74

5 Water quality modelling research

5.1 Contaminant movement and propagation models The contamination of drinking-water in distribution systems is relatively common and may lead to public health consequences (Geldreich, 1996). In a review in the USA Craun (and Colderon (1996) estimated that 26% of waterborne outbreaks were due t contamination of water in distribution systems and Hunter (1997) reported that 15 out of 42 waterborne outbreaks in the UK between 1911 and 1995 were due to distribution contamination. Geldreich (1996) described the importance of distribution contamination in Lima, Peru at the start of the cholera epidemic in the late 1980s. Contamination in distribution systems may result from many factors, including cross-connections to sewers, contamination at service reservoirs, poor hygiene practice during repair and maintenance work and back-siphonage into leaky pipes. Virtually all piped systems leak to some extent and the quality of water may be contaminated by ingress of pollutants through the leaks present in the pipes. Back-siphonage is a widely recognised problem in intermittent supplies, where pipes are not full for many hours of the day. However, it is also increasingly recognised that this is also a problem in fully charged systems, where back-siphonage may be induced by transient pressure waves caused by closing and opening valves within the system (LeChevalier, 2001). Whether back-siphonage occurs depends on a number of factors, for instance Davison et al(2002) note that the relationship between the soil water pressure and residual pressure in an intermittent system that retains some water within the pipe must favour movement of water from the soil to the pipe. Pathogens originate in faecal material within the environment. This may include excreta transport and treatment systems (e.g. sewers, septic tanks) as well as contaminated surface water bodies and solid waste dumps. Apart from direct cross-connections, pathogens must move from the source of faecal material to the source of water (the source-pathway-receptor model). During this process, ARGOSS (2001) and Shijven (2001) show that pathogens may be eliminated through die-off and predation and will may be exposed to attenuation processes (adsorption, filtration) that remove or retard them. Modelling these processes is important in order to understand the risk of different situations and to plan how risks can be managed, Once contaminated water enters into a pipe, it is also important to understand the movement of pathogens through distribution systems and the interactions between pathogen (or index organisms) with chlorine residual, pipe material and flow patterns. For instance, Kirmeyer and LeChevalier (2001) conclude that because fully charged distribution systems have plug flows, there would be little potential for mixing of pathogens entering the system and therefore a flow of water containing significant numbers of pathogens and depleted chlorine residual to move through the system even when the remaining water had significant levels of residual. Kirmeyer and LeChevalier (2001) note that although mixing is an important factor in primary disinfection, its role in secondary disinfection has not been evaluated. The research work currently being undertaken as part of this project attempts to model the processes described above. Specific research questions being addressed are:

75

1. The movement of contaminants from sources of pathogens through the soil and into the water supply pipes (this will help establish the potential polluted area beneath the drains and estimate the magnitude of pollution when the contaminants enter the drinking water pipes).

2. The hydraulic transient behavior of the water pipe network during the charging up

process in intermittent systems (i.e. when the pipes are filling-up immediately after supply has been resumed and just prior to reaching a fully pressurised state).

3. The movement of contaminants in the pipe network during the charging up process

(i.e. how the contaminants propagate in the network during the filling-up of the pipes immediately after supply has been resumed).

4. The impact of hydraulic surges on fully charged systems and the behaviour of

pathogens once introduced into a fully charged system.

To date work has been performed on the modelling in relation to (1) and (2) above and details of the model development are given below.

5.1.1 CONTAMINANT MOVEMENT AND LOADING.

Initial works has focused on development of the hydraulic models and movement of non-discrete particles through water and to use this to develop a model for pathogen movement. This area of the research is divided into three main components: Firstly, contaminant zones (CZ) are developed around foul water bodies based on the seepage process. This model predicts the variable concentration of the contaminants through the CZ. At present, this model has used non-discrete pollutants in the model to establish the major criteria for the model, the next stage will be to develop this model further using the work by Shijven (2001) and other water quality models developed for microbial transport through soil. Secondly, potential areas of pollution around a water supply pipe are established by considering the intersection of the contamination zone with the path of the water supply pipe. Spatial analysis techniques are used in this respect. Thirdly, contaminant loading into the water supply network are calculated using the intersection regions calculated above coupled with estimations related to the age and condition of the pipe network.

5.2 Component 1: Seepage Model Pollutant transport through the soil is a topic that has attracted significant interest and research from the hydrogeological communities as this has a significant impact on the microbial quality of groundwater and of the public health risks that will be caused through consuming this water (ARGOSS, 2001). Contaminants from ditches, polluted water bodies, and leaking sewers are the main source for groundwater and water supply networks pollution. As part of this research a seepage model has been developed to simulate contaminant seepage from foul water bodies and through the soil. This model has been developed and coded in C++. Herein is an introduction to this

76

model, but more details can be found in Yan, et al. (2002). As noted above, at present the model uses chemical parameters as the basis of the seepage model, but future work will focus on modelling of microbial transport. When considering seepage, important parameters include dimensions and shapes of the boundaries of pollution sources. In the developed seepage (refer to Figure 1), the distance and depth of the liquid in the ditch are B and H, respectively. Contaminant in the ditch seeps into soil from the bottom of the ditch. As the depth (y) increases, the distance (x) will increase, which means that the seepage envelope will enlarge during the process of seepage. There is also a change in concentration of contaminant during seepage due to filtration of soil. Therefore the model consists of two components, the first modelling the seepage process (which accounts for the potential polluted area), and the second modelling the variable concentration of contaminant migration through the soil. Seepage equation In order to solve this flow problem, Harr (1991) examined Zhukovsky’s function:

aw

Aekwiz =+=θ

(1) yixz +=

iw ψϕ += Where α = a parameter; A = a real constant; k = permeability of soil; ϕ = potential function; ψ = stream function; w = potential complex; and z = spatial complex.

