1 Introduction

The importance of water for domestic consumption has been long recognised, to the point that the availability of water resources is nowadays acknowledged as both a global development goal and a human right (Jeandron et al. 2019; Hope and Ballon 2019). However, population growth and urbanisation are increasing the number of users and uses of water, making water resources scarcer, and laying the basis for main future challenges within the framework of water resources management in several countries across the world (Siddiquee and Ahamed 2020; Zanfei et al. 2022; Cassiolato et al. 2024; Santamarta et al. 2024;). In this context, understanding the characteristics of water consumption – along with its variations over space and time and their driving factors – is the basis for providing detailed information for decision-makers and authorities (Li and Song 2023; Morain et al. 2024; Zarreh et al. 2024).

Within this framework, end-use analysis allows the identification of the domestic water fixture(s) consuming water at a given time, thus providing useful information on the characteristics of water consumption at the appliance level (Mazzoni et al. 2023; Aliewi et al. 2024). Therefore, the knowledge of consumers’ behaviours at the level of individual end uses can play a crucial role in the development of water-reuse strategies (Dixon et al. 1999; Quon and Jiang 2023) or strategies aimed at increasing consciousness and awareness of water consumption (Romano et al. 2014; Liu et al. 2016; Stewart et al. 2018) and raising users’ sensitivity to the issue of water saving (Luciani et al. 2018; Cominola et al. 2021).

In some cases, the end uses of water can be directly explored by means of modern metering technologies (Savic et al. 2014) through the installation of smart meters on all the domestic water fixtures. However, this method is often time-consuming and expensive, or may be unfeasible for its intrusiveness (Mazzoni et al. 2024). The limitations raised from the direct monitoring of individual end uses have led to the development of non-intrusive techniques allowing the segmentation (disaggregation) of the household-level water-consumption time series into the contribution related to each water-use event, and the subsequent labelling (classification) of each event to the respective end-use category (Cominola et al. 2017; Clifford et al. 2018). End-use disaggregation and classification are generally feasible only if an adequate system including at least one meter installed at the household inlet point – and paired with a tool for data logging – is available. Most of the automated methods developed for this aim rely on flow measurements (Mayer et al. 1999; Fontdecaba et al. 2013; Nguyen et al. 2013; Bethke et al. 2021) gathered at a sufficiently fine resolution (e.g., from 1 s to 1 min) (Heydari et al. 2022) and optionally paired with other data, e.g., electricity, noise, or vibration (Srinivasan et al. 2011; Vitter and Webber 2018). However, the obtainment of fine-resolution water-consumption data could be difficult due to practical limitations. On the one hand, most of the commercial smart flow meters are in-line instruments and therefore may require an intrusive installation phase. On the other hand, alternative solutions (e.g., clamp-on meters) may have strict requirements concerning the material, the diameter, and the layout of the pipeline making up the domestic service line.

In this regard, the installation of pressure sensors may be preferred because they are generally less expensive than flow meters and easier to install (Soldevila et al. 2019; Marzola et al. 2022; Shao et al. 2024). The information provided by pressure data is generally easy to read and can be representative of the local pressure stress due to changes in boundary conditions and users’ activity in the WDN considered (Marsili et al. 2021, 2022). In light of that, methods for end-use analysis relying on pressure data (Froehlich et al. 2009, 2011) have been developed as an alternative to those exploiting flow data. In greater detail, Froehlich et al. (2009, 2011) developed HydroSense®, a software based on pressure data collected with a super-fine (1 kHz) temporal resolution. In fact, since domestic water-use events produce pressure drops in the household plumbing system (Marsili et al. 2023), opening and closure manoeuvres of fixture valves can be detected based on the search of time instants with abrupt pressure variations. Valve opening and closure events are then labelled to individual water fixtures based on transient signatures and a data-driven, hierarchical classifier.

