EPANET uses the Lagrangian time-driven simulation method for water-quality routing in a network discussed in . In general, the method may not always provide exact results. In particular, if the water-quality time step is too long, concentration errors can occur; time steps should be less than the time required for a water parcel to move through the network pipe segment (link) having the shortest travel time for the simulation. In principle, errors can be avoided if a sufficiently short time step is used. However, such time steps may not be practical or feasible from a computational perspective. For example, pumps and valves have zero length in EPANET. Also, EPANET allows a minimum time step of only 1 s, which in some cases may not be sufficiently short. Finally, water parcels can move through only one link in a water-quality time step. Although relevant improvements, for example the use of parallel processing and graphics processing units , can be expected, the approach currently used in EPANET can be, and will likely remain, computationally challenging because of the possibility of a large number of links in a network and the need to use a short time step to minimize concentration errors.

The algorithm used in EPANET to route water quality through a network can result in situations in which constituent mass is gained or lost. These imbalances can occur because of the manner in which water volume and constituent mass are accumulated at nodes and the manner in which volume and concentration are determined for subsequent releases to downstream links. Constituent mass can be generated during the accumulation step and lost during the release step at locations for which the volume of water being moved during a water-quality time step exceeds the volume of the link in which the water is being moved. When there is a spatial gradient in constituent concentration at such locations, the mass generated during the accumulation step and lost during the release step will not be the same and a net generation or loss of constituent mass can occur. A detailed example is provided in Appendix B illustrating how the time-driven algorithm used in EPANET can fail to conserve constituent mass in such situations.

Restricting the movement of water parcels to only one link per water-quality time step means that, when longer time steps are used, a longer time is required for a parcel to reach a particular downstream location. For example, the arrival time of a contaminant pulse at some location following an upstream injection will be delayed if a longer time step is used. This results in concentration errors even if the shape of the pulse is unaffected. If delays are sufficiently long, the potential exists for changes in hydraulic conditions, which could also affect concentrations. Errors in concentrations due to the effects associated with allowing water parcels to move through only one link per water-quality time step can occur even if mass is conserved.

The examples presented in this section were obtained using EPANET's time-driven water-quality routing algorithm; they demonstrate that large mass imbalances can occur, that mass-balance and concentration results can be sensitive to the water-quality time step used, and that a very short time step may be necessary to avoid significant mass imbalances and to minimize concentration errors. These results indicate that to model contaminant intrusion events more accurately a more robust algorithm is needed for use with EPANET that can ensure conservation of mass during water-quality simulations.

For a contaminant injection at a selected node in Network N3, Fig.  shows how the various components of mass-balance change as the water-quality time step is varied using the time-driven simulation method in EPANET. Note that different vertical scales are used in each plot in the figure. The mass that is injected or removed is a cumulative mass; the mass in pipes and tanks is the mass in those locations at each time in the simulation. For an injection at Node 100 in Network N3, a significant mass imbalance can occur for water-quality time steps of 60 s or longer. At the end of the 168 h simulations, the MBRs are 7.11, 4.88, 1.38, and 1 for water-quality time steps of 900, 300, 60, and 1 s, respectively. For the time steps equal to or longer than 60 s, considerably more mass was removed from the network than was injected. A time step less than 60 s is necessary to conserve mass in this case.

Changes in MBRs following injections at Node IN1029 in Network N2 (Micropolis) are shown in Fig.  for the time-driven simulation method and four different water-quality time steps. Considerable time is required before the ratios stabilize for the longer time steps. Only for a time step of 1 s does the MBR approximately equal 1.0. This network contains 196 valves, which, as noted, have zero length in EPANET, and, therefore, zero travel time, which likely contributes to the large MBR values shown in the figure. Note in Fig.  that the mass imbalances are larger for a time step of 300 s than for a time step of 900 s. The location of Node IN1029 is shown in Fig. .

Using two different water-quality time steps, Fig.  shows how the MBR varies during simulations for an injection at Node JUNCTION-3064 in Network N4 (BWSN), again using the time-driven method. Although the ratio stabilizes after about 20 h at 1.006 for a time step of 300 s and at about 1.000 for a time step of 60 s, the ratio can be significantly different from 1.0 during the early portion of the simulations, even with a water-quality time step of 60 s. Mass is first lost from the system, then gained, then lost again, before the ratio approximately stabilizes. Note that the vertical scales on the two plots in Fig.  are different.

