modified lag rang ian method for modeling water quality in

8/14/2019 Modified Lag Rang Ian Method for Modeling Water Quality In

1/16

Water Research 38 (2004) 29732988

Modified Lagrangian method for modeling water quality in

distribution systems

G.R. Munavalli, M.S. Mohan Kumar

Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India

Received 31 May 2002; received in revised form 19 January 2004; accepted 16 April 2004

Abstract

Previous work has shown that Lagrangian methods are more efficient for modeling the transport of chemicals in a

water distribution system. Two such methods, the Lagrangian Time-Driven Method (TDM) and Event-Driven Method

(EDM) are compared for varying concentration tolerance and computational water quality time step. A new hybrid

method (EDMNET) is developed which improves the accuracy of the Lagrangian methods. All the above methods are

incorporated in an existing hydraulic simulation model. The integrated model is run for different network problems

under varying conditions. The TDM-generated solutions are affected by both concentration tolerance and water quality

time step, whereas EDM solutions are dependent on concentration tolerance. The EDMNET solutions are less sensitive

to variations in these parameters. The threshold solutions are determined for all the methods and compared. The hybrid

method simulates the nodal concentrations accurately with least maximum segmentation of network and reasonable

computational effort as compared to the other Lagrangian methods.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Bulk decay; Distribution system; Dynamic modeling; Quality time step; Concentration tolerance; Wall decay; Water

quality

1. Introduction

It is well known that the quality of drinking water can

change within a distribution system. The movement or

lack of movement of water within the distribution

system may have deleterious effects on a once acceptable

supply. These quality changes may be associated withcomplex physical, chemical and biological activities that

take place during the transport process. Such activities

can occur either in the bulk water column, the hydraulic

infrastructure, or both, and may be internally or

externally generated [1]. The ability to understand these

reactions and model their impact throughout a distribu-

tion system will assist water suppliers in selecting

improved operational strategies and capital investments

to ensure delivery of safe drinking water [2].

Basically water quality modeling is simulated in a

steady or a dynamic environment. In steady-state

modeling, the external conditions of a distribution

network are constant in time and the nodal concentra-

tions of the constituents that will occur if the system is

allowed to reach equilibrium are determined. These

methods can provide general information on the spatialdistribution of water quality. In dynamic models the

external conditions are temporally varied and the time

varying nodal concentrations of the constituents are

determined. The algorithms developed include steady-

state [37] and dynamic [1,812] models.

Rossman and Boulos [13] have given a comprehensive

description of dynamic modeling and the existing

numerical solution methods hence a review will not be

repeated here. Instead the treatment given to the

reactions by these methods are presented and also the

advantages and limitations of existing Lagrangian

methods are discussed. In the Time-Driven-Method

ARTICLE IN PRESS

E-mail addresses: [email protected]

(G.R. Munavalli), [email protected] (M.S. Mohan Kumar).

0043-1354/$- see front matterr 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.watres.2004.04.007


2/16

(TDM) the constituent concentration of a segment is

subjected to reaction at every water quality time step

(Qstep). The Qstep is a computational time step at which

the quality conditions of the entire network are updated.

In the Event-Driven-Method (EDM) procedure the

constituent concentration in all the pipe segments are

subjected to reaction with respect to the length of the

subhydraulic time step [11]. In both methods the kineticreaction mechanism continues with time under the

conditions of zero flow or flow reversal in pipes. Both

the TDM and EDM are free from numerical dispersion

and phase shift errors when compared with Eulerian

methods. Basically the TDM simulation procedure is

carried out in steps of pre-specified Qstep. Hence it is

possible that during any step more than one segment

may be consumed at the downstream node of a pipe. If

the segments consumed have different concentrations

then this leads to an artificial mixing whose effect is

more pronounced in tracing sharp concentration fronts.

In addition, the TDM solutions are affected by a loss of

resolution in concentration and accuracy is dependent

on both Qstep and concentration tolerance used. Even

though the EDM is supposed to be accurate irrespective

of the Qstep used, the concentration tolerance used and

the tolerance dependent subsegmentation process at

changing hydraulic conditions may affect the accuracy

of the method. In the EDM procedure the concentration

conditions at a node are updated only when an event

occurs at that node. Also all the segments and nodes are

updated at the end of a hydraulic time step or output

reporting time whichever occurs first. At the start of the

simulation the event occurrences are dictated by the

travel time in the pipes.Rossman and Boulos [13] tested and compared the

Eulerian (FDM and DVEM) and Lagrangian (TDM

and EDM) methods. They concluded that the Lagran-

gian methods are more efficient for simulating the

chemical transport in a water distribution system. The

testing of the methods was done for analytical solutions,

actual field studies and variable sized networks. The

models are contrasted with respect to analytical solu-

tions for validation at zero concentration tolerance and

a particular Qstep.

It is useful to study the differences exhibited by the

Lagrangian methods for a normally used hydraulic time

step of 1 h as reporting time under varying tolerance and

Qstep values with no restriction on the number of

segments generated. It is also interesting to study how

the analytical solutions are contrasted with respect to

the solutions obtained by these methods under varying

concentration tolerance and Qstep values. It is obvious

that the solution given by TDM and EDM may perform

better against the analytical solution for zero concentra-tion tolerance and reasonably small quality time step.

The relative comparison of the methods with analytical

solutions considering the variations in concentration

tolerance and Qstep brings out the degree of variability

exhibited by the methods with respect to the true

solution for the system. Application of the methods to

real life networks will generate a large number of new

segments and this segmentation can be controlled by

imposing a concentration tolerance.

