium decreased from 24.3 days in Case 1 to 9.4 days in Case 2. Similar impacts as Case 2 were also predicted for Case 3 with 6 h regular pumping (Figure 6(c)). The minimum depth to which the water level drops within the sump tank in Node 1 increased further to 1.66 m, and the tank in Node 3 was predicted to get full. From a vague look, the sump tank profiles in Node 3 and Node 13 under Case 3 resembled those predicted in Node 1 and Node 3, respectively, in Case 1 (Figure 6(a) and 6(c)). Concerning Node 22, the time to attain pseudo-equilibrium decreased further to 4.4 days. An increase in the slopes of the early rising portion of the profile curves corresponding to Nodes 6, 13, 16, and 22 was also predicted in Case 3 compared to Case 1.

Concurrently, analyzing the EPyT-IWS predictions of rooftop tank levels (Figure 7) and demand deficit values (Figure 8) under the three different pumping operation cases delivers a perfect picture of the ability of Type A consumer behavior to cope with the IWS practices. As shown in Figure 7, even in Case 1 with only 1 h daily pumping, the household consumers aggregated in Node 1 were predicted to meet their water demands continually without any demand deficit. However, the consumers aggregated in none of the other nodes could meet their demands without deficit under Case 1 (Figure 7(a)). As was predicted for overhead tanks (Figure 6), the rooftop tank water levels in Nodes 6, 13, 16, and 22 under Case 1 attained a pseudo-equilibrium around the value.

Significant changes in the demand deficit plots became apparent with increased pumping duration (Figure 8). The 3 h pumping practice in Case 2 was found adequate to meet the water demands of all 29 households aggregated in Node 3 incessantly. While the 6 h pumping practice (Case 3) was insufficient to satisfy the water demands of consumers aggregated in every node, the results signified that Type A behavior can facilitate the consumers to lessen the demand deficits by a significant degree (Figure 8(c)). The mean demand deficit values for Nodes 6 and 13 decreased from 99.2 and 97.5% to 70.2 and 30.9%, respectively, after changing the pumping duration from 1 to 6 h. Nonetheless, a similar impact was not predicted for Nodes 16 and 22. The mean deficit values for these nodes over 120 days decreased to 98.3 and 97.9% from a shared value of 99.4%. These disparities, over again, emphasize the importance of the consumer positions relative to the water source in coping with the IWS practices.

The performance metric values estimated under the three cases also verify the previous inferences. As seen in Figure 9(a), the time-averaged = 1 estimated for Node 1 over 120 days under the three cases considered suggests that the household consumers in Node 1 can meet their water demands like that in a continuous WDS by adopting Type A behavior. However, in Case 1, the DSR value estimated for all other nodes, including Node 2, the second-most upstream node, was found to be less than one. The DSR value of 0.446 estimated for Node 2 implied that the 47 households aggregated in this node could fully meet the water demands only ∼45% of the time considered. The increment in the pumping hours induced uneven impacts in the DSR values estimated for the 22 nodes. Although a pumping duration of 6 h was found appropriate in fully meeting the demands in five nodes (Nodes 1, 2, 3, 7, and 10) and in meeting water demands for >90% of the time considered in four nodes (Nodes 4, 5, 8, and 11), it was still not enough for consumers in 44 households, mainly aggregated in the three downstream nodes (Nodes 16, 21, and 22), to fulfill demands for >0.05% of the total analyzed time. Furthermore, the variations in the estimated DSR values over the 120 days window (Figure 9(c)) disclosed that even though the typical (Type A) consumer behavior facilitates an improved demand satisfaction, it cannot entirely avoid the unreliability in water availability associated with IWS practices.

The RW values estimated under the three cases revealed more specifics of equity concerning the IWS operation practices. In Case 1, in 5 of 22 nodes (Nodes 1, 2, 3, 7, and 10), the time-averaged RW values over the 120 days were more significant than the corresponding desirable values (Figure 9(b)). This signifies that the consumers of 154 households aggregated in these five nodes draw more water from the WDS during the supply hours than the remaining 376 households. However, surprisingly, although ∼36% of the total water supplied over the 120 days (four times the desirable) was predicted to be drawn by the households aggregated in Node 2, the value was estimated to be only 0.446. Interestingly, by drawing only ∼21% of the total water supplied over the 120 days in Case 2, the consumers in Node 2 were predicted to be able to fulfill their water demands. This signifies that the RW value only gives a certain sense of the relative distribution of the delivered water among the different nodes and sheds no light on the delivered water volume. As well, looking specifically at Node 1, in Case 1, the 25 aggregated households fully met their demands by drawing ∼13% less than that drawn by consumers of Node 2. Although the time-averaged RW values for both Nodes 1 and 2 were more than four times their respective values, the consumers aggregated in Node 2 could not fill their underground sump tanks, and the tank level was predicted to oscillate between 0.75 and 0.5 m. Due to this, they could not pump water from the underground sump tank to the rooftop tanks most of the time. Thus, it can be implied that the discontinuity of water availability can impede consumers' capacity to fulfill their demands.