Figure  compares contaminant concentrations obtained using the time-driven method for Node 247 in Network N1 (EPANET Example Network 3) following an injection at Node 101, for different water-quality time steps. As the water-quality time step decreases, the magnitude and timing of the contaminant pulses at the downstream node change. The concentrations appear to stabilize when the time step is reduced to 60 s. The MBRs given in the figure are values at the end of a 24 h simulation.

Contaminant concentrations at Node TN1810 in Network N2 following an injection at Node IN1029 were determined using the time-driven method and are compared in Fig.  for different water-quality time steps. Again, the magnitude and timing of the contaminant pulses vary with the time step used. However, in this case, rather than generally decreasing as the time step decreases, the concentration of the pulse increases substantially in magnitude as the time step is reduced from 900 to 300 s, consistent with the MBRs determined for this injection node and shown in Fig. . The maximum concentration then decreases by over 99 % going from a time step of 300 s to a time step of 1 s. The MBRs given in the figure are values at the end of a 168 h simulation.

Using the time-driven method, contaminant concentrations were obtained at Node 200 in Network N3 after an injection at Node 100; they are compared in Fig.  for different water-quality time steps. Again, the timing and magnitude of the contaminant pulse change as the time step is reduced to 1 s, as is the case in the examples presented for Networks N1 and N2. The MBRs given in the figure are values at the end of a 168 h simulation. Figure  shows how the components of mass balance vary following an injection at Node 100 for the same time steps used in Fig. .

Figure  compares contaminant concentrations obtained for a receptor node, TANK-12525, in Network N4 following an injection at Node JUNCTION-2514, using different water-quality time steps. Again, the simulations used the time-driven method. In this case, the timing of changes in concentration is similar for all the time steps used and there is a high degree of correlation between the results for the different time steps. However, concentration increases consistently as the time step decreases, unlike the trends in the previous examples. Concentrations appear to have approximately stabilized by a time step of 60 s, with little increase in concentrations occurring when the time step is reduced to 1 s. The MBRs given in the figure were determined at the end of a 168 h simulation. Note that the MBR is less than 1.0 for time steps greater than 1 s. Substantial losses of contaminant mass can occur for longer time steps.

Statistics on the extent of mass imbalance at the end of simulations that used the time-driven method are provided in Table  for the four networks considered. The results in the table are based on independent injections at most nodes for each of the networks. For example, the statistics for Network N4 are based on independent simulations of injections done for most of the 12 523 nodes in the network; a small fraction of the nodes were excluded, as discussed in the next paragraph. The table provides the range in MBRs determined for each network for each of four water-quality time steps and the fraction of injection nodes for which there were imbalances above some thresholds (e.g., 1, 5, 10 %).

MBRs equal to zero were obtained for injections at some nodes: the numerator in the MBR was zero because no mass was present in the pipes or tanks at the end of the simulation and no mass was removed during the simulation. Such cases can occur when there is no flow at the time of injection (e.g., zero-demand nodes). When there is no flow at the time of injection, no contaminant mass is added to the water in the WDS and the injected mass is effectively lost, although the accounting process considers it to be mass injected when determining an MBR. Injection nodes for which an MBR was equal to zero at the end of a simulation were not included when determining the statistics shown in Table . For Network N4, two additional nodes (JUNCTION-9097 and JUNCTION-12348) also were excluded. These two nodes are in dead-end areas with no demands. Therefore, there should be no flows in these areas. However, the network model had a small initial flow at these nodes, inconsistent with a lack of demands in the dead-end areas. (EPANET determines concentrations of constituents in outflows from a node using the flow-weighted sum of inflow concentrations. If other inflows are very small, such anomalous flows could be significant in relative terms and result in concentration anomalies as well.) For Network N3, five nodes had MBRs near 0.2 for all water-quality time steps; however, when the hydraulic and reporting time steps were changed to 1 min, the MBRs for the 1 min and 1 s water-quality time steps were near 1. MBRs near 1 were used for the five nodes when determining the statistics for Network N3 shown in Table .