Despite the high accuracy achieved in end-use disaggregation and classification, the method proposed by Froehlich et al. (2009, 2011) is poorly transferrable due to some limitations. First, the use of a single pressure sensor does not allow discriminating between pressure drops caused by in-house and external end-use events (Morrison et al. 2015). Second, an intrusive calibration phase requiring the use of a large number of pressure sensors is needed to calibrate the software (Cominola et al. 2015). Third, the authors did not refer to situations where the pressure in the water distribution network is not stable (i.e., variations in the network pressure) and thus did not discuss the possible effects of these variations on the method performance and accuracy (Morrison et al. 2015). Lastly, the super-fine temporal resolution needed for end-use identification leads to a very large amount of data to be managed.

In light of the scarcity of pressure-based methods for the analysis of individual water-consumption events – and given the limitations affecting those available in the literature – the following research question is addressed in this paper: is it possible to obtain an affordable water-consumption time series on which to perform event characterization by exploiting only pressure data? To provide an answer, this study proposes a pragmatic, dual-phase methodology aimed at estimating the end uses of water exclusively based on pressure data. The latter are acquired at two sections of the domestic inlet pipe with a resolution comparable to that of many commercial pressure sensors (i.e., 1 s). The novelty of the dual-phase method proposed – which can be considered as an alternative to the traditional flow-measurement-based methods for event characterization – consists in the exploitation of the pressure-flowrate relationship to discriminate between internal and external water-use events for estimating the water-consumption time series of the considered household. It is worth noting that flowrate evaluation by means of pressure measurements can also be performed by Venturi meters, which allow flowrate to be assessed based on Bernoulli’s principle, i.e., by converting a pressure difference between two sections featuring different cross-sectional areas in terms of velocity (and thus flowrate). However, as opposed to Venturi meters – the characteristics of which are similar to those of traditional flow meters in terms of size and installation intrusiveness – the methodology proposed in this study is based on the evaluation of the pressure difference observed between two not-necessarily-close sections induced by head losses, and its conversion in flowrate by means of the pressure-flowrate relationship.

To test the effectiveness of the method, a real case study including 1-s resolution pressure data observed over one month and a half is considered. In the following sections, the layout of the developed methodology and the characteristics of the real case study considered to test the method are presented (2. Methods and Materials). The results obtained by applying the method are then shown and discussed (3. Results and Discussion). Finally, the main outcomes of the study are given (4. Conclusions).

2 Methodology

The methodology for the characterization of water-consumption events (the layout of which is shown in Fig. 1) consists of a dual-phase approach and relies exclusively on the use of pressure data. Overall, it is based on (a) the assumption that pressure time series are available in relation to two sections of the inlet pipe of a user with no in-between outflow (e.g., the user’s water service line); (b) the fact that time series of the differential pressure between the upstream and the downstream section can provide information about the flowrate through the pipe, when coupled with the hydraulic resistance of the segment between the two sections. The method is composed of two phases: (i) pressure data collection and conversion into flowrate (Phase I); (ii) water-consumption event filtering and analysis (Phase II).

Fig. 1
figure 1

Layout of the proposed methodology

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2.1 Phase I. Pressure Data Collection and Conversion into Flowrate

Phase I is aimed at the obtainment of the household flowrate time series only by means of pressure data collected in the field with 1-s temporal resolution. Pressure data are simultaneously collected at two sections of the domestic inlet pipe, with no in-between outflows (e.g., outflows due to the presence of intermediate water fixtures or leakages). These two sections can be considered either along the plumbing system (i.e., downstream the domestic water meter) or along the water service line, based on the availability of hydraulic connections at which to install pressure sensors.

Let us consider two hydraulic sections of a domestic inlet pipe – the former of which (section\(A\)) upstream the latter (section\(B\)) – at which two pressure sensors are installed. If the household water consumption is zero, the total hydraulic head observed at section \(A\), i.e., \({H}_{A}={z}_{A}+{p}_{A}/\gamma +{v}_{A}^{2}/2g\) – being \({z}_{A}\) (\(m\)) the elevation of pressure sensor, \({p}_{A}\) (\(Pa\)) the pressure observed, \(\gamma\) (\(N/{m}^{3}\)) the water specific weight, \({v}_{A}=0\) (\(m/s\)) the velocity of water through section \(A\), and \(g\) (\(m/{s}^{2}\)) the gravitational acceleration) is equal to that observed at section \(B\), i.e., \({H}_{B}={z}_{B}+{p}_{B}/\gamma +{v}_{B}^{2}/2g\). Conversely, if one or many domestic devices are activated, a flowrate \(Q\) (\({m}^{3}/s\)) along the inlet pipeline is produced, and a head loss \(\varDelta {H}_{A,B}\) is observed between sections \(A\) and \(B\). Based on the energy balance between sections \(A\) and \(B\), head loss \(\varDelta {H}_{A,B}\) can be quantified as shown in Eq. (1):

