THE USE OF MULTIPLE-LINKING-ELEMENT FOR CONNECTING

SEWER AND SURFACE DRAINAGE NETWORKS

Leandro, J.(1), Djordjević, S., Chen, A. S.,

Savić, D.

Centre for Water Systems, University of Exeter,Harrison Building,

North Park Road, Exeter, UK, EX4 4QF, United Kingdom

(1) phone: +44 1392 263600; fax: +44 1392 217965; e-mail: jl245@ex.ac.uk

ABSTRACT

Urban drainage systems can be divided into two networks, the surface network which

mainly includes streets and open channels, and the subsurface network which is mainly the

sewer system. If the hydrograph generated by the rainfall does not exceed the sewer capacity

the flow is drained by the downstream sewer system. However, when the sewer system is not

capable to divert the incoming hydrograph to proper outlets, the sewer system can become

surcharged and flooding will occur. In this study, a new linking element between surface and

subsurface network is presented. The surface-subsurface connections including their

subcomponents are analysed individually before assembly into the multiple-linking-elements.

The Multiple-Linking-Element is implemented on SIPSON 1D/1D model. Theoretical results

and hypothesis are presented. A methodology for a real case application is shown.

Keywords: Multiple-linking-element (MLE) Single-linking-element (SLE), dual drainage,

urban flooding

1 INTRODUCTION

Flooding in urban area is a complex process that enrols three major entities, the public,

the drainage system and the environment. These entities interact with consequences to the

community systems at the social, environmental and health levels. Risk estimates can provide

a common metric to compare risks from different sources and flood damage can been

assessed by combining the outputs of urban flood models and flood depth-damage curves

(Balmforth and Dibben 2005). A study on flood risk attribution was recently done by using

these concepts (Hall et al. 2006). Therefore, enhancing the current models for urban flooding

will improve not only the ability to accurately simulate this phenomenon but also contribute

to overall flood risk assessments.

Urban drainage systems can be divided into two networks, the surface network which

includes streets and open channels (major system), and the subsurface network which is the

sewer system (minor system). The complex interactions between surface and subsurface

networks raise a particular challenge to engineers. Only with the recent advances in computer

technology was it made possible to start modelling this complex phenomenon. The sewer

flow is commonly routed by using 1D models in traditional approaches. This 1D

simplification had the advantages in computational efforts decrease and simple numerical

schemes, with few losses in accuracy of he results. The surface flow is usually calculated by

either 1D or 2D models. 3D models are less popular due to their large increase in

computational time and complexity. 1D models can be considered a good approximation up to

some extent (Mark et al. 2004). Where the 1D models cannot be used there are already

models able to perform 2D simulations (Chen et al. 2005). Coupled models are capable of

dealing with both surface and subsurface flows by either 1D/1D or 1D/2D. Commercial

packages such as MOUSE or Info-Works can be used as 1D/1D models. Examples of 1D/2D

models are, MIKE FLOOD that couples the 1D MOUSE model with the 2D MIKE 21 model

(Carr and Smith 2006), or SOBEK that couples a 1D flow and a 2D overland flow modules

(Bolle et al. 2006), or TUFLOW (Phillips et al. 2005).

A recent study (Kaushik 2006) was carried out to compare the modelling of urban areas

by 1D and 2D approaches using the above commercial packages. Recent work on FRMRC

(Flood Risk Management Research Consortium) has been carried out in order to do a full

comparison between the integrated models, which are research applications SIPSON (1D/1D)

(Djordjević et al. 2005) and SIPSON/UIM model 1D/2D (Chen 2007). All the studies

developed so far calculate the flow between subsurface and surface interface by either a weir

type formula and/or an orifice formula (Mark et al. 2004; Nasello and Tucciarelli 2005) or

simply as a sink (Aronica and Lanza 2005). Almedeij (2006) considered the possibility of

multiple inlets and the reduction of clogging factor for the study on grate sag inlets. Muslu

(2002) studied the flow features on side weirs.

Since the main objective in developing coupled models is to increase the overall accuracy

of flood modelling, one of the key issues is therefore how to improve the linking between the

surface and the subsurface network. Surprisingly, there is a lack of research on this subject.

