Spill
• 22 Sep 2022

# Spill

Article summary

The Spill calculates the flow over a jagged or irregular weir. It can be used to model in-line flows over irregular weirs, as well as lateral flows, such as those over embankments between two open channels or between an open channel and a flooded area.

## Data

Field in Data Entry Form

Description

Name in Datafile

Upstream

First spill label

Label1

Downstream

Second spill label

Label2

Weir Coefficient

Weir coefficient (includes discharge, velocity and calibration coefficients)

Cd

Modular Limit

Modular limit (for example 0.9)

m

n/a

Number of ensuing data lines

n

Chainage

Chainage along weir crest at which yi is defined (m)

xi

Elevation

Elevation of weir crest at chainage xi (mAD)

yi

## Theory and Guidance

The Spill calculates the flow over a jagged or irregular weir. It can be used to model in-line flows over irregular weirs, as well as lateral flows, such as those over embankments between two open channels or between an open channel and a flooded area (modelled using a Reservoir unit for example).

The Spill calculates flow over an irregular weir by splitting the calculation into determining flows over segments using an integrated form of the weir equation for dry, free and drowned, forward and reverse modes. These flows and associated matrix coefficients are then summed together subject to stability constraints, to give values over the entire spill.

The Spill can involve two, three, or four connected nodes. The two-node or in-line application operates in much the same way as a weir, the two nodes being the upstream and downstream nodes.

The three-node application usually relates to flow between a channel and an area of static storage such as a reservoir. The four-node application is used to model lateral flow between two channels.

To use a three or four noded unit in the datafile it is necessary to place the labels associated with the spill unit adjacent to the upstream label of the corresponding river reach, as discussed in the RIVER unit section.

Figure 1: Overhead view of a common spill arrangement

The Spill requires you to specify a set of offset/elevation pairs describing the crest of the weir or bank. If the unit is being used to model flow over the bank of a channel then it is usual for the total distance (offset) of the spill to be equal to the distance between the adjacent River Sections.

Figure 2: Side on view of the above spill arrangement

Further discussion of this method can be found in Evans E.P. and von Lany P.H. (1983).

### Equations

#### Mode 0 - Zero Flow

Condition

The upstream (highest) water level is below the bank level defined in the spill, or the water levels are above the bank level but almost exactly equal.

Equation

 qs = 0 (1)

where:

qs = flow over segments

#### Mode 1 - Free Flow (positive sense)

Condition

(y21 + y22) / (y11 + y12) <= m

Equation

 (2)

where:

y11 = upstream water depth in channel 1

y12= downstream water depth in channel 1

y21 = upstream water depth in channel 2

y22= downstream water depth in channel 2

m = modular limit

b = width of spill section

#### Special Case

Water surface nearly parallel to bank (y12» y11 )

q s  = Cd b y 11 Ö y 11

#### Mode 2 - Drowned Flow (Positive Sense)

Condition

(y21 + y22) / (y11 + y12) > m

Equation

 qs = A b { (2/3) yk D - (4/15) yi (dy22 + dy12) } (3)

where:

D = y12 dy21 - y11 dy11

b = width of spill section

A = -Cd / yk2 Ö (1-m)

yk = y12 - y11 - y22 + y21

yi = y12 - y11

dy21 = (y12 - y22)3/2

dy22 = (y12 - y22)5/2

dy11 = (y11 - y21)3/2

dy12 = (y11 - y21)5/2

#### Special Cases

The following special cases may occur:

1. y12 - y11 = y22 - y21
qs = -½ A b yk2 (y11 + y12)Ö (y11 - y21)

2. 2. y11 - y21 << y12 - y11 - y22 + y21
qs = - 2 A b yk5/2 {y11/3 + (y12 - y11)/5}

3. y12 - y21 << y12 - y11 - y22 + y21
qs = -½ A yk2 (2y11 + yi + yk y11/ym)Ö (y11 - y21)

where:

yk = y12 - y11 - y22 + y21

ym = y12 - y21

#### Mode 3 - Free flow (Negative Sense)

The same formulae apply as for mode 1 but with y21 interchanged with y11, and y12 interchanged with y22.

#### Mode 4 - Drowned flow - (Negative Sense)

The same formulae apply as for mode 2 but with y21 interchanged with y11, and y12 interchanged with y22.

### General

The Spill is a complex and powerful tool in river modelling and care must be taken in its application. If a large number of segments have been defined, then a large number of mode changes can occur as a hydrograph passes through the system. Every effort has been made to smooth these transitions but sometimes it may be necessary to reduce the timestep size to avoid large changes in flow being generated during a single timestep. Another scenario that may generate a problem is when a potentially large side spill discharges into a small reservoir.

In order to minimise the numerical problems associated with spills, local under relaxation is applied to the spill unit. The user can control the amount of relaxation through the use of the weight parameter which is entered in the Advanced Options of the Flood Modeller run simulation dialog. The default value is 0.1 and users may wish to reduce this if instability problems occur which are associated with spill units. It is possible that by lowering the value you may reduce the rate of flow over spill units and therefore it is important to check the sensitivity of the results to any changes in this parameter.

#### Derivation of Spill Coefficient Cd

The basic weir equation for free flow used in the Spill is:

 Q = Cd b h1.5 (4)

Thus typical Cd values are 1.85 for sharp crested weirs and 1.7 for round nosed horizontal-crested weir. Where the 'weir' feature is less efficient than man-made concrete weirs (for example flood embankments) then the Cd value should be reduced. Accurate guidance on the selection of Cd values for maintained grass embankments is not available although current good practice suggests values in the range 0.8 to 1.2. If the Spill is being used to model flow over heavily overgrown natural ground, for instance, which is less efficient than a flood bank then lower values may be applicable. These lower values may include an allowance for friction losses for overland flow (see below for an equation which may aid determination of coefficients under such circumstances).

When a Spill is being used to model flood plain flow (for example between Reservoirs), it is possible, by analogy with the equation for fully rough flow, to derive an approximate formula for the spill coefficient. The formula requires the average depth of flow, an estimate of Manning's n and the distance between the spill source and sink:

 (5)

where:

d = average depth of flow (m)

DX = distance between spill source and sink (m)

n = Manning's n for region of flow (eg 0.1)

m = user defined modular limit (eg 0.8)

This formula has the necessary properties that the level difference increases for larger values of n and DX whilst it decreases for larger values of d. Ideally Cd should be a function of flow depth, however as it is fixed, users should use a single representative value of d thus giving a constant Cd for the run duration. It is also worth noting that the formula applies for drowned spill flows, for which the assumption of fully friction dominated flow is valid.

### Modelling a Breach

Flood Modeller has the option to overlay a breach on top of an existing Spill unit in order to model a breach in an embankment, bank or dam. The breach formation is modelled as a time series of deformation dimensions, with time specified as an offset either from the model start time or from when a trigger level at the spill/breach unit is exceeded. This can therefore be used to represent the overtopping or piping failure of an existing dam or embankment, the outflow from which can then be propagated through a 2D or 1D model.

The Breach unit provides functionality to model two different types of breach:

• An open breach models an overtopping failure of the structure, with the breach emanating from the crest of the spill.
• A piping breach starts as a small [enclosed] hole in a structure, enlarges over time, and may ultimately collapse into an open breach.
More information about modelling a breach including the datafile format for the breach unit can be found at the help page Modelling a Breach