Figure 1. Contamination seepage from ditch Separating this expression into real and imaginary parts gives:

)sin(

)cos(

aAex

k

aAey

k

a ψψ

ψϕ

ϕ

αϕ

=+

=−

Substituting ψ− for ψ and x− for x in Eqs. (2), we see that the system of streamlines defined by ψ in these equations is symmetrical about the y-axis. Hence, the y-axis can be

B

H

x

y

(2)

77

taken as the streamline 0=ψ . The free surface must satisfy the condition 0=+−k

y ϕ , and

2q−=ψ , from the first of Eq. (2) we find

απα

)12(

0)2

cos(

+−=

=−

nq

q

In particular, taking 0=n and substituting Eq. (3) with

2q−=ψ and ky=ϕ into the second

of Eqs. (2), we obtain for the free surface

yq

k

Aek

qxπ

−−=−

2 (4) letting 0=y in Eq. (4), we obtain for the half width of the ditch

A

kqBxy −=== 220

(5) Now taking 0=ϕ in Eq. (2), as 0=ψ at the bottom of the ditch, from the parametric equation for the perimeter of the ditch, we find HAy =−= , where H is the maximum depth of water in the ditch. Hence, the quantity of seepage from the ditch section is found from Eq. (5) to be

)2( HBkq += . Rearranging Eq. (4), we can find the seepage free surface equation:

yBHHeBHx +

−−+= 2)2(

21 π

(6) Variable concentration The concentration of contaminant will vary during movement through soil. The mechanism of transport and filtration of pollutant through soil is advection, hydrodynamic dispersion, and interactive processes between pollutant and soil surface (Harvey, 1991). A simple one-dimensional transport of pollutant dissolved in water through soil can be described by the following equation (PESTAN, 1994):

ck

ts

nycv

ycD

tc b

12

2

−∂∂−

∂∂−

∂∂=

∂∂ ρ

(7) Where c = Liquid-phase pollutant concentration ( lmg / ); t = time ( hr ); y = depth along the flow path ( cm ); D = dispersion coefficient ( hrcm /2 ); v = pore-water velocity ( hrcm / ); bρ = bulk density ( 3/ lmg ); n = porosity; s = solid-phase concentration ( lmg / ); and 1k = first-order decay coefficient in liquid phase( hr/1 ).

(3)

78

The term ∂s/∂t is the rate of loss of solute from liquid phase to solid phase due to sorption. Under the assumption of linear, instantaneous sorption, ∂s/∂t can be evaluated as ∂s/∂t =kd ∂c/∂t. For a continuous steady flow with initial concentration 0c seepage into the soil, the steady state outflow concentration, governed by Eq. (7) with ∂c/∂t=0 (Harter, 2000). The boundary condition of Eq. (7) governed by, 0,0 ccy == and 0, =∞= cy . With the assumption of ∂c/∂t =0 and the above boundary condition, we find the solution of Eq. (7)

yD

Dkvv

ecc 2

4

0

12 +−

= (8)

Where c = Pollutant concentration at depth y , ( lmg / ); and 0c = Initial pollutant concentration ( lmg / ). Illustration of seepage model To illustrate the developed seepage model, we need to provide estimates for the soil conditions including density, porosity, etc., and the pollution source dimensions, etc. The input data is listed in Column (1) of Table 1. Using the seepage equation developed above, the seepage flow nets with different stream functions and potential functions are calculated. Eq. (8) is one-dimensional with the relationship of concentration c and depth y . However, calculating the concentration across the flow region is a two-dimensional problem. Therefore, we divide the flow region into many flow pathways, consisting of streamlines, and then, each flow pathway is subdivided into elements by equipotential lines. The plane flow region is made into elements of flow lines and equipotential lines of curvilinear squares, as shown in Figure 2a. (a) (b) Fig. 2. Flow nets and variable concentration The one-dimensional Eq. (8) can be used in each element. In the elements, the average velocity and average depth are used to calculate the concentrations. Figure 2a shows flow nets for the data given in Column 1 of Table 1. The figure clearly illustrate that as depth

0

20

40

60

80

100

120

140

160

-100 -50 0 50 100

x (cm)

y (c

m)

C/C0

C/C0

Ei ri

Ei+1

vi

vi+1

Ei-1

79

increase, the streamlines approach its vertical asymptote, and equipotential lines approach straight lines. Figure 2b illustrates the seepage process in a single flow pathway, the velocity vi of the flow entering an element was calculated from the quantity. The distance ri of the element is calculated using the co-ordinate of flow pathway. The Eq. (8) is calculated in a loop from i=0 (the concentration of contaminant in the ditch) to i=n where n is the number of element in the flow pathway. To calculate the change concentration along the selected flow pathway, Eq. (8) yield the concentration of each element. The process is repeated for each flow pathway across the entire flow nets, to yield the concentration profile across the entire seepage envelope. Using the calculated concentration profile, we can find the concentration profile at the bottom and centre of the flow region. Figure 2a shows the change in relative concentration in both the x direction and y direction. At the bottom in Figure 2a, the curve shows that the relative pollutant concentration at depth 117=y cm has a maximum at the centre ( 0=x ) and is symmetrical about the ditch. The data is listed in Column (2) to (5) of Table 1. On the right of Figure 2a, the curve represents the variation of relative pollutant concentration at the centre ( 0=x ), we can see that the relative concentration diminishes as the depth increases. The data is listed in Column (2) to (5) of Table 2. Table 1. Relative concentration (c/c0) at average depth 117=y (cm)