Also there is a need to develop a methodology which

can nearly eliminate the limitations discussed earlier in

the Lagrangian models for the transport of chemical

species. A hybrid methodology developed herein utilizes

the better features of existing Lagrangian methods. The

performance of all the methods is tested against

available analytical solutions under varying conditions

of concentration tolerance and Qstep for both reactive

and non-reactive constituents. The methods are also

applied to network problems of varying size and a set of

solutions is obtained for a range of concentration

tolerance and Qstep values. An attempt is made to

compare the representative solutions given by existing

methods and the proposed hybrid method at selected

nodes of a network problem. The results are interpreted

in terms of the maximum number of segments generated(maximum segmentation of the network) at any time

during the simulation and the solution time.

2. Governing equations

The methodology is predicated on the assumptions of

one-dimensional flow, single or consecutive steady-state

(extended period simulation) network flow hydraulics,

complete and instantaneous mixing at the nodes, ideal

plug flow with reaction, dispersion being negligible,

single constituent with one or more feed sources and

ARTICLE IN PRESS

Nomenclature

CE external source concentration of constituent

(M/L3)

ci concentration of constituent in pipe i (M/L3)

cnj chlorine concentration at node j (M/L3

)I number of incoming pipes at a node

Is set of links with flows into the tank

Js set of links withdrawing flow from the tank

Njn total number of nodes in the network

QE external flow into a node (L3/T)

Qi flow in pipe i (L3/T)

Qstep water quality time step (T)

R(ci) first-order reaction rate expression for pipe i

R(Cs) first-order reaction rate expression for tankt time (T)

ui mean flow velocity in pipe i (L/T)

Vs volume of storage tank (L3)

G.R. Munavalli, M.S. Mohan Kumar / Water Research 38 (2004) 297329882974


3/16

reactions based on first-order kinetic characteristic

functions.

2.1. Network model

A water distribution system comprises of links (pipes,

pumps, valves) interconnected by nodes (junctions,

storage points) in some particular branched or looped

configuration. The network model is represented by

node-link system. A network water quality model

determines how the concentration of a dissolved

substance varies with time throughout the network

under a known set of hydraulic conditions and source

input patterns.

2.2. Hydraulic model

The hydraulic simulation model [14] is modified to

handle the extended period simulation and is applied togenerate the dynamic flows in pipes during the specified

hydraulic time steps (normally 1 h).

2.3. Water quality model

The water quality model formulation is from Ross-

man et al. [10].

Transport of the constituent along the ith pipe is given

by the classical advection equation:

@ci@t

ui@ci@x7Rci; 1

where, ci is the concentration of constituent in pipe i

(mg/l) as a function of distance x and time t; ui the mean

flow velocity in pipe i (m/s); and Rci the reaction rate

expression (equals zero for conservative constituent).

Instantaneous and complete mixing at the node is given

by the equation:

cnj

PIi1 Qici QECEPI

i1 Qi QE; j 1;y; Njn; 2

where, I is the number of incoming pipes at node j; Njnthe total number of nodes in the network; QE the

external source flow into node j m3

=s; and CE theexternal source concentration into node j (mg/l).Mass balance at storage tanks is given by

dVsCs

dtX

iAIs

QiciX

jAJs

QjCs RCs; 3

where, Is is the set of links with flows into the tank; Jsthe set of links withdrawing flow from the tank; Vs the

volume of storage tank (m3); Cs the concentration of

constituent (mg/l) within a storage tank; RCs the first-

order reaction rate expression for a tank; Qi the flow

(m3/s) in pipe i; and ci is the concentration of constituent

(mg/l) in pipe i:

3. Numerical methods for water quality modeling

The existing Lagrangian methods are described in

detail by Rossman and Boulos [13] and hence are not

discussed here. The proposed hybrid method is de-

scribed in the following sections:

3.1. Numerical hybrid method (EDMNET)

3.1.1. Terminology

Parcel: It is an imaginary finite volume of water

within a pipe.

Segment: It is the portion of a pipe volume considered

to be made up of a number of discrete parcels of water.

It is assigned with a constituent concentration as a

parameter and is represented by two separators at each

end.

Separator: It is a line that separates two segments and

is assigned with distance travelled with respect to theupstream end of the pipe (DT), time of creation (TC),

time of arrival at its downstream node (TA) and

effective residence time (ERT) as parameters. The two

separators of a segment are associated with the most

downstream and most upstream discrete parcels of

water within that segment.

Activity: An activity is said to occur when a separator

in any of the pipe reaches its downstream node.

Effective residence time (ERT): It is defined for both

the separators and the discrete parcels of water. It is the

total time taken by any discrete parcel of water/

separator to reach the downstream node from the

upstream node of a pipe. For any interior discrete

parcel, (ERT) can be calculated using the time slope of a

segment and the location of the parcel within the

segment.

Time slope (TS): If ERT of a separator is represented

by an ordinate, then the line joining ordinates of two

separators for a segment represents the time slope.

Generation of new separator/segment: A new separa-

tor/segment in all outgoing pipes from a upstream node

of the pipe is generated when the difference in

constituent concentration at that node and in most

upstream segment of the pipe exceeds imposed specified

concentration tolerance.

