Hydraulic Theory
Reservoir Routing


From the MIDUSS Version 2
Reference Manual  Chapter 8
(c) Copyright Alan A. Smith Inc. 
Reservoir routing
involves the application of the continuity equation to a storage
facility in which the storage volume for a particular geometry is a
dependant only on the outflow. This can be viewed as a special case
of the more general kinematic wave routing
procedure described in section 8.3 Kinematic
Flood Routing in which the weighting coefficients are assigned
values of
a
= 0.0 and
b
= 0.5 (see Figure 8.5).
Figure 8.6  A spacetime
element for reservoir routing.
With reference to Figure 8.6 the continuity
equation can be averaged over a timestep
Dt
as follows.
where QI
= time series of inflow values
QO = time series of outflow values
S = storage volume
1,2 = subscripts corresponding to
times t and
t+
Dt
respectively.
Equation [8.46] can be expanded as follows to
yield an indirect solution for the outflow
QO2.
or
where
Figure 8.7  Graphical
illustration of equation [8.48]
The application of
equation [8.48] is illustrated graphically in Figure 8.7. Starting
with some initial, known value of
QO1 = QI1
the corresponding value of
f(QO1)
is found by interpolation or otherwise. The inflow hydrograph
QI(t)
provides an upstream boundary condition from which
QI2
can be found. Then from equation [8.48] a value is obtained for
f(QO2)
and finally by back interpolation
QO2
is calculated and the process continues for other time increments.
Estimating the
Required Pond Storage
At the start of the Pond command MIDUSS
estimates the required volume by making the assumption that the
reservoir is linear. This means that the storage volume
S is a linear function of the
outflow QO and defined in
terms of a lag coefficient K.
Thus:
For this special case the storage terms can be
eliminated from equation [8.47] and an explicit solution is obtained
for QO2 as follows.
With reference to Figure 8.8, MIDUSS initially
assumes that the lag coefficient K2
has a value of 0.2tb
(where tb
is the time base of the inflow hydrograph) and the corresponding peak
outflow QO2 is obtained by
applying equation [8.51] to the inflow hydrograph. Arbitrary
assumptions are also made for the lag which will attenuate the peak
outflow to 1/100th of the peak inflow. Using a secant method as
illustrated in Figure 8.8 the estimate of
K is successively improved.
The relation is shown in equation [8.52].
Figure 8.8  An iterative
solution for K.
where
X
= log(K)
F
= log(QOmax)
This procedure
converges very rapidly on the necessary lag
Kopt to produce the
desired peak outflow
QOspec.
The corresponding storage volume is given as
Sopt
= Kopt
x
QOspec.
Numerical
Stability in Reservoir Routing
The storage indication
method is traditionally assumed to be inherently stable. However,
this complacency is not justified in situations where the hydrograph
is sharply peaked and the dischargevolume functions are poorly
conditioned or exhibit pronounced discontinuities or points of
contraflexure. In such circumstances, use
of an arbitrary timestep can result in a computed outflow peak that
is larger than the peak inflow  a condition that is physically
impossible.
The search for a criterion to avoid this anomaly
can start with the assumption that:
where e
> 0
Now by substituting [8.54], equation [8.49] can
be written as:
or
This provides an upper limit on the routing
timestep to be used which is shown in equation [8.57].
The peak outflow must lie on the recession limb
of the inflow hydrograph, so that:
The routing timestep is then defined
approximately as:
To implement this
check, MIDUSS scans the storagedischarge function to determine the
flattest part of the curve and uses this to determine an appropriate
submultiple of the timestep to be used. Figure 8.9 shows a typical
situation which can give rise to problems of this type.
Figure 8.9 
Storagedischarge function for a typical outlet control device.
Outflow Control
Devices in Ponds
MIDUSS provides a
number of tools to assist in the creation of the necessary table of
stage, discharge and storage values which form the basis for
evaluating the function
f(QO)
of [8.48]. This section describes how these flow estimates are made
for two basic types of outflow control device.
·
Orifices
·
Weirs
·
Outflow pipes
·
Horizontal orifices
Figure 8.9 shows a
simple but typical device which incorporates an orifice for low flow
control and a weir for less frequent flood events. Up to 10 weirs and
10 orifices can be defined. In addition, MIDUSS has a special tool to
assist in the design of Rooftop Flow Control and Storage for onsite
control.