$$\varDelta {H}_{A,B}={H}_{A}-{H}_{B}=\left({z}_{A}+\frac{{p}_{A}}{\gamma }+\frac{{v}_{A}^{2}}{2g}\right)-\left({z}_{B}+\frac{{p}_{B}}{\gamma }+\frac{{v}_{B}^{2}}{2g}\right)={R}_{A,B} {Q}^{2}$$
(1)

where \({R}_{A,B}\) (\({s}^{2}{m}^{-5}\)) is the equivalent hydraulic resistance of the pipeline between sections \(A\) and \(B\), which depends on pipe length, diameter, material, and layout (e.g., presence of elbows or other obstacles to the flow), which result in distributed and minor losses. Assuming that sections \(A\) and \(B\) have the same area (e.g., sections are along the same inlet pipe) and that there are no in-between outflows, the kinetic component of the total head does not vary between sections \(A\) and \(B\) being the flow and thus the water velocity the same. Thus, Eq. (1) can be simplified as:

$$\varDelta {H}_{A,B}=\left({z}_{A}-{z}_{B}\right)+\left(\frac{{p}_{A}}{\gamma }-\frac{{p}_{B}}{\gamma }\right)$$
(2)

Equation (2) reveals that, if pressure values and sensor elevation at sections \(A\) and \(B\) are available, the head loss \(\varDelta {H}_{A,B}\) can be assessed, hence the water flow through the pipeline considered (if its hydraulic resistance \(R\)is known). It is worth noting that the use of Eqs. (1)–(2) also allows discriminating between water-consumption events occurring in the household considered and events generated by end uses located in other households (e.g., nearby houses) or pressure variations due to changes in network operation. In fact, only internal events would result in positive head losses \(\varDelta {H}_{A,B}\) along the domestic inlet pipeline, whereas events due to external users or pressure changes in the network would alter the total head at sections \(A\) and \(B\) of a comparable amount leading to zero \(\varDelta {H}_{A,B}\)-values. Overall, the use of two pressure sensors instead of one makes the methods able to consider only internal events – while neglecting the external events – unlike the method developed by Froehlich et al. (2009, 2011).

Based on the above considerations, a five-step procedure can be applied for the obtainment of the domestic flowrate time series on which to subsequently apply water-use event analysis. The key-elements of each step are illustrated, by way of example, in Fig. 2.

  1. 1.

    Installation of pressure sensors in proximity to the two sections considered (A and B) and pressure monitoring with 1-s resolution, i.e., obtainment of synchronous pressure signals \({p}_{A}\left(t\right)\) and \({p}_{B}\left(t\right)\) (\(t=1,\dots ,T\), being \(T\) the length of the overall monitoring period) (Fig. 2a).

  2. 2.

    Evaluation of the difference \({z}_{A}-{z}_{B}\) in the elevation between sensors placed at sections \(A\) and \(B\). This can be obtained either through direct measurements – such as altimetric surveys – or estimates, e.g., by considering the offset between signals \({p}_{A}\left(t\right)\) and \({p}_{B}\left(t\right)\) in relation to a period with no domestic water consumption (i.e., when \({H}_{A}={H}_{B}\)and therefore \({z}_{A}-{z}_{B}={p}_{A}/\gamma -{p}_{B}/\gamma\), as shown in Eq. (2)). Specifically, knowledge of the sensor-elevation difference \({z}_{A}-{z}_{B}\) also allows a direct comparison between the total head at sections \(A\) and \(B\) (Fig. 2b).

  3. 3.