As expected, increasing the pumping hours has positively impacted reducing the disparities between the calculated and expected time-averaged RW values (Figure 9(b)). However, the effects were pronounced only at the nodes closer to the storage tank, and at downstream nodes, particularly in Nodes 16, 21, and 22, the impacts were relatively nominal. For instance, between Cases 1 and 3, the time-averaged RW value calculated for Node 22 only increased from 0.002 to 0.003, while that Node 1 decreased from 0.231 and almost reached the desirable value. The 120-day variations in the RW values also verify the aforementioned inference (Figure 9(d)). Altogether, the EPyT-IWS outcomes corresponding to Cases 1–3 imply the importance of simultaneously examining the performance metrics (DSR and RW) in arriving at the optimal pumping schedule for any IWS system.

Interestingly, substantial disparities were evident in the time-averaged DSR values estimated for the 22 nodes over 120 days under Cases 4–7 (Figure 10(a)). Although such differences for the time-averaged DSR under Case 7 from the rest were foreseeable, the apparent dissimilarities between the values estimated under Cases 4–6 were unexpected. For Node 1, as anticipated, the value estimated (0.931) was the same under all four cases of irregular pumping. However, the value was less than the maximum estimated under Cases 1–3 (Figure 9(a)). This signifies that the amount of water that can be stored within the underground sump tank is insufficient to prevail over the nonavailability of water caused by the shutting down of pumps for 15 days (Figure 10(c)). For Nodes 2 and 3, the time-averaged DSR values were almost equal under Cases 4–6. However, from a closer look, the DSR values under Case 4 were slightly higher than the other two and were equal to that estimated in Case 7 for Nodes 2 and 3 (0.828 and 0.794, respectively). Between the values estimated in Cases 5 and 6, the values in Case 5 were marginally high.

However, for the remaining 13 nodes, the demand satisfaction of consumers in Case 5 was higher than that in Cases 4 and 7. These differences were predicted to be more profound in the downstream nodes. Specifically, looking at Nodes 6, 16, and 22 (Figure 8(a) and 8(c)), operating the pump from 9 PM to 12 AM for the last 60 days (Case 5) induced a substantial rise in the ability of consumers to fulfill their demands compared to operating the pumps from 11 AM to 2 PM (Case 4) and 4 PM to 7 PM (Case 6). The estimated time-averaged DSR values for these three nodes were 0.266, 0.157, and 0.185 in Case 5, while values in Cases 4 and 7 were significantly lower (0.033, 0.009, and 0.010 and 0.039, 0.009, and 0.011, respectively). The reason for the same can be attributed to the diurnal variations in water demand. In Case 5, the pump was operated during the period when the water demands of the consumers were relatively minimal (Table 4). Due to this reason, the consumers aggregated in the upstream nodes (mainly Nodes 1–3), who experience a somewhat uninterrupted demand satisfaction, seldom operate their pumps during the late-night pumping hours. Hence, outflows to the underground sump tanks of households aggregated in the upstream nodes were predicted not to occur for extended periods in Case 5 compared to that in Cases 4 and 7. This led to more volume of delivered water becoming available to withdraw for the underground sump tanks aggregated in the downstream parts of the WDS, eventually enabling them to fill up their rooftop tanks more frequently and fulfill their demands. In Case 7, by increasing the pumping hours from 3 to 13 h for the last 60 days, the time-averaged DSR values for all the nodes of the test network were predicted to attain the maximum value of 1 (Figure 10(c)). This implies that adopting the typical Type A behavior enables the consumers to enjoy uninterrupted demand satisfaction even under the IWS operation of WDS if the water supply is reasonably prolonged and regular (predictable intermittency).