Table  shows that as the water-quality time step decreases, the maximum MBRs for each network decrease towards 1.0 and the minimum MBRs generally increase. However, for Network N2 the minimum MBR increased only to 0.08 for the 1 s time step and for Network N4 it reached only 0.83. For all four networks considered, the fraction of injection nodes with imbalances above the thresholds listed in the table decreases consistently as the time step decreases. For a time step of 1 s, only about 2, 1, and < 1 % of the nodes had imbalances greater than 1 % for Networks N2, N3, and N4, respectively. There were no mass imbalances greater than 0.01 % for this time step for Network N1 (excluding three nodes for which the MBR was zero). This is in contrast to the sizable fraction of nodes in all the networks that have imbalances for a time step of 300 s, although the imbalances for Networks N1 and N4 are relatively minor for this time step, with only about 1 and 2 % of nodes in these networks, respectively, having an imbalance greater than 10 %.

For a small fraction of the injection nodes in Networks N2, N3, and N4, about 0.8, 0.5, and 0.2 % of all nodes, respectively, the MBR did not change as the water-quality time step decreased or did not change so that the MBR converged toward 1.0. The lack of convergence of the MBR for these nodes had limited influence on the statistics in Table ; it did influence the values shown for the minimum MBR for Network N2, particularly for a time step of 1 s. The non-convergence can be the result of several factors, including cases involving nodes in dead-end areas, as noted above. In addition, some cases had flows at the time of injection that were unexpected, given a lack of demands; the flows disappeared for subsequent time steps. Some problems appear to be related to the hydraulic solution; these were eliminated if a short (60 s) hydraulic time step was used.

Mass balances are sensitive to the injection time used. An acceptable mass balance for a particular application for a given injection node does not guarantee an acceptable mass balance if hydraulic conditions are changed. As an example of the extreme changes that can occur when hydraulic conditions change, consider an injection at Node VN1263 in Network N2 (see location in Fig. ), a 168 h simulation, and a 900 s water-quality time step. For a 1 h injection starting at 06:00, the MBR at the end of the simulation was 467. For a 1 h injection starting at 00:00, the MBR at the end of the simulation was 0.007. In the first case, much more mass was removed from the network by nodal demands during the simulation than was injected; in the second case, very little mass remained in the system at the end of the simulation or was removed from the system by nodal demands during the simulation. In both cases the extreme values for the MBR are the result of errors in the estimated concentrations downstream from the injection location, too high in the first case and too low in the second. These errors resulted in the erroneous numerical generation or loss of constituent mass in the system.

The cases considered to this point used relatively short, 1 h injections. However, major mass imbalances also occur for injections with long durations. For example, with a 300 s water-quality time step, the largest MBR for any injection node in Network N2 for a 1 h injection at 00:00 is about 23; for 6 and 12 h injections at that time the largest MBRs are 1495 and 971, respectively. About 17, 22, and 22 % of injection nodes have mass imbalances greater than 50 % for the 1, 6, and 12 h injections, respectively. About 32, 39, and 35 % have imbalances greater than 10 %. The statistics for mass imbalance are relatively insensitive to injection duration for this network. However, mass balances for a particular injection node can be sensitive to the injection duration. For example, for Node VN1263 in Network N2 the MBRs are 3.3, 1495, and 971 for injections at 00:00 with durations of 1, 6, and 12 h, respectively.

EPANET 3, the development version of EPANET , also yields results with mass imbalances. Consider a case involving chlorine decay with both bulk and wall reactions and EPANET Example Network 1 with default input parameters, except for the water-quality time step. No water-quality sources were used; however, initial chlorine concentrations were specified. The network is very simple, with only 9 junctions and 12 pipes. A chlorine mass imbalance of 0.85 % (< 1 %) was obtained for a water-quality time step of 900 s and a 24 h simulation. For time steps of 300, 60, and 1 s, the imbalances were 0.77, 0.58, and 0.49 %, respectively. Mass imbalance was determined in EPANET 3 in the same manner as discussed in this paper except that the initial mass of chlorine in the system also was considered, as was the mass of chlorine lost due to chemical reactions. Execution time increased from about 0.02 s to about 0.03, 0.04, and 1 s when the time step was decreased from 900 s to 300, 60, and 1 s, respectively, an overall increase in execution time of about 50 fold. To obtain mass imbalances well below 0.1 %, a time step well below 1 s may be needed, along with additional increases in execution time. This is an extremely small, simple network and large imbalances are not expected. For this example, the EPANET 3 code was compiled using the GNU Compiler Collection .