    Evaluation of the head-loss time series \(\varDelta {H}_{A,B}\left(t\right)\) through Eq. (2), i.e., based on the difference between the measured pressures – i.e., \({p}_{A}\left(t\right)\) and \({p}_{B}\left(t\right)\) – and the offset in sensor elevation, i.e., \({z}_{A}-{z}_{B}\) (Fig. 2c).

  4. 4.

    Application of Eq. (1) for head-loss conversion into a flowrate time series \(Q\left(t\right)\) based on the hydraulic resistance of the domestic inlet pipeline between sections \(A\) and \(B\) (Fig. 2d).

  5. 5.

    Flowrate time series filtering to remove negative values along with those values which are below a given threshold.

With specific reference to step 4, it is worth noting that the obtainment of the \(Q\left(t\right)\) time series by means of the relationship between head loss and flowrate is dependent on the knowledge of the hydraulic resistance \({R}_{A,B}\) of the system. From an operational standpoint, the \({R}_{A,B}\)-value can be assessed by: (i) generating a series of \(n\)-individual water-consumption events \(\left\{{E}_{1},{E}_{2},\dots ,{E}_{n}\right\}\) of known duration \(\left\{{D}_{1},{D}_{2},\dots ,{D}_{n}\right\}\) in the household selected, in turn; (ii) reading the domestic water meter either at the beginning and at the end of each event in order to estimate the consumed volumes \(\left\{{V}_{1},{V}_{2},\dots ,{V}_{n}\right\}\) and, consequently, their related flowrates \({Q}_{i}={V}_{i}/{D}_{i}\) (\(i=1,\dots ,n\)); (iii) calculating head losses \({(\varDelta {H}_{A,B})}_{i}\) (\(i=1,\dots ,n\)) through sensor-elevation difference \({z}_{N}- {z}_{M}\) and the difference between pressure signals \({p}_{A,i}\) and \({p}_{B,i}\) observed in relation to the periods of duration \(\left\{{D}_{1},{D}_{2},\dots ,{D}_{n}\right\}\) over which events are generated, in turn; (iv) applying a linear-regression method (e.g., the least-square method) to the cloud of n-points of \({\left\{{Q}^{2},\varDelta {H}_{A,B}\right\}}_{i}\) coordinates in order to find the \({R}_{A,B}\)-value so that the function\(\varDelta {H}_{A,B}={R}_{A,B} {Q}^{2}\) is the best approximation of the \(n\)-points.

With specific reference to step 5, it is worth noting that the flowrate obtained at the end of step 4 may be affected by a level of uncertainty as a consequence of: (i) the uncertainty affecting pressure signals; and (ii) the uncertainty associated with the system hydraulic resistance (the value of which is obtained by means of a regression method). In greater detail, the uncertainty affecting the flowrate time series can result in a background noise, also featuring sporadic negative values. Given this, the flowrate time series is preliminary processed by removing all negative values along with those values which are lower than a given threshold \({Q}_{min}\) (dependent on pressure sensor accuracy, the uncertainty related to sensor elevation, and the uncertainty related to the \(R\)-value). This threshold value, \({Q}_{min}\), can be obtained by applying the relationship between head loss and flowrate Eq. (1), in which a threshold head loss \(\varDelta {H}_{min}\) is assumed based on all the above uncertainties (e.g., \(0.10 m\)).

Overall, at the end of Phase I, the household-level water-consumption time series \(Q\left(t\right)\) is available.

2.2 Phase II. Water-Consumption Event Filtering and Analysis

The aim of Phase II is to gather information about water consumption at the level of individual events, by exploiting the 1-s flowrate time series obtained in Phase I, which includes both individual events (e.g., toilet flush) and combined events (e.g., toilet flush simultaneous to a tap use). The segmentation of the latter events into a series of individual events may provide insight into the characteristics of the end uses included in the selected household despite the lack of detailed information about end-use categories. In light of the above, the flowrate time series is subjected to an automated method for signal stabilization and the segmentation of combined events into individual events. The method is based on the assumptions according to which: (i) individual water demand events can be reasonably approximated by rectangular pulses in the flowrate time series (as demonstrated by Buchberger and Wells (1996); and therefore (ii) combined events can be well approximated by a composite rectangular signal, due to event overlaps in time (Mazzoni et al. 2023). The structure of the signal-stabilization and segmentation method (an example of which is illustrated in Fig. 2d-e) is described below:

  1. i.