3.1.2. Basic concept of the method

It is a fact that the discrete parcels within a segment

from the downstream end to the upstream end have

linearly varying effective residence times which are

represented by the time slope. At any stage during the

simulation, a discrete parcel of a segment in a pipe

reaches its downstream node. An ERT of that discrete

parcel can be computed using the time slope and its

position in the segment. As the constituent concentra-

tion of that segment is known, the reacted concentration

for that parcel of water can be determined. Thus the

ARTICLE IN PRESS

G.R. Munavalli, M.S. Mohan Kumar / Water Research 38 (2004) 29732988 2975


4/16

constituent concentration at any node can be computed

by knowing these reacted concentrations of the discrete

parcels reaching that node from the incoming pipes. The

method is either governed by a specified water quality

time step or the system activity. The process of

computing the concentration and generating the new

segments (if and when required) is carried out at all thenodes irrespective of whether the Qstep or the system

activity governs the simulation. The proposed method is

described in detail in the following subsections.

3.1.3. Initialization

At the start of the simulation, each pipe has a single

segment with the first separator at the downstream node

and second separator at the upstream node. This

segment is assigned with the constituent concentration

of the downstream node. The second separator (at the

upstream node) has an ERT equal to the travel time of

the pipe while the first separator (at the downstreamnode) has zero value. The line joining the ERT of these

two separators represents the time slope as shown in Fig.

1. Also the second separator has time of arrival equal to

the pipe travel time whereas the first separator is already

at the downstream node. These times of arrival

constitute the scheduled activity times till any change

in the hydraulic conditions occur. The second segment

for this pipe as and when it is created will follow the

second separator and it carries the concentration of its

upstream node till any change occurs in the concentra-

tion at that upstream node of the pipe. Also this second

segment has zero time slope indicating that all thediscrete parcels in this segment will have same ERT

equal to pipe travel time till any change in hydraulic

conditions occur.

3.1.4. Time step computation

The first activity is scheduled to occur at a time equal

to the least of all the travel times of separators in the

entire network. Till this time the second separator of all

the pipes carry the concentration (which has not

changed since no activity occurred in the entire network

at any node) of the upstream node forward. But the

concentration at all the nodes is continuously changing

and this change needs to be carried forward. Hence in

the case of a large time gap between two successive

activities than Qstep, it is required to update theconcentrations at all the nodes in between intervals of

Qsteps also. Thus the simulation clock is either moved

to the next scheduled system activity time or the

previous time is increased by Qstep.

3.1.5. Sequence of steps at any time

Case (a): If System activity governs the simulation.

In this case a separator in one of the pipes

corresponding to that activity reaches its downstream

node and the separators in other pipes do not reach their

downstream nodes. But in rare cases two activities occur

simultaneously. In such cases the algorithm handles theactivities one by one. The separators in all the pipes are

moved forward by a length corresponding to the time

period equal to the difference of current system activity

time and previous time, and the distance moved by them

is updated.

First consider the pipe in which a separator has

reached its downstream node. The arrival of a separator

at its downstream node indicates that the first discrete

parcel of the next segment in line has reached the

downstream node effecting a change in concentration.

As the ERT of the separator (and hence the discrete

parcel) and the concentration of the segment are known,

the reacted concentration contribution of this discrete

parcel to its downstream node can be calculated. The

time of creation for this discrete parcel is the difference

between the current time and its ERT. If the reaction

coefficient is considered to be varying with hydraulic

conditions then the hydraulic time periods through

which this discrete parcel has passed should be

identified. And the discrete parcel is subjected to a

change in concentration with the appropriate reaction

coefficient and the corresponding time period. This

process of computing the reacted concentration con-

tribution is illustrated in Fig. 2. In this figure the

computation is illustrated for a discrete parcel which hasreached its downstream node at a time of 1370 min. By

knowing its ERT (computed using TS) the TC can be

computed as 1224 min. The discrete parcel has passed

through different hydraulic time steps each having a

different reaction constant. Then using the first-order

reaction rate expression the reacted concentration

contribution can be determined as represented in the

figure. The separators and segments are reordered for

this pipe. And the time slope for the most downstream

segment is computed.

Next all the pipes where the separators have not

reached their downstream nodes are considered. As the

ARTICLE IN PRESS

Fig. 1. Definition sketch: initialization (EDMNET).



5/16

time slope and length of the most downstream segment

are known, the ERT of the discrete parcel reaching the

node in that segment can be determined. Then the time

of creation of this discrete parcel is the difference

between current time and the ERT of the discrete parcel.

The reacted concentration contribution of this discrete

parcel can be computed in a similar way as explained

earlier.

The concentrations at all the nodes are then computed

with these reacted concentration contributions from all

incoming pipes and using Eq. (2). Then new separators/

segments are created at each node by comparing the new

nodal concentrations with the concentration of most

upstream segment in all the outgoing pipes from that

node depending on the specified concentration tolerance

imposed.

Case (b): If Qstep governs the simulation.

In this case no separator in any pipe reaches itsdownstream node. All the separators are moved forward

by a time period equal to Qstep and the distance

travelled by them is updated. Then ERT and time of

creation for all the discrete parcels are computed as

illustrated earlier. The remaining steps namely determin-

ing the reacted concentration contributions, nodal

concentrations and creating the new separators/seg-

ments are also same as illustrated earlier.

The above sequence is carried out till the end of a

hydraulic time step. It usually happens that the

scheduled activity time or the simulation clock time

increased by the Qstep does not coincide with the end of

a hydraulic time. Hence it is necessary to determine that

last time step (usually less than Qstep) and carry out all

the steps as applied for case (b) above.