Although differences in the time-averaged RW values estimated under Cases 4–6 for the 22 nodes are not as significant as those for DSR, noticeable disparities become apparent from a thorough look (Figure 10(b)). Like what we have observed in Cases 1–3, the time-averaged RW values estimated in Cases 4–6 in the upstream nodes were larger, and the time-averaged RW values corresponding to the downstream nodes were significantly lower than the respective desirable values (Figures 9(b) and 10(b)). Although Case 5 can be deemed the most effective pumping schedule among the three irregular pumping cases (Cases 4–6) from the DSR values, it is challenging to make such a clear-cut inference based on the RW values. However, based on the results, it is hard to disregard the growth in the equity of the water supply introduced by the increase in the pumping hours (Figure 10(b) and 10(d)). For Nodes 1, 3, 6, 13, 16, and 22, with desirable values of 0.047, 0.055, 0.034, 0.060, 0.028, and 0.043, respectively, the average RW values corresponding to the final 60 days in Case 5 were 0.071, 0.160, 0.009, 0.029, 0.002, and 0.005. By increasing the water supply duration, the average RW values over the final 60 days improved to 0.040, 0.073, 0.035, 0.066, 0.026, and 0.039, respectively, and became very close to the desirable values (Figure 10(d)). Thus, it can be implied that Type A consumer behavior stimulates a virtually equitable water distribution among consumers under a predictable and regular water supply routine.

By switching from Type A behavior to Type B, the value for the consumers aggregated in Node 1 declined substantially from 0.931 (Figure 10(a)) to 0.007 (Figure 11(a)). This specified the ineptness to move to Type B behavior if the pumping operation continues the irregular and unreliable routine specified as Case 5. The results from EPyT-IWS simulations showed that the storage tank, acting as the source to the consumers of the test network, is prevented from becoming full under Case 5a due to the constant outflows to the underground sump tanks during the periods of water availability. Therefore, the static pressure at the ferrule point(s) of Node 1 never reaches the needed value to deliver water directly to the rooftop tanks located at a higher elevation (7 m above the ferrule point). The rooftop tank in Node 1 was predicted to fill for only 35 min (from 0 to 1.05 m) from the start of the simulation, concurrently with the storage tank emptying (from the initial level of 2.85 m). As a result, the consumers aggregated in this node could only satisfy their demands for ∼21 h (over 120 days) by withdrawing water from the rooftop tank till it fell to the level corresponding to (Figure 11(c)). A noticeable decrease in the time-averaged RW values for Node 1 (from 0.126 under Case 5 to 0.013 under Case 5a) also indicated that adopting Type B behavior by the consumers aggregated in Node 1 is not worthwhile under the IWS practice considered (Figures 10(b) and 11(b)).

The inability of households in Node 1 to satisfy demands brought upon by switching over to Type B behavior did not reflect considerably on the remaining 505 households aggregated in the remaining 21 nodes (Figures 10(a) and 11(a)). No noticeable change in the DSR values in Case 5a compared to that in Case 5 was predicted in the downstream nodes (Figures 10(c) and 11(c)). At the same time, a visible increase was evident in two intermediate nodes – Nodes 7 and 10. The time-averaged DSR values in these two nodes increased to 0.693 and 0.677 from 0.621 and 0.610, respectively. Unlike the DSR values, the RW values predicted for the intermediate nodes – specifically Nodes 4, 7, 8, 10, 11, 13, 15, 17, and 18 – displayed an obvious rise between Cases 5 and 5a. The time-averaged RW values for the aforementioned nine nodes increased from 0.063, 0.056, 0.053, 0.143, 0.043, 0.028, 0.016, 0.034, and 0.019 to 0.074, 0.070, 0.064, 0.169, 0.053, 0.034, 0.020, 0.043, and 0.022, respectively (Figures 10(b) and 11(b)).

Switching from Type A to Type B behavior for Node 1 was predicted to be more viable if the IWS practice followed the pumping routine specified by Case 7. The predicted increase in the average DSR values for Node 1 over the last 60 days (from 0 in Case 5a to 0.748 in Case 7a) verifies the above statement (Figure 11(c)). The viability of switching from Type A to Type B was also reflected in the RW values (Figure 11(b) and 11(d)). However, switching to Type B consumer behavior in Node 1 impeded the consumers from achieving complete demand satisfaction (Figure 11). Besides, as predicted (Figure 10), complete demand satisfaction and virtually equitable water distribution can likely be attained by adopting Type A behavior in every node. Considering the aforementioned facts, altering the consumer behavior in Node 1 was deemed unreasonable for the test network under the cases analyzed.