Known parameters (1)

Distance (cm) (2)

c/c0 (3)

Distance x(cm) (4)

c/c0 (5)

-54.97 0.11 3.62 0.27 -47.52 0.13 10.88 0.26 -40.11 0.16 18.14 0.24 -32.75 0.19 25.43 0.22 -25.43 0.22 32.75 0.19 -18.14 0.24 40.11 0.16 -10.88 0.26 47.52 0.13

B=60 cm H=30 cm k=0.004 D=0.0006 cm2/hr k1=0.00022 /hr n =0.32

-3.62 0.27 54.97 0.11 Table 2. Relative concentration (c/c0) at distance 0=x (cm) Known parameters (1)

Depth y (cm) (2)

c/c0 (3)

Depth y (cm) (4)

c/c0 (5)

30.00 1.00 69.43 0.60 31.08 0.99 75.92 0.54 33.71 0.96 82.59 0.48 37.20 0.93 89.40 0.43 41.41 0.89 96.34 0.39 46.21 0.83 103.38 0.34 51.49 0.78 110.50 0.31 57.16 0.71 117.69 0.27

B=60 cm H=30 cm k=0.004 D=0.0006 cm2/hr k1=0.00022 /hr n =0.32

63.16 0.66

80

.2.1.1 Component 2: Potential Areas of Pollution.

This model uses spatial analysis techniques to identify the intersection of the CZ with the path of water distribution pipes After the intersection regions have been identified contaminant load into the network is predicted using estimations related to the age and condition of the pipe network. At present the model has been tested using data from the literature and estimating conditions of the pipes based on reasonable estimates. Combined with the output from the seepage model mentioned above, a spatial analysis algorithm is developed to aid in locating of potential areas of contamination in water distribution systems. At present this is based on the proximity and layout of sewer systems and other key sources of pollution, but has not yet addressed septic tanks and other forms of on-site sanitation. Leaking sewer pipe Figure 2a shows an example flow net. For practical purposes, the free surface of the flow net can be considered to approach its vertical asymptote, and the equipotential lines can be taken as horizontal at a depth of 2/)2(3 HBy += (Harr, 1991). From Eq. (6), when

2/)2(3 HBy +> , 2/)2( HBx += , and the width between the two vertical asymptotes is HB 2+ . In order to identify the potential polluted area affecting the water distribution system,

we are required to establish the intersection of the potential polluted envelope (calculated earlier), with the paths of the water pipe. To identify the intersection requires extensive computational efforts, as the envelope is 3-D in nature. However, by making some assumption, it is possible to simplify the process without sacrificing accuracy of the result. It assumes that the potential polluted boundary of sewer is uniform from top to bottom, the problem can be assumed to be two-dimensional (rather than three-dimensional). In Figure 3, the boundary of the potential polluted areas approximate to rectangles made by asymptotes instead of the parabola. In the rectangle, the breadth represents the distance between the two asymptotes, and length represents the length of sewer. The developed computer program can locate the pipes which are in the potential polluted boundary, i.e., the parts of the pipes which are prone to pollution due to leaking sewers. The program finds all the potential polluted areas in water distribution systems. Other sources of faecal material Apart from sewer pipes, there are other pollution sources such as wastewater disposal pound, buried waste, spills or landfills etc., from which water distribution system may be contaminated. The boundaries of these foul water bodies can be simplified to polygons. When considering water bodies whose area is vast and shallow in depth, we can assume that the seepage boundary is uniform during the movement of contaminant through the soil. This assumption simplifies the computational process involved in determining the potential polluted area. Otherwise, numerical methods such as Finite Element Method (FEM) must be employed to calculate the seepage process. The developed computer program assumes the boundaries of water bodies as polygons. Re-entrant polygons (right in Figure 4) are divided into several convex polygons as shown in Figure 4, and the potential polluted areas arising from each individual polygon calculated.

81

Fig. 3. Simplification of computational process Fig. 4. Water body as polluted source

1

2

3

Water body

Water pipe

Sewer

Water pipe

Sewer

Water Pipe

Ground

5 B

82

Illustration of model Based on the mathematical models a computer program has been developed in C++ to compute the relative pollutant concentration through the soil, and locate the potential pollution area in water distribution systems. The program can show visually the pollution area in water distribution system given that the layout of both sewer systems and water distribution systems are given. A sample water distribution system (shown in Figure 5) coupled with sewer system and foul water bodies, is illustrated. The output of the modelling is shown in Figure 6.

Fig. 5. Water supply network with sewer system and water bodies

Fig. 6. Potential polluted area in water supply networks

83

5.2.2 Component 3: Loading Model

The models described above produces the potential pollution area of water supply pipe, i.e., the segment between the boundary point along water supply pipes. The next step is to calculate contaminant loading (CL) between the boundary points. The boundary point will be of the form of (x1, y1), (x2, y2), where “1” and “2” represent the point along a pipe for which contaminant will occur (i.e., CL will be between point 1 and 2). We begin by assuming that the planes in which the water supply pipe and sewer exist are parallel (the vertical distance between sewer and water pipe is fixed, Fig. 7), the derivation of the contaminant loading can be given by the following (Yan, et al., 2002).

∫×= 2

1 )sin()()(x

xdxxVxCCL

α (9)

Where CL = Contaminant loading, (mg/s); 1x , 2x = intersection points between water pipe and CZ; )(xC = Concentration profile in perpendicular position, (mg/l); )(xV = flow velocity, (m/s); and α = Angle between the projections of water pipe and sewer in horizontal plane. a. Vertical projection b. Horizontal projection Fig. 7. Water supply pipe under sewer at specified depth If the planes in which the water supply pipe and sewer exist are not parallel (the vertical distance between sewer and water pipe is variable). Additional terms are required to account for its angle in vertical plane, as shown in Fig. 7 ( oo 900 <=< α ) and its angle in right hand side plane, as shown in Fig. 8 ( o0=α ).