3.1.6. Sequence of steps at the start of any hydraulic time

stepAt the start of a next hydraulic time step a new set of

flows are computed. Now the parameters ERT and time

of arrival of the separators have to be changed in the

pipes where the flows are affected. The computation of

new set of these parameters is done depending on

whether the flow in the pipe reverses or not. If the flow

does not reverse in a pipe then the computation of these

parameters is simple. For any separator the new ERT

and TA are given by

ERT Current time TC Time reqired to reach

the downstream node with the current velocity;

TA Current time Time reqired to reach

the downstream node with the current velocity:

This is illustrated in Fig. 3 and the ordinates indicate

ERT of separators. Note that a new separator is

introduced at the upstream node of the pipe. This is

essential as the discrete parcels (yet to enter the pipe)

from new segment will have a different ERT (equal to

pipe travel time). Also note that the ordinates of

ARTICLE IN PRESS

Fig. 2. Computation of reacted concentration contribution

(EDMNET).

Fig. 3. Definition sketch: handling no flow reversal (EDM-

NET).



6/16

separator one (sep 1) in old flow and new flow are the

same.

But the computation of ERT and TA for separators in

pipes with flow reversals is entirely different. The

discrete parcel which is about to reach the downstream

end with respect to old flow has to travel back towards

the current downstream end and hence its ERT in thepipe is longer. Hence it is necessary to distinguish

between the two ERT values of a discrete parcel in the

most upstream segment (current) and a discrete parcel

about to enter the pipe. It is done by introducing a

dummy segment of zero length (seg 4 in Fig. 4) at the

upstream end of the pipe. And similarly the discrete

parcel which has just entered the upstream end with

respect to old flow has in effect zero ERT in the pipe. It

is illustrated in Fig. 4. The new ERT and TA for the

separators are computed using the same relations

quoted earlier. It should be noted that a new separator

(sep 5) is introduced at the upstream of the pipe andseparator one (sep 1) has zero ERT.

These two sequences viz. at any time and at the start

of a hydraulic step are continued till the end of a total

simulation time.

3.1.7. Analysis and discussion on the proposed hybrid

method

The main objective of the proposed hybrid method is

to consider and carry forward the effect of changes in

the nodal concentrations as much as possible. That is

why all the nodal conditions are updated regularly either

at activity occurrence times or at times increased by

Qstep. This helps in proper simulation of existing

concentration conditions in the pipe unlike the EDM

procedure. It should be noted that moving the simula-

tion clock by a Qstep does not always result in finersegmentation of the pipe. It is only used to update the

conditions regularly in case the occurrence of an activity

is delayed. The generation of the new segments is mainly

controlled by the difference in the concentrations at a

node and in the most upstream segment of an outgoing

pipe from that node being greater than the tolerance. As

all the nodal concentrations are updated and new

segments are created at any time (unlike EDM), rarely

the two successive activities differ by the specified Qstep

of 5 min in this method. However the effect Qstep on

EDMNET results are shown in the application exam-

ples. Also the treatment given to the reaction term isentirely different from the other methods. The concen-

tration of a parcel of water reaching its downstream

node is subjected to reaction for its ERT in that pipe.

The ERT consists of time periods either having a

constant or varying (in case of chlorine) reaction

constant. The method has many advantages in changing

flow conditions. Also it is possible that a little more

solution time may be needed in some cases as more

number of events are covered by the method.

4. Testing of methods against analytical solutions

All the methods are tested against the analytical

solutions for two test problems. The objective of

analytical testing is to study how closely the solutions

given by the methods agree with analytical solutions as

the concentration tolerance values are varied between

0.05 and 0 mg/l. In addition, the effect of Qstep is also

tested against the analytical solution for TDM.

Test problem 1. The schematic of test problem 1 is

shown in Fig. 5 and is a modified version of the problem

used by Boulos et al. [11]. The Tables 1 and 2 summarize

the pipe and node characteristics, respectively. All the

pipes have a roughness coefficient of 120. The supplysources A, B and C represent pumping wells with a total

head of 50.0, 56.0 and 60.0 m, respectively. The three

well pumps are identical and the operating data is

presented in Table 3. The control valve in pipe 2 has a

minor loss coefficient of 10.0. The chlorine concentra-

tion of 1.0, 2.0 and 1.5 mg/l are injected constantly at

sources A, B and C, respectively. The wall reaction

parameter is set to zero for all the pipes. This test

problem is meant to validate the EDMNET model

against an analytical solution, and to illustrate the effect

of concentration tolerance and Qstep values on the

performance of TDM. As EDM and EDMNET

ARTICLE IN PRESS

Fig. 4. Definition sketch: handling flow reversal (EDMNET).



7/16

solutions are same for constant flow conditions this test

problem does not produce an effective comparison of

these two methods. The nodal concentrations are

obtained with a reporting time of 3.0 min. For TDM

solutions, Qsteps of 3 and 1 min are used along with

concentration tolerances of 0.0 and 0.05 mg/l. The

analysis of the solutions is done for the nodes 1 and 2.

The TDM fails to simulate the concentration fronts

correctly as shown in Fig. 6(a) for zero tolerance and

Qstep of 3 min. This fact can be seen at the sharp change

in concentration fronts for both the nodes. Fig. 6(b)shows a TDM and analytical solutions are identical for a

concentration tolerance of 0.0 mg/l and Qstep of

1.0min. But TDM solution exhibits oscillations for

concentration tolerance of 0.05 mg/l even for a smaller

Qstep of 1.0 min as shown in Fig. 6(c). All these

observations show that TDM solution is affected by

both the concentration tolerance and Qstep values. The

EDMNET solutions are obtained at Qstep of 3 min and

a concentration tolerance of 0.05 mg/l. In contrast to

TDM solution the EDMNET/EDM solution is indis-

tinguishable from analytical solution even for a con-

centration tolerance of 0.05 mg/l as shown in Fig. 6(d).Test problem 2. The test problem 2 is shown in Fig. 7

with all node details. It was previously used by Rossman

and Boulos [13]. The pipe characteristics are tabulated in

Table 4. The problem is meant to test how well the

method can track a reactive substance (chlorine) in a

network subjected to flow reversals. Initially all water in

the network is at a concentration of 0.50 mg/l and is fed

from A (with a concentration of 1.0 mg/l) reservoir.