    Event isolation. Water-consumption events (i.e., portions of the flowrate time series characterized by continuous, positive values) are considered in turn (Fig. 2d).

  2. ii.

    Event stabilization. A moving window of limited width (e.g., 5 s) is applied to stabilize the flowrate signal making up each event. All time instants \(t\) of the event period for which an abrupt variation in the moving-average flowrate is observed (e.g., 1 L/min) are considered, because these are possibly related to the occurrence of a new opening or closing manoeuvre. Event portions between the identified time instants \(t\) are smoothed based on the average flowrate value observed over each of these sub-periods (Fig. 2e).

  3. iii.

    Event segmentation. The selected event is segmented into a series of individual events based on the matching of the opening and closuring manoeuvres identified. This is done by coupling opening and closing manoeuvres of comparable magnitude (Fig. 2e).

All individual (i.e., segmented) events making up the aforementioned dataset can also be described based on their features (i.e., duration, volume, flowrate, daily frequency of occurrence) to provide the analyst further information for event labelling or the application of automated methods for end-use disaggregation and classification (i.e., end-use parameters required as input). Specifically, the characteristics of individual events can be first investigated by identifying the number of water uses belonging to each cell of a duration-volume mesh. It is expected that: (i) some events (i.e., those related to human-controlled end uses) fall over lines of a given slope, meaning that these are due to water uses of different duration (and volume) but constant flowrate; (ii) some others (i.e., those related to fixed-volume or automated end uses) are lumped into single hotspots, meaning that these repeat over time with no changes in duration or volume. In addition, the probability-density-function (PDF) of water-use event features can be used to estimate the statistical distributions of water-use parameters, which allow the calibration of water-demand models (e.g., Blokker et al. 2010) or methods for automated end-use classification based on stochastic input (e.g., Mazzoni et al. 2024). From an operational standpoint, empirical PDFs are obtained through the Kernel Density Estimator by applying the MATLAB’s R2019a® ksdensity function. Subsequently, five analytic PDF curves are assumed to fit each empirical PDF: normal, lognormal, exponential, Weibull, and Gamma. Among these curves, the best-fitting PDF is then selected based on the distribution type related to the lowest Root-Mean-Square-Error between the empirical and the analytical PDF, along with its related parameter values.

Fig. 2
figure 2

Main steps of the proposed methodology: (a) pressure data collection; (b) total-head time-series obtainment; (c) head-loss time-series obtainment; (d) conversion into flowrate time series and events isolation and filtering; and (e) end-use event segmentation

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3 Case Study

The case study considered to validate the method is a household supplied by the WDN of a seaside resort located in the municipality of Comacchio (northern Italy), along the Adriatic coast (Mazzoni et al. 2022). Specifically, the household concerned is a single-family two-story house provided with a 40-m long domestic plumbing system – composed of DN20 polyethylene pipelines – along which a number of domestic water fixtures are installed (five taps, two showers, two toilets, a dishwasher, and a washing machine). The 4-m long DN20 household service line includes a mechanical water meter (Figure S1, Supplementary Material) and is connected to a 45-m long WDN branch supplying a total of ten dwellings.

Pressure monitoring was conducted for 43 days over the summer period, when the household was inhabited by three people. Specifically, two service-line sections were considered: section \(A\), upstream the water meter and near the junction connecting the service line with the main WDN, and section \(B\), immediately downstream the mechanical water meter (panel a and b of Figure S1). Sections \(A\) and \(B\) were selected for their in-line position on the household inlet pipe and due to accessibility reasons. At each section, an acquisition system composed of a STS® pressure sensor with full scale (FS) equal to 10 bar – and accuracy of 0.5% FS – and a MV155 ISOIL® data logger was installed, allowing the simultaneous monitoring of pressure with 1-s time resolution.