Sewer

Water

Ground

Sewer

Water pipe

x

84

We consider a three dimensional problem related to the relative position of water supply pipe and sewer pipe. The three projection diagrams, i.e., the vertical projection, the right hand side projection and the horizontal projection are shown in Fig. 8 and Fig. 9. Fig. 8a gives the relative position of water supply pipe and sewer pipe on vertical plane. The right hand side projection, shown in Fig 10a, gives the relative position of water supply pipe and sewer pipe on right hand plane, and the horizontal projection, shown in Fig. 8b and 10b, which gives the relative position of water supply pipe and sewer pipe on horizontal plane. a. Vertical projection b. Horizontal projection Fig. 7. Water supply pipe under sewer at various depth ( oo 900 <=< α ) a. Right hand side projection b. Horizontal projection Fig. 9. Water supply pipe under sewer at variable depth ( o0=α )

Sewer

Water

C Sewer

Water pipe

Ground

H0

Q

Sewer

Water

Sewe

0H

β

Water

85

As stated above, Eq. (9) is valid when the planes that water supply pipe and sewer exist are parallel(below a sewer at a specific depth). Otherwise, The contaminant loading can be given by,

+×+

−×−

=

∫

∫2

1

2

1

)tan,()tan,(

)tan,(sin

)tan,(

00

00

x

x

x

x

dxxHxVxHxC

dxxHxVxHxC

CLββ

θα

θ

(10a)

Where 0H = base distance from water pipe to sewer, (m); θ = angle of water pipe in vertical plane (i.e. pipe below sewer at a fix depth if it equal to 0); and β = angle of water pipe in right hand side plane (i.e. water supply pipe below sewer at a fix depth if it equal to 0). To simulate the loading of contaminant ingress into a water pipe, we need to consider the condition of the pipe, in particular the factors that will affect the amount of pollutant entering given the CL (i.e. cracks and leaky joints in pipes measured from leakage, pipe age, pipe material, etc.). The relationship between pipe conditions and percentage of CL that enters a pipe would need to be established, which clearly is a difficult task, and is a part of the ongoing research. However, the aim of this section is not to go into the specifics of such a relationship but to argue for the relevance of it in the context of vulnerability assessment. In this context, we will estimate ingress as a percentage of CL, where the exact value of percentage will be established from pipe conditions. Considering the above factors, the CL in polluted water pipe can be given by,

TPiCLPCL ××= (10b) Where PCL =contaminant loading in polluted water pipes, (mg); Pi = percentage of contaminant ingress into water pipe; and T = time since flow resumes, (s). From the above, contaminant load ingress into a water supply pipe can be established. This would serve as input into a contaminant propagation model, this research is also being undertaken.

oo 900 <=< α

o0=α

86

5.3 Contaminant propagation model Water quality simulations use network hydraulic solutions as part of their computations where the results of the hydraulic simulation are the starting point in developing water quality models. Kirmeyer and LeChevalier (2001) note the importance of hydraulic models for water quality models. Therefore in order to develop a water quality propagation model the hydraulics within the distribution system needs to be fully understood. The initial work in this area has focused on the development of models that would be applicable for intermittent supplies. Further work is planned to review and develop models suitable for fully charged systems. In intermittent systems there are three main hydraulic states that need to be considered, namely the empty state, charging state and full charged state. Hydraulic States Empty State: This is the period where there is no supply and at which time contaminants enter the distribution system. During this period the pipes in the network are partially filled and pressure, flows are very low or negligible. The water quality loading in this state is predicted by means of the contaminant loading model (described previously). Charging State: The charging process starts just after the supply resumes in the distribution system, the water column moves through the pipes with varying velocities, picking up contaminants from pollution points. This is an important state and differentiates intermittent systems from continuous. The majority of the research done to date has focussed mainly on developing detailed hydraulic model to predict the velocity changes in the pipe network during the charging process. The details of this are given below. Full Pressure State: The network has reached full pressurisation and behaves similarly to a continuous system (except the demands are pressure related). This component of the model will be based on existing models for continuous supply.

5.4 Water Quality Simulation Once the hydraulic analysis is completed the results of the hydraulic simulations are used as the staring point in performing water quality analysis. At present most computer models, including EPANET from the USEPA, are designed to model either the decay or changes in concentration of chemicals such as chlorine and disinfection by-products. Far less work has been undertaken in modelling the movement of microbes introduced into the distribution system. The discussion below therefore focuses primarily on chemical water quality models. For chemicals contaminants, transport, mixing and decay are the fundamental physical and chemical processes. The equations described in these processes can be found in Bocceli, et al. (1998), Clark and Grayman (1998) and Grayman et al (2000)). Transport in pipes Most water quality model make use of one-dimensional advective –reactive transport to predict the changes in the constituent concentrations due to transport through a pipe, and to account for formation and decay reactions. The concentration within a pipe i as a function of distance along its length x and time t is given below:

87

)( ii

i

ii Cx

CAQ

tC θ+

∂∂

=∂

∂, pi ...1= (11)

Where

−iC Concentration in pipe i −iQ Flow rate in pipe i −iA Cross sectional area of pipe i

−)( iCθ Reaction term In this research the above equation (coupled with the boundary conditions) will be solved using Discrete Volume Method. In DVM each pipe is divided into a series of equally sized completely mixed volume segments at the end of each successive water quality time step, the concentration within each volume segment is reacted first and transferred to the adjacent downstream segment. If the adjacent segment is a junction node, Mass and flow entering the node is added to the already existing mass at the node. After the reaction steps are completed for all the pipes, resulting mixture concentrations at each junction node is computed and released into the first segments of the pipe with flow leaving the node. This process is repeated until a new hydraulic condition occurs. Mixing at Nodes Water quality simulation uses a nodal mixing equation to combine concentrations from individual pipes described by the advective transport equation, and to define the boundary conditions for each pipe as referred to above. The equation is written by performing a mass balance on concentrations entering a junction node. Such an approach is efficient for chemicals, although has more limited application for microbial quality where such approaches would not be valid.