The importance of accurate linking between the surface and subsurface network is that it

ultimately determines and regulates the extent of flood damage on the ground surface due to

the surcharge from subsurface network. These previous studies simplify the manhole

connections by using a weir or an orifice equation without providing guidance for appropriate

ranges for setting the parameters in each equation. The inlet characteristics, including the

number of inlets, the discharge pattern of the different inlet components, and how these

factors are related to each other, are all neglected in those studies.

In this study, a new linking element between surface and subsurface network is

presented. The Multiple-Linking-Element (MLE) is implemented on SIPSON model to

improve the simulation of subsurface and surface exchange. Theory and assumptions are

described and a methodology for real case applications is presented.

2 MULTIPLE-LINKING-ELEMENT (MLE)

General references on this subject disregard the interaction between the two networks.

They assume the inlet capacity to be controlled only by the type of inlet and flow on the street

network (Akan and Houghtalen 2003; Butler and Davies 2000).

The linking element should be able to cope with the flow exchange between these two

networks, and adjust the rate exchanged and direction depending on the local time-varying

features in both networks. Due to the complexity of surface-link-subsurface the objective is to

isolate its components, to analyse them separately and finally reassemble them all in a unique

element (MLE).

The components identified are the following:

• Geometry types (Manholes, inlets, pipes).

• Multiple inlet connections.

• Difference in inlets ground levels.

• Street geometry (slope, shape, roughness).

• Blockage.

• Delay in time of full capacity reach (only relevant in multiple inlet connections).

• Control sections.

The MLE will be defined by a given number of Single-Linking-Element (SLE) therefore

we will first define the SLE (See Figure 1).

2.1 SLE – GEOMETRY1

Considering a generic inlet defined by: a rectangular gutter, a boxed shape inlet, a pipe

connection and a manhole. (See Figure 1)

Figure 1 SLE-Geometry schematization

2.2 SLE- THEORY AND ASSUMPTIONS

The concept is to define the control section that restricts the flow rate. The result should

be a more accurate discharge curve that reflects the actual type of connection between

surface-pipe-surface as a whole.

A steady-state flow condition is assumed in the analysis, and the derived equations for

the control sections are described:

CS1

• From the gutter to the inlet, the Bernoulli equation is considered valid. The flow enters

the inlet at critical depth. In this way the flow is controlled by a weir type formula.

Although the approximation used is the same as for broad crested weir, there is a need

to consider a discharge coefficient2.

H1g3

2LiH1

3

2CdQcs1 ××××= , Valid for: 0H1 ≥ (1)

Where:

Cd- Discharge coefficient

H1 - The surface water depth (m)

)yWx(W2Li +×= - Perimeter length of inlet box (m)

Wx- Length of inlet box (m)

Wy- Width of inlet box (m)

g- Gravity acceleration (9.8m/s2)

CS2 and CS4

• From the inlet to the vertical pipe, the Bernoulli equation is again used. It is

considered that it will only start working when the water level in the inlet box reaches

the ground level. This will enable us to plot this control section along with the

previous one. This is a realistic assumption because only when the volume of the inlet

is filled the vertical shaft can start to influence the flow discharge from the weir. This

will have the appearance of an orifice type formula.3 CS2 is used for the discharge

1 This geometry can be adapted for other cases. 2 There are different discharge coefficients, assessed experimentally for other applications. A default value of

0.5 is used, but this should be adjusted as the experimental data is made available.

from surface to subsurface, for the reverse discharge CS2 is switched to CS4.

( )HiH12gApCdQcs2 +××= , Valid for: 0HiH1 ≥+ (2)

( )HsHiH1H22gApCdQcs4 −−−××−= , Valid for: HsHiH1H2 ++≥ (3)

Where:

/42pDπAp ×= – Section area of pipe connection (m2)

Hi – Height of inlet box (m)

Hs – Height of vertical shaft (m)

H2 – The manhole water depth (m)

CS3 and CS5

• From the orifice to the manhole, a friction energy loss will be used to assess the full

discharge capacity of the pipe. Like in the previous control, it is considered that it will

only start working when the water in the inlet box is at least at the ground level. CS3 is

used for the discharge from surface to subsurface, for the reverse discharge it is used

CS5.