During the monitoring, water-meter readings were done with the aim of evaluating the volume of water consumed in the household over a given time window, and thus validating the proposed method. In greater detail, several readings were done daily or sub-daily for the first two weeks of pressure monitoring, to initially test the method on short periods (i.e., Periods 1–11). In addition, a conclusive reading was done at the end of the overall monitoring (i.e., at the end of Period 12), to evaluate method effectiveness in providing water-consumption data on a longer period (Table 1). Readings revealed that, on average, about 129 L/p/d (liters per person per day) were consumed in the household considered over the 43-day period of pressure monitoring.

Table 1 Duration, observed water consumption, estimated water consumption (i.e., obtained by applying the Phase I of the proposed method) and percent deviation between estimated and observed water consumption for each water-meter reading period
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In addition, during the first week of monitoring (Periods 1–6), residents were asked to fill in reports by specifying the time of activation of showers, toilets, and appliances (i.e., dishwasher and washing machine) in order to identify these end-use categories on the segmented-event time series. Tap events were not included in the above report due to their generally higher frequency of use. However, these can be easily identified based on the individual events not matched to the above end-use categories. Overall, the comparison between report results and the segmented-event time series allows evaluating end-use features and thus enables water-demand models or end-use classification methodologies to be applied over subsequent monitoring periods.

4 Results and Discussion

4.1 Phase I. Pressure Data Collection and Conversion into Flowrate

With reference to the domestic case study considered, the pressure monitoring at the two sections \(A\)and \(B\) with 1-s resolution leads to the obtainment of synchronous pressure signals \({p}_{A}\left(t\right)\) and \({p}_{B}\left(t\right)\), with \(t=1,\dots ,T\), over a period of 43 days (\(T=43\cdot 86,400\)). The elevation difference \({z}_{A}-{z}_{B}\) between sensors is estimated by considering the offset between signals \({p}_{A}\left(t\right)\) and \({p}_{B}\left(t\right)\) with reference to a time window during which no flow in the inlet pipe is observed (i.e., \({H}_{A}={H}_{B}\)), resulting in a value of 0.54 m. The above sensor-elevation difference is applied to the pressure data collected to obtain the head-loss time series \(\varDelta {H}_{A,B}\left(t\right)\) (see Eq. (2)). The hydraulic resistance \({R}_{A,B}\) of the inlet-pipe segment between sections \(A\) and \(B\) is estimated by activating one domestic water fixture at a time for a period of known duration \(D\), and by evaluating the volume \(V\) of consumed water based on two water-meter readings and, consequently, the event steady-state flowrate \(Q\). The above process is carried out for a total of \(n=40\) events. Event flowrate values and the produced head losses are considered to represent each event \(i\) (\(i=1,\dots ,n\)) as a dot of given coordinates in the \({Q}^{2}\)versus\({\varDelta H}_{A,B}\) plane, as shown in panel c of Figure S1. In light of the linear relationship between \({Q}^{2}\) and \({\varDelta H}_{A,B}\) (Eq. 1), the linear approximation of the cloud of \(n=40\) points of \({{\{Q}^{2},{\varDelta H}_{A,B}\}}_{i}\) coordinates is conducted by applying the least-square method, leading to an estimated \({R}_{A,B}\)-value of about \(4.035\cdot {10}^{7} {s}^{2}{m}^{-5}\).

The flowrate time series is finally processed by removing all negative values along with those values which are lower than a given threshold \({Q}_{min}\) (assumed in this case equal to 3 L/min and corresponding to a head loss of 0.10 m, also considering pressure-sensor accuracy).

In order to first evaluate the performance of the method, the percent deviation of the volume delivered to the household over a series of daily or sub-daily periods is considered (Table 1). As shown in the table, the overall domestic water consumption estimated by the method over the monitoring period deviates from the observed water consumption of about 2.3%, confirming the capability of the method of effectively providing flowrate time series starting from pressure data. Based on the results reported in Table 1, it emerges that: (i) percent deviations present both positive and negative values, suggesting that flowrate time series are not systematically overestimated or underestimated, and (ii) a maximum deviation of 10.2% observed is related to a period showing a rather low consumption (i.e., of about 144 L over 26 h) which is overestimated of only 15 L. Therefore, the methodology is shown to be effective in estimating household water consumption over the monitoring period starting only from pressure data.