EPANET allows four source types to be used to define locations of sources of constituents . As noted above, the simulations in this study used the MASS source type, which allows a source strength to be specified in terms of mass added per unit time. The other source types (CONCEN, FLOWPACED, and SETPOINT) all specify source strength in terms of concentration. All are similar in that they allow the addition of the mass of some constituent to the system. EPANET's failure to conserve constituent mass is not related to how the mass is added to the system. Therefore, the findings presented here related to a failure to conserve mass apply independently of the source type used in the simulation. As the example discussed in the preceding paragraph demonstrates, failure to conserve mass can occur in situations in which no sources are used and the only mass present in the system is the result of the initial water quality specified.

Changes in a WDS do not occur at regular time intervals. For example, the time required for a water parcel to move from node to node in the system varies from pipe to pipe and also within a pipe as conditions change. In addition, some pipes can be short, have a high flow rate, and require only a short time for a water parcel to move through them. This transit time can be too small to be practical for use as a water-quality time step in a simulation. A situation in which events (changes) occur at irregular intervals suggests use of an event-driven simulation.

A preliminary event-driven algorithm is outlined here and used to obtain results for comparison with those provided by EPANET using the current time-driven approach. The event-driven approach used is conceptually similar to, but was developed independently of, the Lagrangian event-driven simulation method discussed in . An event-driven method updates the state of water quality in the system only when a change (an event) occurs during the simulation, in contrast to the current time-driven method in EPANET, which updates water quality across the entire network at fixed time steps. Numerous events can occur during a time step, as water moves through the network. Various event-driven approaches have been presented previously, for example by . The preliminary event-driven algorithm discussed here is included as an option in the current version of TEVA-SPOT (TEVA-SPOTInstaller-2.3.2-MSXb-20170110-DEV) and is being made available to EPANET developers to obtain community support and assistance with improving and evaluating the algorithm.

The event-based, water-quality routing algorithm used here moves homogeneous volumes of water (water parcels with a uniform concentration of a water-quality constituent) through a network. Initially, water parcels are accumulated at all nodes where water enters the system. Nodes with accumulated water parcels from all inflow links are processed in an arbitrary order. Mixing or combining of water parcels occurs at nodes based on the inflow rates of the links flowing into the nodes. Water parcels are combined if the absolute difference between their concentrations is less than some specified amount (the quality tolerance), consistent with the approach used in EPANET 2. After parcels are combined at a node, any nodal demand is removed; the remaining water parcels then are split based on the flow rates of the links flowing from the nodes. These parcels are added to lists of parcels for the downstream links. Any volume in excess of the volume of a link is removed from the leading parcels and placed at the downstream node for further processing. That node is then added to the set of nodes with accumulated water parcels waiting to be processed. Due to recirculating flows, situations can occur in which none of the nodes waiting to be processed has accumulated water parcels on all inflow links. In such cases, an incomplete parcel with the volume that will be moved, but an unspecified concentration, is created for each inflow link that does not have an accumulated inflow. These incomplete parcels are moved, combined, and split in the same manner as parcels for which constituent concentration has been determined; however, internal references are maintained that allow concentrations to be updated when parcels for which concentrations have been determined arrive at a node for which incomplete parcels were created. Flow reversals between hydraulic time steps are accommodated in the same manner as in EPANET 2. The event-driven simulation method provides results that do not depend on the water-quality time step if it is equal to or shorter than the hydraulic time step. The method actually does not require an independent water-quality time step: the simulation is event-driven as long as the hydraulic conditions do not change. Because by construction the method accounts for every individual water parcel, its resulting MBR will always be 1.0. An example illustrating the operation of the algorithm using a case with recirculating flow is provided in Appendix C. Working through the example provided in the appendix will provide a better understanding of the approach outlined in this paragraph.

Concentrations obtained using EPANET's time-driven algorithm tend to converge toward those obtained using the event-driven algorithm as the water-quality time step used in the time-driven algorithm decreases. For short water-quality time steps (e.g., 1 s) with the time-driven approach, the results for the two methods can be very similar and differences can be difficult to see in the plots used in this paper. Therefore, to better examine this convergence, least-squares fits were determined relating (1) the concentrations obtained using the time-driven approach with a water-quality time step of 1 s (TD1) and the concentrations obtained using the same approach with a 60 s time step (TD60), (2) TD1 and the concentrations obtained using the time-driven approach with a 300 s time step (TD300), and (3) TD1 and the concentrations obtained using the event-driven approach with a 3600 s time step (ED3600). These least-squares lines have the form y^=ax+b, where a and b are the slope and intercept of the fitted least-squares line, x is the value of TD1, and y^ is the fitted value of TD60, TD300, or ED3600, depending on which is being used. The results of fitting least-squares lines are shown in Table  for the four cases examined in Figs.  to .