( )Lp

H2HsHiH132

RApKQcs3−++

×××= , Valid for: H2HsHiH1 ≥++ (4)

( )Lp

HsHiH1H232

RApKQcs5−−−

×××−= , Valid for: H2 H1 Hi Hs≥ + + (5)

Where:

K - Roughness coefficient (m1/3/s)

R - Hydraulic radius (m)

Lp- Length of pipe connection (m)

2.3 MODEL APPLICATIONS

Figures 2 and 3 show the results considering H2=0 (no surcharge in the manhole is

allowed) and H2=0.60m, respectively. The following default values are used in the study:

Li=0.80m, Hi=0.40m, Hs=0.60m, Lp=5.00m, Dp=80mm (DN80), Dm=1.00m, K= 80 m1/3/s.

0 0.2 0.40

0.05

0.1

.15

0

Q1 h( )

Q2 h( )

Q3 h H2,( )

.50 h

0 0.2 0.40

0.05

0.1

.15

0

Q1 h( )

Q2 h( )

Q3 h H2,( )

.50 h

Figure 2 - H1=[0,0.5],H2=0 Figure 3 - H1=[0,0.5],H2=0.6

It is shown in Figure 2 that CS2 starts to dominate CS1, at approximately H1 = 0.18. In

Figure 3 it is shown a global change from CS2 to CS3 just by changing the flow depth in the

manhole.

It is possible to plot these variations as function of both depths H1 and H2. Figure 4 is the

plot considering only flow from surface to subsurface network, and Figure 5 considers both

directions.

Q (m3/s)

Q (m3/s)

H1 (m) H1 (m)

Legend: CS1

CS2

CS3

Figure 4 – Flow surface to subsurface Figure 5 - Flow bi-directional

What is interesting to gain from the graph on Figure 4, is the perception that full pipe

flow is getting much more restrictive than any other control as soon as the manhole starts to

get surcharged. Both CS1 and CS2 are insensitive to the manhole surcharge. We can also see

the flow tending to zero as the water level in the manhole reaches the water level in the

surface. Now let us consider that the water level in the manhole can increase above the water

level in the surface, we would then have flow from the manhole to the surface. In this case

equations (2) and (3) from CS2 and CS3 need to switch to equations (4) and (5) from CS4 and

CS5 and CS1 is no longer valid. In Figure 5 we can see that both CS4 and CS5 are very close,

this means it will depend greatly on the specific field characterises to determine which one

would be predominant. With this final graph we conclude a generic formulation that can be

applied to other case studies. After defining the geometry properly it is possible to define the

flow exchange based on the water levels at surface and manhole.

Just as an illustration, this procedure can then be applied to blockage/partial opening or

full opening. In the first case this could be done by considering, a manhole without any inlets

and the flow entering through the manhole cover considering a small percentage of the cover

area as an inlet. The manhole could be considered to have 1m of internal diameter (Dm), and

an orifice with a small percentage of the total manhole area (ex. 1%). In the second case, this

could be done simply by increasing the percentage value (ex. 60%). In the two cases the Ap,

in Eq.2, Eq.3, Eq.4 and Eq.5 is replaced by a percentage of the section area of the manhole

(Am) and in Eq.1 the Li is replaced by a percentage of the perimeter of the manhole3.

With the SLE defined it is now possible to consider the MLE as combination of several

SLE, by applying a multiplication factor to the SLE.

3 IMPLEMENTATION

In order to compare the results between MLE and SLE, a sensitivity analysis is carried.

Four typical real cases are set up in the SIPSON model with the number of inlets being the

corresponding case number. The objective is to compare the performance of considering the

real number of inlets through the use of SLE with the use of a single inlet through the use of

MLE (See Figures 6 and 7).

3 This paragraph is intended to show the flexibility of the methodology, but this can only be considered after

calibration over existing data.

Q (m3/s)

CS5

CS2

CS1

CS3

Q (m3/s)

H1

(m)

CS3

CS4

CS1

CS2

H2

(m) H2

(m)

H1

(m)

Figure 6 – SLE Figure 7 – MLE

In the MLE methodology, an irregular cross section is used to join the two side channels

used in the SLE methodology. It is interesting to notice that, the MLE methodology not only

reduces the number of inlets used but also reduces the number of channels in the street

network, thereby decreasing the number of loops in a network (See Figure 8).