∑

∑

∈

∈

+=

j

j

i

j

OUTii

INijnii

OUT Q

UCQC

,

(12)

Where −

jOUTC Concentration leaving the junction node j

−jOUT Set of pipes leaving node j −jIN Set of pipes entering node j

−iQ Flow rate entering the junction node from pipe i −

iniC , Concentration entering junction node from pipe i −jU Concentration source at junction node j

The nodal mixing equation describes the concentration leaving a network node (either by advective transport into an adjoining pipe or by removal from the network as a demand) as a function of the concentration that enter it. The equation describes the flow weighted average of the incoming concentrations. If a source is located at a junction, constituent mass can also be added and combined in mixing equation with the incoming concentrations.

88

Mixing in tanks Pipes are some times connected to reservoirs and tanks as opposed to junction nodes. Again, a mass balance of concentrations entering or leaving the tank or reservoir can be performed.

)())(( , kknpik

ik CCtCVQ

dtdC θ+−= (13)

Where −kC Concentration within tank or reservoir k −iQ Flow entering the tank or reservoir from pipe i −kV Volume in tank or reservoir k

−)( kCθ Reaction term The above equation applies when a tank is filling. During a hydraulic time step in which the tank is filling, the water entering from upstream pipes mixes with water that is already in storage. If the concentrations are different, blending occurs. The tank mixing equation accounts for blending and any reactions that occur within the tank volume during the hydraulic step. During a hydraulic step in which draining occurs, terms can be dropped and the equation be simplified

)( kk C

dtdC θ= (14)

Specifically the dilution term can be dropped since it does not occur. Thus the concentrations within the volume is only subject to chemical reactions. Furthermore, the concentration draining from the tank becomes a boundary condition for the advective transport equation written for the pipe connected to it. To date the work undertaken as part of this research project has focussed on developing the hydraulic charging model and details of this model will now be given.

5.5 Charging model To date a hydraulic analysis model that simulates the ‘charging’ process within intermittent systems is still being developed. The model developed integrates both the momentum equation and the water-column velocity equation to generate the positions of the column water-front in the network at any time. This hydraulic model will form the basis of the contaminant propagation model. As mentioned above, the focus of this component of the research is to develop an analysis tool that simulates the charging up process of the distribution system, immediately after supply has been resumed and just prior to a fully pressurised state. This is short period (first 30 minutes or so), after supply is resumed when the pipes are filling from a near empty state to a fully charged pressurised state. The model presented herein is novel in that there has been no attempt to model the charging process in intermittent water distribution systems previously.

89

The objective of the model developed here is to try and predict the time at which different users get water and from this try and estimate the actual effective duration of supply received by consumers at different locations. For example (as will be demonstrates later), the model will show that in a 1.5 hour duration of supply some consumer will receive water for almost the entire duration while others will only receive water for 55 minutes because they are positioned at the tail-ends of the system and fail to receive water until after 30 minutes after supply resumes. Clearly this results in inequity in the quanta of water delivered to different consumers. The model developed is used to simulate the behaviour of starved intermittent systems (Srinivasa-Reddy and Elango 1989, Vairavamoorthy 1994, Akinpelu 2001). The term “starved” is used to describe those systems where consumers are unable to collect sufficient quantities of water to fill available household storage. Generally in intermittent systems consumers are not connected directly to the network, but via some form of storage. The storage may be in the form of underground tanks which are filled during supply hours and are then pumped to overhead tanks; or vessels which are filled during supply hours. The storage facilities act as buffer storage providing water to the consumers during non-supply hours. As mentioned already, the term “starved” describes those systems where consumers are unable to fill available household storage. Starved networks are those where the hours of supply and pressure are so limited that the consumers are forced to collect as much water as possible during supply hours. Almost all the networks studied by the authors in and around the city of Madras, India fall into this category. They operate either for 1 to 2 hours per day, or between 3 and 4 hours every third day. In line with observations it assumed in the development of the model that all outlets in a starved network are not controlled and are always open.

5.6 Network charging model The charging model developed attempts to simulate the charging up process of the water network after supply resumes. This model predicts the time at which different users receive water after supply resumes and highlights the time lag experienced by tail-end users in the network. The charging model gives the flow velocity during charging process, which is needed in contaminant propagation model. Fig. 10 shows typical results obtained from the developed model. Clearly this model is important in that enables engineers to estimate the potential diversity in supply of an intermittent distribution system. THEORY In the development of the model the following four assumptions are made (Liou and Hunt 1996): The pipe remains full so that there is a water column with a well defined front The pressure at the front face is atmospheric The water-pipe system is incompressible (i.e. a rigid water column) Frictional flow-resistance relationship for steady flow may be used.

90

In addition to the assumptions above, it is also assumed that all consumer outlets are permanently open and there is no attempt to manipulate their characteristics. Therefore as the waterfront moves all air in the system is expelled. As in a paper (Liou and Hunt 1996), we will first consider a straight pipe and will then develop the model further for a complete distribution network.