After 6 h, pipe 1 is closed and the network begins to

receive water from the B (with a concentration of

0.50 mg/l) reservoir thus causing the flow reversal in

pipes 2 and 3. The chlorine is decaying with a bulk decay

constant of 2.0 d1 with no wall reaction. The network

nodal concentrations are simulated by all the methods

with a Qstep of 5 min and varied concentration

tolerances (0.0 and 0.05 mg/l). The TDM solutions for

Qstep of 5 min and concentration tolerance of 0.00 and

0.05 mg/l are shown in Figs. 8(a) and (b). The solution

appears to be too sensitive for the concentration

tolerance variations. It shows that the solution fails to

simulate the concentration fronts even for zero tolerance

at a Qstep of 5 min. The TDM solution is found to track

the fronts better at a Qstep of 2 min and 0.00mg/l

concentration tolerance, but the introduction of

0.05 mg/l concentration tolerance results in an oscillat-ing solution. The EDM (0.00 and 0.05mg/l) and

EDMNET (0.00 and 0.05 mg/l at Qstep of 5 min) results

are shown in Figs. 8(c)(f), respectively. The EDM

solution simulates results which are indistinguishable

from the analytical solution at 0.00 mg/l concentration

tolerance. But the EDM solutions also exhibit variations

between concentration tolerance values of 0.05 and

0.00 mg/l with the analytical solution. This is due to the

fact that EDM needs subsegmentation at the sixth hour

due to the change in hydraulic condition and the

subsegmentation depends on the concentration toler-

ance used. If the concentration tolerance used does not

ARTICLE IN PRESS

Fig. 5. Network of Test problem 1.

Table 1

Pipe data (Test problem 1)

Pipe no. Length

(m)

Diam

(mm)

Reaction

coefficient

(day1)

Flow (l/s)

1 300.0 480 3 320.39

2 600.0 350 5 21.40

3 300.0 480 3 557.70

4 650.0 400 5 52.89

5 400.0 350 20 58.20

6 300.0 480 10 671.91

7 600.0 300 20 210.81

8 400.0 350 20 389.19

Table 2

Node data (Test problem 1)

Node no. Demand (l/s) Elevation (m) Initial

concentration(mg/l)

1 400.0 120.0 0.6

2 200.0 120.0 0.7

3 350.0 120.0 0.8

4 600.0 120.0 0.6

5 0.0 50.0 0.6

6 0.0 56.0 0.7

7 0.0 60.0 0.8

Table 3

Pump characteristic data (Test problem 1)

Head (m) Flow rate (l/s)

130.0 0.0

120.0 1000.0

100.0 2000.0



8/16

divide the existing segment into sufficient number of

subsegments required for representing the concentration

profile then the EDM results in such a solution. But this

variation (Fig. 8(d)) is less when compared to TDM

(Fig. 8(b)). The EDMNET solutions are virtually

identical with analytical results for the extreme tolerance

values used.

The contrasting of all the methods as done above with

analytical solutions shows that TDM solutions are

concentration tolerance and Qstep dependent, the

EDM solutions are dependent on concentration toler-

ance (during subsegmentation) and EDMNET is less

sensitive to both these parameters in the range used. For

these examples, the EDMNET solutions obtained by

using a coarser tolerance and a coarser Qstep are

comparable with those obtained from other methods for

a finer tolerance.

ARTICLE IN PRESS

Fig. 6. Analytical validation of Test problem 1: (a) TDM for tolerance=0.00mg/l and Qstep=3.0 min; (b) TDM for

tolerance=0.00 mg/l and Qstep=1.0min; (c) TDM for tolerance=0.05 mg/l and Qstep=1.0min; (d) EDMNET for toleran-

ce=0.05 mg/l and Qstep=3.0 min.


Table 4

Pipe data (Test problem 2)

Pipe no. Length (m) Diam (mm) Roughness Flow (l/s)

06 h >6 h

1 3048 457 100 147.33 0.00

2 1524 457 100 134.73 12.60

3 61 457 100 122.12 25.20



9/16

5. Application: results and discussions

In this section all the methods are applied to two

networks of varying sizes subjected to dynamic condi-

tions. The objective of this section is to evaluate thesensitive behavior of the methods under varied concen-

tration tolerance and Qstep values. Both concentration

tolerance and Qstep are varied for TDM and EDM-

NET, whereas only the concentration tolerance is varied

for EDM. A concentration tolerance range of 0.0025

0.05 mg/l is considered and sets of solution are obtained.

The TDM and EDMNET solutions are obtained for

Qstep of 5, 3 and 1 min at each of the concentration

tolerance in the above range. In all, seven sets consisting

of three TDM, three EDMNET and one EDM solutions

are obtained at each node for both the test problems.

Then analysis of each set of solutions is done at selected

nodes of each test problem. The main thrust of

analyzing the results is to study the relative variation

of the solutions within each set and to find out a

threshold solution. A threshold solution is a solution at

a concentration tolerance below which there is nosignificant improvement in the solutions. The threshold

solutions of each set are compared with each other. The

comparison is also made in terms of the maximum

segmentation and the solution time for each method at

this threshold solution. The maximum segmentation

refers to the highest number of segments by which the

network is divided at any time during the simulation

process.