4.2 Phase II. Water-Consumption Event Filtering and Analysis

The application of automated method for signal stabilization and segmentation to the flowrate time series \(Q\left(t\right)\) obtained resulted in 7704 events, 1355 of which (17.6%) overlapped in time.

Duration and volume of all the individual events obtained from filtering are considered to locate them in the duration-volume mesh (Fig. 3a) based on their frequency of occurrence. The darker the hotspot(s) shown in Fig. 3a, the higher the frequency of use. The individual events related to human-controlled end uses, such as taps and showers, are likely to be characterized by a constant flowrate and fall over lines of a given slope. Otherwise, the events related to fixed-volume or automated end uses are typically more lumped into hotspots.

Based on the results of the automated method for signal stabilization and segmentation, the statistical distributions of four event features are investigated: (i) duration per use (s/use), (ii) volume per use (L/use), (iii) average flowrate per use (L/min) and (iv) daily per capita frequency of occurrence (uses/p/d). The empirical and (best-fitting) analytical PDFs of these parameters are shown in Fig. 3b-e in relation to all water-use events observed over Periods 1–12 (i.e., about 43 days of monitoring). The figure reveals that:

  • Duration and volume per use are well fitted by a lognormal probability distribution, with right-skewed PDF curves and peak values very close to the vertical axis (i.e., related durations of a few seconds and volumes of less than 1 L). This suggests that the majority of detected and segmented events is tied to the use of taps, which are typically the most frequently activated end uses, but they also have the lowest durations and volumes per use, as reported by several studies (Mead and Aravinthan 2009; Redhead et al. 2013). Average duration and volume of the individual events detected are about 30 s and 2.5 L, respectively.

  • The flowrate-per-use distribution is fitted by a slightly right-skewed Weibull distribution. However, the fitting of flowrate empirical PDF by means of the five analytical functions here considered is less accurate due to the presence of several peak values (e.g., around 2 L/min, 4 L/min and 7 L/min) in the empirical PDF, because of the specific flowrates typically characterizing different end uses. This further confirms what emerges from the analysis of Fig. 3a, i.e., events belonging to different end-use categories tend to fall over lines of a given slope and thus flowrate.

  • The daily per capita frequency of occurrence is well fitted by a nearly symmetrical Weibull distribution, showing a peak value of about 45 uses/p/d and an average of about 50 uses/p/d. The results obtained are comparable with those available in the literature for the residential sector (Mazzoni et al. 2023), i.e., a total of about 20–50 uses/p/day, the majority of which related to taps and toilets.

Fig. 3
figure 3

(a) Duration-volume mesh obtained for the household considered, and probability-density-functions (PDFs) of event duration (b), volume (c), flowrate (d) and per capita daily frequency (e)

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It is worth noting that the availability of water-use reports filled in by household residents during the first week of monitoring (Periods 1–6) also allows statistical analyses to be performed in relation to individual end-use categories, the results of which are shown in Figure S2 (Supplementary Material). The figure reveals that: (i) dishwasher and washing-machine events are characterized by constant flowrates – along with durations and volumes the values of which are lumped into one or few narrow intervals as a consequence of the regular behavior in terms of water withdrawal; (ii) events tied to manually-regulated fixtures (i.e., showers and taps) show a greater variability in durations and volumes per use, as a consequence of different ways in which they are activated; (iii) toilet events show very regular flowrates but the empirical PDFs of duration and volume per use include two peaks due to the presence of dual-flush systems; (iv) daily per capita frequency of use is the highest in the case of taps and toilets (with an average of about 25 and 8 uses/p/d, respectively) followed by showers (about 2 uses/p/d), whereas it is the lowest in the case of appliances (about 1.5 inflows/p/d, i.e., about 0.3 uses/p/d considering on average 5 inflows/use). In addition, as far as the overall water volumes consumed by each end use are concerned, the analyses conducted in relation to Periods 1–6 reveal that the largest daily per capita water consumption are tied to the use of showers, toilets and taps (i.e., 50 L/p/d, as shown in panel a of Figure S3, Supplementary Material), whereas a lower consumption is due to washing machine and dishwasher use (i.e., 12 and 5 L/p/d, respectively).