5.6.1 SINGLE FILLING PIPE

Fig. 11 depicts a straight and empty pipe connected to a reservoir. Water advances in the pipe upon the sudden opening of the valve. As the pipe is being filled, the governing momentum equation and water column front velocity can be written as: (15)

Ri

RiRi

AQ

dtdL = (16)

Where RiH is the (total) head at reservoir R; RiH∆ is the head-loss in pipe Ri; RiL is the water column length in pipe Ri; and RiQ , RiA are the flow and cross-sectional area of pipe Ri respectively. If the Hazen-Williams equation is used to for head-loss then RiH∆ is given by: (17)

Where

( ) 85.163.254.278 −= RiRiRi DCK (18) and RiQ is the flow in litres per second; RH and RiH∆ are in metres; RiL is in metres; and

RiD , RiC are diameter (metres), and ‘C’ value (Hazen-Williams friction constant), of pipe Ri respectively. If the pipe segment has a slope then additional terms to account for the varying elevation, as a function of ‘L’ must be included, i.e. Equation (15) becomes: (19)

Where

*

)()(Ri

iRRiRRi L

hhLhLf −−= (20)

and Rh and ih are the elevations of node R and i respectively and *

RiL is the total length of pipe Ri.

0=−∆−dt

dQgALHH Ri

Ri

RiRiR

85.1RiRiRiRi QLKH =∆

0)( 85.1 =−−−dt

dQgALQLKLfH Ri

Ri

RiRiRiRiRiR

91

Equations (16) and (19) can be integrated to find L(t) and Q(t) given the initial conditions L(0) = Lmin and Q(0) = 0.

5.6.2 Network junctions

Now consider a junction node in a water distribution system. The node can be in three states as shown in Fig. 21. Below the equations required to describe any of these three states is given. It should be noted that a combination of these equations can be used to describe the entire network at any particular time and the resulting equations integrated to establish pressure at junction nodes and the position(s) of the waterfront in pipes undergoing filling. State 1 (Fig. 21a) Here the pipe ij is filling and the water column front has not yet arrived at junction j. The equations for this state are identical to those (16) and (19) except that RH is replace by iH (where iH is the total head at node i).

0)( 85.1 =−−−dt

dQgAL

QLKLfH ij

ij

ijijijijiji (21)

ij

ijij

AQ

dtdL

= (22)

State 2 (Fig. 21b) Here the pipe ij is full and pipes jk and jl are filling. The length of the water column in pipe ij is equal to the length of the pipe ( *

RiL ), and the terms iH and jH are included in the equations. Also an additional equation is required to ensure continuity of flow at the junction (algebraic sum of flow at a node is zero). The equations describing State 2 are as follows:

0*

85.1* =−−−dt

dQgAL

QLKHH ij

ij

ijijijijji (23)

0)( 85.1 =−−−dt

dQgAL

QLKLfH jk

jk

jkjkjkjkjkj (24)

0)( 85.1 =−−−dt

dQgAL

QLKLfH jl

jl

jljljljljlj (25)

jk

jkjk

AQ

dtdL

= (26)

jl

jljl

AQ

dtdL

= (27)

0=−− jljkij QQQ (28)

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State 3 (Fig. 21c) Here all pipes connected to the junction node are full and all heads at the nodes connected to the junction include pressure head terms. The water column lengths in all pipes are equal to the physical lengths of the pipes ( *

ijL *jkL and *

jlL ). The equations describing State 3 are as follows:

0*

85.1* =−−−dt

dQgAL

QLKHH ij

ij

ijijijijji (29)

0*

85.1* =−−−dt

dQgAL

QLKHH jk

jk

jkjkjkjkkj (30)

0*

85.1* =−−−dt

dQgAL

QLKHH jl

jl

jljljljllj (31)

0=−− jljkij QQQ (32)

5.6.3 Outflow junctions

In a water distribution system it is likely that a junction node will also be an outflow (demand) node. Hence if the pressure is sufficient at a junction node then water will be delivered to the outlets connected to this node. If a junction has associated with it a demand (i.e. if it delivers water to outlets), then the continuity equations needs to include an outflow term. For example if there is an outflow jC at junction j (Fig. 13), then the continuity equation is:

0=−−− jjljkij CQQQ (33) The model being developed in this research specifically concerns intermittent water distribution systems, where the duration of supply and pressures are so limited that consumers are forced to collect as much water as possible during supply hours. Because of the low supply rate of water and the intermittent nature of supply, the demand for water at the nodes in the network are not based on notions of diurnal variations of demand related to the consumers behaviour, but on the maximum quantity of water that can be collected during supply hours. This will be dependent only on the driving pressure heads at the nodes (Srinivasa-Reddy and Elango 1989, Vairavamoorthy 1994, Akinpelu 2001). Therefore, in the continuity equations (33), the consumption jC , at node j, is not a constant but a function of the pressure head )( jj hH − ; where jH and jh are the nodal head and elevation at node j respectively, i.e.:

)( jjjj hHC −Ω= (34)

93

Where jΩ is the pressure dependent demand (PDD) function. The nature of the PDD can be assumed to be similar to the relationship of discharge out of a free flowing outlet as it can be assumed that the consumer will try to collect as much water as possible and hence there will be no attempt to manipulate outlet characteristics and therefore the outlets will behave as uncontrolled free flowing outlets (Srinivasa-Reddy and Elango 1989, Vairavamoorthy 1994, Akinpelu 2001) The theoretical relationship between flow from an outlet and the pressure causing discharge is the power law, which states that the discharge rate is proportional to the pressure head at the outlet raised to a power between 0 and 1, i.e.:

jnjjjj hHAC )( −= (35)