Figure 8- SLE and MLE concept

3.1 RESULTS

For each case, two simulations are conducted in order to assess the overall performance

of the MLE methodology: The flow direction is predominantly from subsurface network to

surface network (pipe-street) in the first simulation, whereas it is predominantly from surface

to subsurface (street-pipe) in the second. This is achieved by using a trapezoidal hydrograph

with elevated inflow in the network where flow is required to be predominant combined with

a constant hydrograph with lower inflow in the other network.

Figure 9 shows the flow curve of the four case studies. The dark lines are the results

obtained by using SLE. The pink lines are the ones by MLE.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 50 100 150

Flow (m3/s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 50 100 150

Flow (m3/s)

SLE's

MLE

Test case:1,

RCd:0.15,

MLE:1xSLE

-0.2

-0.15

-0.1

-0.05

0

0.05

0 50 100 150

Flow (m3/s)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 50 100 150

Flow (m3/s)

SLE's

MLE

Test case:2,

RCd:0.08,

MLE:2xSLE

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 50 100 150

Flow (m3/s)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 50 100 150

Flow (m3/s)

SLE's

MLE

Test case:3,

RCd:0.03,

MLE:2.3xSLE

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 50 100 150

Flow (m3/s)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 50 100 150

Flow (m3/s)

SLE's

MLE

Test case:4,

RCd:0.04,

MLE:4xSLE

Figure 9- Results for cases 1,2,3 and 4

Test case 1 has identical results in both SLE and MLE since no simplification is used.

The case is used to verify the MLE procedure.

Test case 2 shows a good agreement between SLE and MLE results. The slight increase

in flow street-pipe (SP) by using the MLE methodology can be explained by the channel

downstream boundary conditions change due to the vary in channel geometry. Although the

channel area remains the same, the IRG algorithm to compute the new cross-section (double

rectangular channel) uses the same formulation to compute the Froude number as in the single

independent rectangular channel. Since the Froude number is given by BAgVFr // ×= , and

the downstream boundary condition is critical depth; hence Fr=1. The increase of the channel

width reduces the Fr number; hence it has to be balanced by raising the water depth. This rise

in water depth may explain the rise in flow seen on the graph.

Test case 3 reveals to be the most difficult case to model through the use of this

methodology, without any further change. The non symmetry of the test case produces a

different behaviour in pipe-street (PS) and SP. If in PS this is not so notorious since no major

simplification is introduced in the pipe network (the main responsible for flow in this

situation). In the SP the difference is enhanced once the simplification in the street network

was done. The non symmetry of the inlet distribution made inlets to work in different flow

ranges; hence the equivalent element could not achieve the same results by considering only

Time (min)

Time (min)

Time (min)

Time (min)

one element as the summation of three elements working on the same flow point (delay in

time of full capacity reach). The best fitting results were obtained using a number of

equivalent elements of 2.3 rather then 3.

Test case 4 shows a good agreement in both PS and SP. This case is the more complex to

SIPSON simulation, since it involves a quite complex surface network. In order to get

accurate results and to prevent an unrealistic start, the “Hot Start” method is used to determine

the initial condition.

3.2 DIFFERENCE OF INLET LEVELS

The cases 2 and 4 will be chosen to study the influence of variable ground levels at gutter

inlets (See Figure 10).

a) case 2 b) case 4 c) case 4

Figure 10 – Difference in Z’s inlets

For the SLE method, the study cases will be referred as the SLE-sets, and these are:

• For case 2 three different heights are studied. The difference between them is 15 cm.

Set C corresponds to the previous Case 2. Set B the inlet is 15 cm higher, and set A

is30 cm higher than C.

• For case 4 five different heights are studied. Set C corresponds to the previous Case 4.

The heights in Sets A and B are raised 30 and 15 cm, respectively. Sets E and D have

three inlets heights raised 15 and 30 cm, respectively.

For the MLE method, the study case will be referred as the MLE-sets, and these are:

• For case 2, the same as in previous SLE method but considering only one inlet.