Test problem 3. The methods are next applied to a

system for which field sampling of water quality

behavior had been made by Environmental Protection

Agency (EPA) and American Water Works Association

ARTICLE IN PRESS

Fig. 8. Analytical validation of Test problem 2: (a) TDM for tolerance=0.00mg/l and Qstep=5.0 min; (b) TDM for

tolerance=0.05mg/l and Qstep=5.0 min; (c) EDM for tolerance=0.0 mg/l; (d) EDM for tolerance=0.05 mg/l; (e) EDMNET for

tolerance=0.00mg/l and Qstep=5.0 min; (f) EDMNET for tolerance=0.05 mg/l and Qstep=5.0 min.



10/16

and Research Foundation (AWWARF). The system, the

Brushy plains zone of the south central Connecticut

Regional Water Authority, has been used many times in

the past to validate and test network water quality

models [2,15]. The network schematic is shown in Fig. 9.

The bulk decay factor and wall decay factor used are

0.55d1 and 0.15 m/d, respectively. The bulk decayfactor in the tank is assumed to be 0.55 d1. The input to

the network has a constant chlorine value of 1.15 mg/l.

All the numerical methods are applied to simulate the

chlorine concentrations at all the nodes with a wide

range of concentration tolerance as specified above. For

each of the Qstep values the TDM simulations showed

wide variations in the nodal concentrations with respect

to the concentration tolerances used. In case of the

TDM the threshold solution is identified at a concentra-

tion tolerance of 0.005 mg/l for all the Qstep values used.

Similarly the EDMNET simulations are also analyzed

and in contrast to the TDM simulations EDMNETexhibited no such variations except at few locations for a

concentration tolerance of 0.05 mg/l, and the threshold

solution corresponds to a concentration tolerance of

0.015 mg/l for all the Qsteps. The EDM solutions also

showed variations at a number of periods along the

profile in the coarser part of the concentration tolerance

range, but in the finer range the variation between the

solutions is less. This is due to the fact that the coarsertolerance used results in an insufficient number of

subsegments to represent the concentration profile of

an existing segment. The threshold EDM solution is

obtained at a concentration tolerance of 0.0075 mg/l.

Table 5 shows the maximum network segmentation and

the solution times for all the methods at each of the

threshold solutions. It can be seen that as Qstep

decreases the number of segments used by TDM

increases; whereas the variation in Qstep values has a

least effect on the number of segments generated

indicating that a 5 min Qstep is sufficient to fill the gap

between delayed activity occurrence times. The EDMsolution required a maximum segmentation of the

network and comparatively more solution time. It can

be noted that the EDMNET solutions discretize the

network into the least number of segments with reason-

able computational effort. Also if all the methods are

compared at the same concentration tolerance of

0.01 mg/l and a Qstep of 3 min the EDMNET performs

in between the TDM and EDM as far as the segment

generation and solution times are concerned. But it

simulates the conditions better at these parameters as

the variations exhibited by the EDMNET solutions are

less compared to the other methods. The nodal

concentrations simulated by the threshold solutions of

the TDM at a Qstep of 1 min, the EDMNET at a Qstep

of 5 min and the EDM are given for nodes 3, 11, 19 and

34 in Fig. 10. The solutions given by all the methods

represent the general pattern of the observed chlorine

levels at these nodes. The system was also analyzed using

a wall decay factor of 0.457 m/d, but the variation

between the observed and simulated chlorine levels is

large. A sensitivity study with respect to the wall decay

factor is needed to match the general pattern of chlorine

levels in a better way.

The objective of this case study is to illustrate how the

various water quality models developed in the presentstudy predict the fluoride levels (conservative constitu-

ent) for a well calibrated extended period hydraulic

simulation model and to compare with field observed

values. Using a finer concentration tolerance and water

quality time step of 3 min all the models are run and the

results are obtained. The input fluoride concentrations

at the source are taken from EPANET Network 2. The

fluoride concentrations predicted by all the models are

shown in Fig. 11. All the models have resulted in

identical fluoride concentrations and the results for the

nodes 3, 10, 19 and 34 are shown in Figs. 11(a)(d),

respectively. Also the observed fluoride concentrations

ARTICLE IN PRESS




11/16

ARTICLE IN PRESS

Table 5

Segments and solution time at threshold tolerance (Test problems 3 and 4)

Test problem Method Concentration tolerance (mg/l) Qstep (min) Segments Solution time (s)

3 (Reactive) TDM 0.005 5.0 830 3.07

TDM 0.005 3.0 990 3.24

TDM 0.005 1.0 1260 3.24TDM 0.01 5.0 553 3.02

TDM 0.01 3.0 661 3.13

TDM 0.01 1.0 737 3.68

EDM 0.0075 1193 9.66

EDM 0.01 997 7.74

EDMNET 0.015 5.0 678 5.16

EDMNET 0.015 3.0 670 5.17

EDMNET 0.015 1.0 671 5.22

EDMNET 0.01 5.0 854 5.44

EDMNET 0.01 3.0 846 5.55

EDMNET 0.01 1.0 855 5.61

3 (Conservative) TDM 0.00005 3.0 1308 3.74

EDMNET 0.00005 3.0 804 4.29EDM 0.00005 3.0 790 7.47

4 TDM 0.0025 5.0 1296 11.60

TDM 0.0025 3.0 1685 13.98

TDM 0.0025 1.0 2773 14.52

EDM 0.005 2205 30.71

EDMNET 0.005 5.0 1228 18.60

EDMNET 0.005 3.0 1228 19.38

EDMNET 0.005 1.0 1228 19.32

Fig. 10. Comparison of threshold solutions for Test problem 3 at nodes (a) 3, (b) 11, (c) 19 and (d) 34.