Where jA and jn are constants for the particular outlet. The resulting continuity equations will then be given by

0)(321 =−−−− jnjjj hHAQQQ (36)

5.6.4 Method of solution

Single Filling Pipe As stated earlier, the Equations (16) and (19) represent flow in a single pipe filling. The method proposed herein to integrate these equations are the same as given elsewhere (Liou and Hunt 1996), which is the Runge-Kutta method with an adaptive step size. Therefore given L(0) = Lmin and Q(0) = 0, Equations (16) and (19) are integrated to obtain L(t) and from this Q(t). Figs 14 - 16 shows the simulated results for a horizontal pipe of length 5000m, diameter 0.5 - 4.0m, and a ‘C’ value of 100, filling from empty. If acceleration is ignored when considering the filling of the pipe, Equations (16) and (19) can be replaced by the single equation:

RiRiRi

RiRiRi

ALKLfH

dtdL 1)(

54.0

−= (37)

Figs 14 – 16 also show a comparison when including and ignoring acceleration. Fig. 14 investigates the impact of changes to the time interval for a pipe of diameter D = 0.5m and constant water elevation H = 20m. The results from Fig. 14 show that there is no impact of variations in time interval with or without acceleration. Fig. 15 investigates the impact of varying the water level in the reservoir for a time interval t∆ of 1 second. Again it can be seen from Fig. 15 that there is little impact in excluding acceleration from the analysis. Fig. 16 investigates the impact of changing pipe diameter for fixed time interval ( t∆ = 1 second), and reservoir elevation (H = 20m). Again there is very little impact when excluding

94

acceleration and it only becomes significant for very large diameter pipes (not realistic in a water distribution system). Figs 14 – 16 clearly shows that excluding acceleration does not have a major impact in the simulated velocity profile. It should be noted however, that there are minor deviations in the early part of the analysis (i.e. 60 seconds not shown in Fig. 14 because of the time resolution used), but this initial interval is not important when analysis water distribution systems. The research work currently being undertaken is concerned with the analysis of large water distribution networks with a view of estimating the different times of discharge at the various outlets in the network. Hence, in the analysis it is not important to consider the behaviour of the network in the initial minutes of supply but to focus more on the filling process in the system at large that takes place after several minutes. Therefore in the analysis of the entire network the acceleration term will be ignored. To further demonstrate this, a simulation is performed using experimental data given by Liou and Hunt (1986). In Fig. 17, the measured average velocities of the water column (as given by Liou and Hunt), are compared with the model developed (variations in the measure average velocities are shown by the error bars at one standard deviation above and below the mean values). From Fig. 17 it can be seen that the results of the model developed herein are very close to that obtain experimentally. Note that the velocity head terms play a significant role in this analysis as at the beginning of supply for such a small system cannot be ignored. Network As stated earlier the state of an entire network can be expressed in terms of Equations (21) to (36). These equations can be expressed in finite difference form and solved. For example consider Fig. 21b, Equations (23) to (28) and (36) can be expressed in finite difference form as follows:

( ) 0*

85.1* =∆

∆−∆+−−t

QgALQQLKHH Ri

Ri

RiRiRiRiRiiRi (38)

( )( ) ( )0)( 85.1 =

∆∆∆+

−∆+∆+−∆+−t

QgA

LLQQLLKLLfH jk

jk

jkjkjkjkjkjkjkjkjkj (39)

( )( ) ( )0)( 85.1 =

∆∆∆+

−∆+∆+−∆+−t

QgA

LLQQLLKLLfH jk

jk

jkjkjkjkjkjkjkjkjkj (40)

( )jl

jljljl

AQQ

tL ∆+

=∆

∆ (41)

( )jk

jkjkjk

AQQ

tL ∆+

=∆

∆ (42)

( ) ( ) ( ) ( ) 0=−−∆+−∆+−∆+ jbjjjjljljkjkijij hHaQQQQQQ (43)

The above system of equations can be solved for Q∆ , L∆ and H, using for example the Newton Raphson’s method. Note that the above equations include the acceleration terms.

95

Also the number of equations to be solved can be reduced by rewriting the equations only in terms of dl and H. As stated earlier it is important to include the acceleration terms for the specific problem being analysed here, as its significance diminishes for the time frames under consideration here. Ignoring acceleration terms also simplifies the governing equations resulting in a less complex solution method (equivalent to what is obtained in standard network analysis). If acceleration is ignored, Equations (21) to (36) for the three states of the network described earlier can be expressed as follows: State 1 (Fig. 21a)

ijijij

ijiij

ALKLfH

dtdL 1)(

54.0

−= (44)

State 2 (Fig. 21b) Equations (37) to (42), are replaced with the following:

jkjkjk

jkjjk

ALKLfH

dtdL 1)(

54.0

−= (45)

jljljl

jljjl

ALKLfH

dtdL 1)(

54.0

−= (46)

( ) 0)()(

54.054.054.0

* =−−

−−

−−

−jb

jjjjljl

jlj

jkjk

jkj

ijij

ji hHaLK

LfHLK

LfHLKHH

(47)

State 3 (Fig. 21c) Equations (29) to (32) are replace with the following:

( ) 054.054.054.0

* =−−

−−

−−

− jbjjj

jljl

lj

jkjk

kj

ijij

ji hHaLKHH

LKHH

LKHH

(48)

Note that in the above formulation the head-loss equations have been substituted into the continuity equations. Note that both Equations (47) and (48) are identical to the nodal formulation form of the continuity equations used in network analysis (Lam and Wolla 1972). There are many efficient network analysis solvers available for solving Equation 34 and can easily be incorporated into the network charging model. During this research NETIS (Vairavamoorthy et al. 2001) has been used for this purpose. NETIS is freely available network analysis program specifically developed for intermittent systems (in particular it includes pressure dependent demand functions), and has been included in the international guidelines for intermittent systems (Vairavamoorthy et al. 2001).