The sensitivity analysis was performed to ascertain how far the assumptions in developing the hydraulic model impact the EPyT-IWS outcomes. In this direction, a deterministic sensitivity analysis procedure was adopted, and EPyT-IWS simulations were performed by varying the value of one uncertain parameter at a time. The parameters selected include the maximum capacity and maximum depth of the underground sump and overhead rooftop tanks, the diameter values of the house connection pipe (connecting ferrule point and storage tank), and the household pumping rate (⁠⁠). The default values for the maximum capacity of the underground sump tank and overhead rooftop tank were selected as 2 and 1 m³, respectively. The default values selected for the maximum depth of the underground sump tank, the maximum depth of the overhead rooftop tank, and the diameter of the house connection pipe were 2.5, 2, and 19.05 mm, respectively. For ⁠, the default value was five times the base demand value at every node. For the test network, EPyT-IWS simulation was performed by selecting the pumping routine specified in Case 1 using the default values with = 60 s and = 120 days. The results obtained were reported as the base values.

The results specify that out of the six parameters, the EPyT-IWS outcomes are most sensitive to the diameter value selected for the house connection pipes. Notable differences between the time-averaged DSR and RW over 120 days were evident in all 22 nodes from varying the house connection pipe diameter, contrasting that distinguishable time-averaged DSR and RW value variations were apparent only in the upstream node, primarily Nodes 1 and 2, for the rest five parameters (Figure 12). The high impact of the diameter value of the house connection pipe in controlling the EPyT-IWS outcomes is attributed to the implication of and values on the pressure-dependent value estimation for every consumer node j (based on Equation (5) or (10)). The second-most artificial string component parameter that is most influential to the EPyT-IWS outcomes was the capacity of the underground sump tanks. This value establishes the total volume of water that the consumers can collect during water supply hours for subsequent pumping and usage, influencing the demand satisfaction and the total withdrawal volume at the consumer points. The household pumping rate was found to be the imaginary string component parameter that significantly controls the time-averaged DSR and RW estimations because of its influence on governing the emptying and filling dynamics of the sump tank and rooftop tanks, respectively. The other three parameters, such as sump tank depth, rooftop tank capacity, and rooftop tank depth, were not quite prominent in administering EPyT-IWS outcomes under the IWS operation scenario specified as Case 1.

The findings corroborate the potential of the EPyT-IWS tool to induce significant contributions toward planning, designing, and operating WDS with IWS characteristics. However, numerous assumptions were made to oversimplify consumer behavior in IWS systems and to develop the hydraulic model, which is yet to be calibrated and verified using real-world data. Leakage is significant in WDS, particularly those operated intermittently (Ghorpade et al. 2021). In addition, considerable volumes of water are wasted within the IWS system consumer premises owing to the overflow of domestic storage tanks during water availability or due to these tank emptying preceding the supply course to avail fresh water (Mastaller & Klingel 2018). Nevertheless, these losses were not accounted for in the present study. During IWS operation, air entrapment may occur in pipes, may amplify the frictional resistance to flow (Kumpel & Nelson 2014), and may cause the flow to get transitioned between free surface and pressurized flows (Kerger et al. 2012; Leib et al. 2016). These complex dynamical features of flows within WDS during IWS operation were overlooked in the EPyT-IWS development. The future directions involve advancing EPyT-IWS toward simulating transient free-surface and pressurized flows and modeling leakage losses.

This article presented the development, functioning, and application of a pressure-dependent volume-driven hydraulic analysis approach and a novel simulation tool (named EPyT-IWS) for WDS toward accounting for characteristic consumer behavior in understanding the performance of IWS systems more realistically. The following specific inferences were drawn after interpreting the EPyT-IWS simulation outcomes obtained by applying it to a representative IWS system:

• Type A consumer behavior (storing water within underground sump tanks and later pumping it into overhead rooftop tanks for use) enhances the capability of consumers of IWS systems to enjoy improved demand satisfaction even under a limited water supply.

• The positioning of consumers relative to the source tank is an essential factor in controlling their ability to withdraw water from WDS to their respective storage tanks.

• Barring the water supply duration, the clock time of water availability in IWS systems can significantly influence consumers' ability to fill their storage tanks and meet water demands.

• The sensitivity analysis results established the inevitability of careful selection of the parameter values concerning the additional artificial and imaginary string connected to every node in the hydraulic model on the reliability of EPyT-IWS predictions.

This research was supported by a grant from the Ministry of Science & Technology of the State of Israel and Federal Ministry of Education and Research (BMBF), Germany.

All authors contributed to the study conception and design. Model development and analysis were performed by G. R. A. The first draft of the manuscript was written by G. R. A., and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.