12/16


13/16

of segments compared to other methods in a reasonable

computational time. Also the EDMNET solutions and

the corresponding segmentation of network are least

affected by the change in the Qstep values. Again it

suggests the fact that a 5 min time step is sufficient to

update the network conditions in the case of delayed

activity occurrences. The EDM results are also accurate

but result in more segmentation compared to EDM-

NET. The TDM solutions are fast but the accuracy is

dependent on both the concentration tolerance and

Qstep values. The concentrations at the nodes 123, and

163 for the threshold solutions are shown in Fig. 13(B).

Next, all the methods are used to simulate the waterage for this network. The results of the run in terms of

segmentation and solution time are given in Table 9 for

different tolerance and Qstep values. Further the water

ages simulated at these tolerances and Qsteps by all the

methods for the nodes 123 and 163 are shown in Fig. 14

(the legend in the figure shows method, tolerance and

Qstep). The results show that the EDMNET results at

coarser tolerance (12 min) and coarser Qstep (5 min) are

comparable with those of TDM results at finer tolerance

(3 min) and finer Qstep (1 min). It can be further noted

that if coarser values of tolerance and Qstep are used for

TDM an incorrect peak at 9 h for both the nodes occurs.

Thus the results indicate that the EDMNET solutions at

coarser tolerance and Qstep values are comparable with

those obtained using finer values from the other

methods.

6. Conclusions

All the water quality modules TDM, EDM and

EDMNET are encoded and integrated in the existing

hydraulic model. The integrated model was run on

different network test problems under varied tolerance

values for each water quality module. An additional setof results are also obtained with TDM and EDMNET at

varied Qstep values. The results of this study showed the

following:

1. Analytical comparison of the TDM solution shows

that it fails to simulate the exact solution for sharp

concentration fronts with coarser Qstep values even at

zero tolerance, but it results in a much closer solution

for a smaller Qstep and finer tolerance. In the

application test problems the method exhibited wide

variation in the solutions between coarser and finer

concentration tolerance values. Also the smaller Qstep

values need to be used even at a finer concentration

ARTICLE IN PRESS




14/16

tolerance in order to get results comparable with the

other two methods. Thus the TDM solutions are

dependent on both the concentration tolerance and

Qstep values.

2. The EDM simulates the analytical solution exactly

for a steady-state hydraulic conditions. But in changing

hydraulic conditions the method requires much finer

tolerance for simulating the exact solution. The EDM

though exhibits however much lower variations thanTDM, and its accuracy is largely dependent on the

concentration tolerance. If the concentration tolerance

used is not enough to divide the existing segment into

sufficient number of subsegments then the method may

result in an inaccurate solution. Otherwise each and

every segment is subjected to a refined subsegmentation.

As the number of subsegments increases the better is the

accuracy of the method is better.

3. The analytical comparison of the modified event

driven method (EDMNET) is excellent even for the

coarser concentration tolerance values used under all the

conditions. The EDMNET is less sensitive to the

ARTICLE IN PRESS

Table 6

Reaction parameters (Test problem 4)

Pipe no. Bulk

(d1)

Wall

(m/d)

Pipe no. Bulk

(d1)

Wall

(m/d)

40 0.31 3.0487 238 0.31 3.0487

50 0.31 1.524 241 0.31 3.0487101 0.31 6.0975 243 0.31 1.524

103 0.31 6.0975 245 0.31 1.524

105 0.31 6.0975 247 0.31 1.524

109 0.31 6.0975 249 0.31 1.524

111 0.31 6.0975 251 0.31 1.524

112 0.31 6.0975 257 0.31 1.524

113 0.31 6.0975 261 0.31 1.524

114 0.31 6.0975 263 0.31 1.524

115 0.31 6.0975 269 0.31 1.524

116 0.31 3.0487 271 0.31 1.524

117 0.31 6.0975 273 0.31 1.524

119 0.31 6.0975 275 0.31 1.524

121 0.31 6.0975 277 0.31 1.524

186 0.31 3.0487 281 0.31 1.524193 0.31 3.0487 283 0.31 1.524

195 0.31 3.0487 287 0.31 1.524

197 0.31 3.0487 289 0.31 1.524

199 0.31 3.0487 291 0.31 1.524

201 0.31 3.0487 295 0.31 1.524

204 0.31 3.0487 297 0.31 6.0975

205 0.31 3.0487 299 0.31 6.0975

213 0.31 3.0487 301 0.31 6.0975

215 0.31 3.0487 303 0.31 6.0975

217 0.31 3.0487 305 0.31 6.0975

219 0.31 3.0487 307 0.31 6.0975

223 0.31 3.0487 311 0.31 3.0487

225 0.31 3.0487 315 0.31 3.0487237 0.31 3.0487

Table 7

Initial chlorine concentrations (Test problem 4)

Node

no.

Conc.

(mg/l)

Node

no.

Conc.

(mg/l)

Node

no.

Conc.