Finally, the flowrate time series \(Q\left(t\right)\) obtained from the application of Phase I – coupled with information about end-use features – allows automated methods for end-use classification to be applied in relation to subsequent datasets, thus making the investigation of the characteristics of residential end uses of water possible on a broader sample. In light of the above, the automated approach proposed by Mazzoni et al. (2024) is applied here to classify the (unlabeled) segmented events related to a subsequent time window (i.e., Periods 7–12) based on end-use parameters defined in relation to Periods 1–6. The results obtained by applying the automated method (shown in panel b of Figure S3) reveal that the greatest portion of water consumption is tied to the use of showers, toilets, and taps, followed by appliances. However, a greater dishwasher use is observed over Periods 7–12 (i.e., 12 L/p/d) as opposed to Periods 1–6 (i.e., 5 L/p/d). This can be due to: (i) a real increase in the frequency of dishwasher use or (ii) the similarity between the characteristics of dishwasher events and those of tap uses, which can lead to misclassification (Attallah et al. 2023).

Overall, the insights emerged from the application of the methodology proposed can support water utilities in characterizing and modeling residential water consumption at the user level. Moreover, the outcomes of the methodology can lay the basis for the design of user-oriented water-demand management strategies, such as the efficient management of water resources, the revision of water tariff and the definition of incentives, or the implementation of campaigns aimed to raise awareness among consumers.

4.3 Study Limitations

Despite the promising results achieved, some limitations emerge in the application of the proposed methodology:

  • The method needs the installation of at least two pressure sensors to discriminate between internal and external water uses.

  • Hydraulic sections at which pressure sensors are installed have to be selected along a segment of the inlet pipe of the household with no in-between outflows. In the case of in-between outflows, the method would mistake the intermediate outflow event for an event downstream both sections.

  • Due to the uncertainty affecting pressure data and sensor elevation, a lower threshold \(\varDelta {H}_{min}\)is considered in head-loss conversion, below which the above head loss is not related to any flowrate. This implies that low-flow events – i.e., events the flowrate of which is lower than a given threshold \({Q}_{min}=\sqrt{\varDelta {H}_{min}/R}\) – cannot be detected by the method if the produced head losses are lower than \(\varDelta {H}_{min}\) or when the hydraulic resistance \(R\) of the system is limited.

5 Conclusions

This study proposed a dual-phase methodology aimed at obtaining information about domestic water-consumption exclusively through pressure data collected at two in-line sections of the household inlet pipe. The results obtained on a real case study highlight that pressure data collected with 1-s time resolution are sufficient to effectively estimate the water-consumption time series and provide useful information about individual water-consumption events.

In conclusion, it worth remarking the following aspects:

  • The method allows obtaining a high-resolution (i.e., 1-s) flowrate time series by means of pressure data and overcoming the technical and the practical limitations affecting the installation of smart flow meters at the domestic inlet.

  • The method is effective in estimating domestic water consumption. With specific reference to the case study considered, the average deviation between observed and estimated water consumption is of only 2.3%.

  • Useful insights into the characteristics of water-consumption events can be provided by the method. If coupled with more detailed information provided by the residents, this can also pragmatically unveil the characteristics of residential end uses of water, i.e., end-use parameter values that are typically required to perform automated end-use disaggregation and classification.

  • Thanks to the two-sensor solution, the method can discriminate between water-use events occurring within the household considered and external water-use events (or changes in network operation), making it dependent only on users’ activity supplied by the selected household. This represents a novelty element as opposed to other pressure-based studies available in the literature.

Future studies will mainly focus on (i) the application of the proposed method to a larger sample of case studies in order to further test method sensitivity to hydraulic-resistance values and pressure-data time resolution, and (ii) the integration of alternative approaches, such as cluster analysis or machine learning techniques, aimed at the refinement of the methodology.