• For case 4, three sets are used for the comparison. Namely A, B and C. Case A uses 30

cm as inlet height, case B uses 15 cm as inlet height and Case C uses 0 cm as inlet

height.

The MLE-sets results will then be compared with the corresponding SLE results. Figure

11 and Figure 12 shows the results obtained, for the flow curve of the case 2 and 4,

respectively.

As we can see the global optimum performance is obtained by using a mean value for the

inlet height. However if the flow is expected to have a predominant direction either PS or SP

we may reduce the error by using a different value for the adopted inlet height.

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 50 100 150Flow (m3/s)

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 50 100 150

Flow (m3/s)

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 50 100 150

Flow (m3/s)

RCd.: Multi 0.18,Single 0.06 RCd.: Multi 0.18,Single 0.06 RCd.: Multi 0.18,Single 0.06

0

0.05

0.1

0.15

0.2

0.25

0 50 100 150

Flow (m3/s)

SLE-a

SLE-b

SLE-c

MLE-a

0

0.05

0.1

0.15

0.2

0.25

0 50 100 150

Flow (m3/s)

SLE-a

SLE-b

SLE-c

MLE-b

0

0.05

0.1

0.15

0.2

0.25

0.3

0 50 100 150

Flow (m3/s)

SLE-a

SLE-b

SLE-c

MLE-c

RCd: 0.08 RCd: 0.08 RCd: 0.08

Figure 11 – Results of PS (upper row) and SP (lower row) for cases 2

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 50 100 150

Flow (m3/s)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 50 100 150

Flow (m3/s)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 50 100 150

Flow (m3/s)

RCd: 0.04 RCd: 0.04 RCd: 0.04

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 50 100 150

Flow (m3/s)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 50 100 150

Flow (m3/s)

SLE-a

SLE-b

SLE-c

SLE-d

SLE-e

MLE-b

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 50 100 150

Flow (m3/s)

SLE-a

SLE-b

SLE-c

SLE-d

SLE-e

MLE-c

RCd: 0.04 RCd: 0.04 RCd: 0.04

Figure 12 – Results of PS (upper row) and SP (lower row) for cases 4

3.3 STABILITY

The SIPSON model applies an implicit numerical scheme (Preissman four-point method).

Although the implicit models do not have to respect the Courant’s stability criteria (Chaudry

1987), they may suffer from instability derived from oscillations in cases the rapid changes of

flow or depth within single time step. This is more prone in cases with small water depth. To

overcome this instability two actions have to be taken4. First a reduction coefficient needs

(RCd) to be applied to an auxiliary equation (weir type) for the proposed single linkage

element. Second, the number of equations used per time step is limited.

4 This is specific of the model used for testing the MLE other models may have other requirements

Time (min)

Time (min)

4 RELATED ISSUES 1D/2D MODELLING

In order to define the MLE in 1D/2D models similar issues need to be accounted for. In

(Chen et al. 2007) the 1D SIPSON model is used for subsurface flow routing and the 2D

Urban Inundation Model for overland flow simulations. The two models are linked through

manholes and the interacting discharges are determined by the water levels within manholes

and surface grids. The runoff and flooding entering and leaving the 1D drainage system at

manholes are considered as sinks and point sources, respectively, in the 2D surface modelling.

5 CONCLUSION

The MLE methodology developed for linking 1D/1D model promises to be a step

forward in improving validation of coupled models, and can be extended to 1D/2D models.

The consideration of a more complex overview of the hydraulic behaviour of an inlet is

successfully implemented in the SIPSON model. This is equally relevant for other models. On

the other hand, the increase in complexity led to some instability problems in the model. Two

possible ways to overcome this were shown. The simplification of the manhole inlets by a

single MLE provides good results. In real cases, where information of manhole inlets is not

available this may be sufficient as a default set of parameters for the simulation, but care has

to be taken in order to judge on the correctness of results. In case information about manhole

inlets is available, a similar study should be performed in order to determine the correct

number of inlets and reduction coefficients.

The presented methodology is currently being implemented and tested on real case

studies. Future research will include comparison between results of 1D/1D and 1D/2D

coupled models.

AKNOWLEDGMENTS

The research presented in this paper is funded from the FRMRC work package 6.1 (Grant

GR/S76304/01).

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