96

The proposed method for solving the above equations for a given t∆ is as follows: State 1: Equation (44) can be integrated using the Runge-Kutta method (with adaptive step size), to obtain tt

ijL ∆+ . State 2: Given t

jH , Equations (45) and (46) can be integrated using Runge-Kutta method (with

adaptive step size), to obtain ttjkL ∆+ and tt

jlL ∆+

Using ttjkL ∆+ and tt

jlL ∆+ solve the non-linear simultaneous equations (Equation (47) coupled with similar equations for other nodes in the network), using a standard non-linear equation solution method (e.g. Newton-Raphson method), to obtain tt

iH ∆+ and ttjH ∆+ .

State 3: Non-linear simultaneous equations (Equation (48) coupled with similar equations for other nodes in the network), can be solved using a standard non-linear equation solution method (e.g. Newton-Raphson method), to obtain tt

iH ∆+ ttjH ∆+ tt

kH ∆+ and ttlH ∆+ .

5.6.5 Computer implementation

The network charging model has been coded in C++ and includes the routines for integration (Runge-Kutta method with adaptive step-size (Press et al. 1992), and routines from NETIS (Vairavamoorthy et al. 2001), for the solution of the sparse linear symmetrical equations (decomposes a real matrix into a product of triangular matrices LU by Gaussian elimination with partial pivoting). The developed program includes a visual interface that animates the charging process against time and indicates if and when nodes are receiving water.

5.6.6 Application of the charging model

The performance of the charging model has been applied to several networks of different sizes and varying complexity and it has been shown to be an efficient and robust technique. Below two examples applications are given. Example 1 is that given by Liou and Hunt (1996). This example considers a 6424 m long, 4.6 cm inside diameter pipe traversing an undulating terrain where the water level of the supply reservoir is 2.9m above the pipe inlet. Initially the pipeline is empty except 0.3m long water column between the reservoir and a closed valve (Lmin). The elevation profile of the pipeline is as shown in Fig. 18. Fig. 19 compares the simulated velocity history generated using the developed model with that given by Liou and Hunt and shows a good match between the results. Example 2 is that given by Vairavamoorthy et al. (2000), and is small standpipe network operating for 1.5 hours each day. The network consists of 81 nodes with 110 pipes and the nodal and pipe data for this network can be found in Vairavamoorthy (1994). Screen captures of the simulation model for different time instances is shown in Fig. 20. From Fig. 20, it can be seen that the animated display showing the charging up process clearly highlights areas

97

within the distribution system when people are receiving water and those areas where people receive water very late in the supply. Fig. 21 shows the time at which selected nodes first receive water after supply has resumed. From Fig. 21, it is interesting to note that while some nodes receive water after 2-3 minutes others only receive water after 35 minutes. With a distribution network operating for 1.5 hours this time lag in first receiving water will clearly affect the quantum received by individuals being served by this system.

98

Figure 1: Charging up Process In Distribution System.

Figure 11: Charging in a Single Pipe

Discharge No

Reservoi

5 i

Pipe RiHR

99

(a) (b) (c)

Figure 12:Charging at Junction ‘j’

i j

k

l

i j

k

l

Lij

i j

k

l

L*ij

Ljl

100

Figure 13: Outflow at Junction ‘j’

Figure 14: Column Length as a function to time for varying t∆

i j

k

l

Cj

∆ t = 120s ∆ t = 60s ∆ t = 1s

0

1000

2000

3000

4000

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time (second)

Dis

tanc

e (m

)

0 500 1000 1500 2000 2500 3000

0 500 1000 1500 2000 2500

Incl. Accel

excl Accel

101

Figure 15: Column Length as a function to time for varying diameters

Figure 16: Column Length as a function of time for varying Heads HR

D = 0.5mD = 1.0mD = 2.0mD = 4.0m

0

1000

2000

3000

4000

5000

0 500 1000 1500 2000 2500 3000

Time (second)

Dis

tanc

e (m

) ∆t = 1s H = 20m

incl. Accel

excl Accel

H = 5mH = 20mH = 40m H = 10m

0

1000

2000

3000

4000

5000

0 1000 2000 3000 4000 5000 6000

Time (second)

Dis

tanc

e (m

)

∆t = 1sD = 0.5m

incl. Accel

excl Accel

102

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7

Distance (m)

Velo

city

(m/s

)

Laboratory test

Proposed model

Figure 17: Comparison of Model with Experimental Data (Liou and Hunt 1996)

940

950

960

970

980

990

1000

1010

1020

0 1000 2000 3000 4000 5000 6000

Pipe length (m)

Elev

atio

n (m

)

Figure 18: Elevation Profile of Pipeline (Liou and Hunt (1996))

103

0

0.5

1

1.5

2

0 1300 2600 3900 5200 6500

Length (m)

Velo

city

(m/s

)

Result of the proposed model

Result by Liou and Huntl

Figure 19: Water Column Velocity as Function of Column Length

104

Figure 20: Animated Display showing charging up process

Time after

Dot (red) indicates node

receiving

105

Figure 21: Predicted time for water to arrive at selected nodes

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

43 46 49 50 52 56 59 61 64 66 69 71

Node Number

Tim

e (m

inut

es)