The number (N) of hourly concentration values used to obtain the results shown in the Table  corresponds approximately to the number of hourly concentration values shown in the figures for the different networks. For Network N1, N was 24, covering the entire length of the simulation. For Network N2, it was 60, the length of the middle portion of the plot in Fig. . For Network N3, N was 39, the length of the period from hour 1 in the simulation to hour 40 (see Fig.  ). For Network N4, results for the entire 168 h simulation were used. The water-quality tolerance in the simulations used to obtain the concentrations needed for the analysis presented in the table was 0.01, except for the event-driven simulations for Network N4, for which 0.1 was used.

If the concentrations obtained using the time-driven method with a 1 s water-quality time step are identical to those obtained for one of the other cases, the slope of the least-squares line relating the concentrations will be 1, the intercept will be 0, the adjusted R2 will be 1, and the residuals for the fit will all be 0. From Table , the results for Networks N1, N2, and N3 show that of the three cases considered, the concentrations for TD1 are closest to those for ED3600. Also, the agreement improves going from TD300 to TD60 to ED3600. This improvement is what would be expected if the results for the time-driven approach are converging to those for the event-driven approach as the water-quality time step used for the time-driven approach decreases. For Network N4 there is a high degree of correlation between the concentrations obtained using the time-driven method with different time steps (cf. Fig. ) and the quality of the fit is similar for all three cases considered. However, because the magnitude of the concentrations increases as the time step decreases, the slope of the least-squares line increases going from TD300 to TD60.

The results in Table  for Network N2 indicate a less-than-perfect correlation between the concentrations obtained using the time-driven algorithm used in EPANET and a water-quality time step of 1 s and those obtained using the event-driven algorithm and a time step of 3600 s. The concentrations obtained using the two approaches are compared in Fig.  (which uses an expanded vertical scale compared to the one used in Fig. ). The results obtained for the two simulations are noticeably different. However, compared to the differences between the results obtained using the time-driven approach using a 1 s time step and those obtained using longer time steps, the differences are quite minor, as can be seen from a comparison of Figs.  and .

Overall, the results presented here demonstrate that substantial mass imbalances can occur during EPANET water-quality simulations. Such mass imbalances tend to disappear and significant changes in constituent concentrations can occur as the water-quality time step becomes small. Also, these constituent concentrations tend to converge toward those obtained using the event-driven simulation method, which conserves constituent mass.

The preliminary event-driven algorithm discussed here currently addresses only those constituents that behave as tracers. The algorithm needs to be expanded to consider constituent decay. The algorithm also needs to be evaluated using a wider range of networks and cases. The accuracy, storage requirements, and computation time for other types of water-quality modeling problems, such as source tracing, water age, and chlorine decay need to be examined. Preliminary results are presented here to help motivate additional efforts to improve water-quality simulations in EPANET.

The results presented here indicate that, in general, a water-quality time step of 1 s may be necessary to obtain acceptable mass-balance results when using the time-driven approach in EPANET. For large networks, such a time step can require considerable computational effort. Statistics for execution times for TEVA-SPOT, using EPANET and the time-driven algorithm, are provided in for several network models, including Network N4 (BWSN, called Network E3 in the reference) for a 1 s water-quality time step. Results in the reference are for a subset of the nodes considered here, only those with a nonzero demand, and include some additional computations beyond those used for this paper but demonstrate substantial execution times. For a single injection, execution times were about 70 min using a 2.3 GHz processor. For injections at all nonzero demand nodes for the network, the execution time was about 16 days for a server with four such processors using 32 cores, with 32 simultaneous simulations being performed. The event-driven algorithm is not fully developed; however, because a water-quality time step as long as the hydraulic time step can be used, it is expected to generally require less computational effort than the time-driven algorithm if conservation of mass is required. Table  compares execution times for the time-driven and event-driven algorithms for examples used in this paper. As noted above, these examples were selected to demonstrate mass imbalances. A time step of 1 s was used for the time-driven algorithm and 3600 s was used for the event-driven one. To provide a second injection location for Network N3, Node 300 is included, although no example is presented using this node. The execution times were obtained using a single 2.8 GHz processor. In general, for the cases considered here, the event-driven approach requires substantially shorter execution times than the time-driven approach in EPANET when a 1 s water-quality time step is used.