(mg/l)

10 0.15 157 0.20 215 0.21

15 0.15 159 0.20 217 0.2120 0.15 161 0.20 219 0.21

35 0.15 163 0.20 225 0.21

40 0.15 164 0.20 229 0.21

50 0.15 166 0.20 231 0.21

60 0.15 167 0.23 237 0.21

61 0.23 169 0.26 239 0.21

101 0.15 171 0.15 241 0.21

103 0.25 173 0.15 243 0.15

105 0.25 177 0.15 247 0.15

107 0.02 179 0.15 249 0.15

109 0.02 181 0.15 251 0.13

111 0.18 183 0.15 253 0.15

113 0.05 184 0.15 255 0.15

115 0.18 185 0.15 257 0.13117 0.15 187 0.26 259 0.13

119 0.24 189 0.15 261 0.13

120 0.15 191 0.21 263 0.15

121 0.25 193 0.15 265 0.23

123 0.25 195 0.21 267 0.21

125 0.15 197 0.21 269 0.15

127 0.15 199 0.21 271 0.15

129 0.15 201 0.21 273 0.21

131 0.15 203 0.21 275 0.28

139 0.15 204 0.26 601 0.25

141 0.15 205 0.15 Tank 1 0.10

143 0.15 206 0.28 Tank 2 0.05

145 0.15 207 0.21 Tank 3 0.05147 0.15 208 0.28 Lake 1.50

149 0.15 209 0.21 River 1.25

151 0.15 211 0.21

153 0.15 213 0.21

Table 8

Concentration pattern at river source (Test problem 4)

Time

period (h)

Concentration

(mg/l)

Time

period (h)

Concentration

(mg/l)

01 1.25 1213 1.50

12 1.25 1314 1.50

23 1.30 1415 1.25

34 1.30 1516 1.25

45 1.50 1617 1.25

56 1.50 1718 1.25

67 1.30 1819 1.50

78 1.30 1920 1.50

89 1.30 2021 1.50

910 1.00 2122 1.50

1011 1.00 2223 1.25

1112 1.00 2324 1.25



15/16

different concentration tolerance values used for simu-

lating the nodal concentrations. The Qstep value has the

least effect on the EDMNET solutions. In other words,

the method provides a solution at coarser concentration

tolerance which is comparable to the solutions of other

methods but with finer concentration tolerance or

smaller Qstep values. The EDMNET eliminates defi-

ciencies such as artificial mixing of segments and loss of

resolution in concentration as in the case of TDM, and

concentration tolerance dependent subsegmentation as

in the case of EDM. The maximum segmentation of the

network is least (for all simulations) with a reasonable

computational effort. Hence the method provides a

good tool for analyzing accurately reasonable size

networks for simulating the water quality within

distribution systems.

ARTICLE IN PRESS

Fig. 13. (A) Effect of Qstep on TDM solutions for Test problem 4 at nodes 123 and 163 and (B) comparison of threshold solutions for

Test problem 4 at nodes 123 and 163.

Fig. 14. (a) and (b) Water age simulations for Test problem 4 at nodes 123 and 163.



16/16

References

[1] Boulos PF, Altman T, Jarrige PA, Collevatti F. Discrete

simulation approach for network water quality models. J

Water Resour Plann Manage ASCE 1995;121(1):4960.

[2] Rossman LA, Clark RM, Grayman WM. Modeling

chlorine residuals in drinking water distribution systems.

J Environ Eng ASCE 1994;120(4):80320.

[3] Males RM, Clark RM, Wehrman PJ, Gates WE. Algo-

rithm for mixing problems in water systems. J Hydraul

Eng ASCE 1985;111(2):20619.

[4] Clark RM, Grayman WM, Males RM. Contaminant

propagation in distribution system. J Environ Eng ASCE

1988;114(4):92943.

[5] Shah M, Sinai G. Steady state model for dilution in water

networks. J Hydraul Eng ASCE 1988;114(2):192206.

[6] Wood DJ, Ormsbee LE. Supply identification for water

distribution systems. J AWWA 1989;81(7):7480.

[7] Boulos PF, Altman T, Sadhal K. Computer modeling of

water quality in large networks. J Appl Math Modeling

1992;16(8):43945.

[8] Liou CP, Kroon JR. Modeling the propagation of

waterborne substances in water distribution networks.

J AWWA 1987;79(11):548.

[9] Grayman JA, Clark RM, Males RM. Modeling distribu-

tion system water quality: dynamic approach. J Water

Resour Plann Manage ASCE 1988;114(3):295311.

[10] Rossman LA, Boulos PF, Altman T. Discrete volumeelement method for network water quality models. J Water


[11] Boulos PF, Altman T, Jarrige PA, Collevatti F. An event-

driven method for modeling contaminant propagation in

water networks. J Appl Math Modeling 1994;18(2):

8492.

[12] Islam MR, Chaudhary MH. Modeling of constituent

transport in unsteady flows in pipe networks. J Hydraul

Eng ASCE 1998;124(11):111524.

[13] Rossman LA, Boulos PF. Numerical methods for water

quality in distribution systems: a comparison. J Water


[14] Niranjan Reddy PV. General analysis, paramter estima-tion and valve operational policy in water distribution

networks. PhD thesis, Indian Institute of Science, Banga-

lore, 1994.

[15] Clark RM, Grayman WM, Males RM, Hess AF.

Modeling contaminant propagation in drinking water

distribution systems. J Environ Eng ASCE 1993;

119(2):34964.

[16] Clark RM, Rossman LA, Wymer LJ. Modeling distribu-

tion system water quality: regulatory implications. J Water


[17] Rossman LA. EPANET users manual. Risk Reduction

Engineering Laboratory, US Environmental Protection

Agency, Cincinnati, Ohio, 1994.

ARTICLE IN PRESS

Table 9

Segments and solution time for water age simulations (Test

problem 4)

Method Tolerance

(min)

Qstep

(min)

Segments Solution

time (s)

TDM 12 5 674 13.95TDM 3 1 2452 14.63

EDM 12 1307 17.96

EDM 3 2658 24.77

EDMNET 12 5 1075 16.26


modified lag rang ian method for modeling water quality in

Documents