Orifice Flow for
Pond Control
The stage discharge
equation for the orifice is calculated for two cases which depend on
the relative value of the specific energy H relative to the invert of
the orifice and the diameter of the orifice
D.
In Case 1, H > D
and the orifice is fully submerged.
where H
= head relative to the invert of the orifice
D
= orifice diameter
g
= gravitational acceleration
Cc
= coefficient of contraction
In Case 2, H
£
D and the orifice acts as a broadcrested weir of circular
shape. The critical discharge can be approximated by equation [8.61]
where
Figure 8.10  Critical
flow through a segment of a circle.
As shown by the
comparative plot of Figure 810, equation [8.61] is a very reasonable
approximation to the critical discharge through a segment of a circle.
Weir Flow for Pond Control
Figure 8.11  Definition
sketch of a trapezoidal weir.
The weir control is assumed to have a general
trapezoidal shape as illustrated above. The critical discharge is
calculated for the central rectangular section and two triangular
sections. For any value of head H
greater than the weir sill elevation
Z the critical discharge can be calculated using the
general criterion for critical flow of equation [8.8]
and calculating the crosssection properties
A and
T in terms of the parameters
shown in Figure 8.11.
For a rectangular section:
where
For a triangular section:
where
Typical Storage
Components for Detention Ponds
In addition to control
flow estimation tools, MIDUSS provides a few methods for estimating
the available volume in various standard storage facilities. These
assist you in setting up the stage, discharge and storage values which
form the basis for evaluating the function
f(QO)
of [8.48]. This section describes how these storage estimates are
made for three basic types of storage facility.
(1)
Rectangular ponds
(2)
Super Pipes
(3)
Wedge (or Inverted Cone) ponding
In addition, MIDUSS
has a special tool to assist in the design of Rooftop Flow Control and
Storage for onsite control.
Rectangular Pond
Storage
Figure 8.12  Schematic
of a 3stage rectangular pond
Detention ponds are
usually constructed with side slopes which are dictated by
consideration of maintenance (e.g. grass cutting) and safety. It is
common for the side slope to be different at different water surface
elevations. If the pond has a permanent storage component (e.g. for
quality) it may be desirable to maintain a flat slope of 4:1 or 5:1
for 3m/10ft both below and above the permanent water surface
elevation. Even if the pond is a "dry" pond it may be necessary to
have a flatter slope at higher depths in order to get a suitably
nonlinear stagestorage curve.
Figure 8.12 shows an
idealized pond with three stages. The shape in plan is approximated
by a series of rectangles corresponding to different elevations and
which have an aspect ratio
L/B
which reduces with increasing height.
In practice it is most unlikely that the pond
geometry correspond closely to this idealized shape but the
rectangular pond method provides a useful design tool to estimate the
general dimensions (volumes, land area etc.) required to achieve a
required level of flow peak reduction.
The volume is calculated using Simpson's rule so
that;
where
MIDUSS provides an approximate estimate for the
base area A1. This is
calculated from the estimate of required storage volume and assumes
that only a single stage is used and that the depth is 2/3 of the
depth range specified, the base aspect ratio is 2:1 and the side slope
is 4H:1V.
SuperPipes for
Pond Storage
Figure 8.13  Schematic
of Super Pipe Storage
Figure 8.13 shows a
typical arrangement of a single superpipe with a simple outflow
control device installed at the downstream end. The control can be
installed either in the pipe barrel or in a manhole structure. The
latter is convenient if more than one super pipe converges at a
junction node.
You should remember to
avoid using too steep a gradient as this can seriously limit the
available storage volume since the water surface is likely to be
nearly horizontal.
In MIDUSS the volume is obtained by calculating
the crosssection of the storage at 21 equally spaced sections along
the length L and then
using Simpson's Rule. The section area is given as a function of the
relative depth y/D from
the following equations.
in which
f is
obtained by [8.11]
where
The volume is then obtained as:
MIDUSS provides an initial default length for a
single super pipe assuming that (1) the diameter is approximately half
the depth range, (2) the slope is zero (3) the pipe is full and the
volume is equal to the estimated required storage.
Wedges (or
inverted Cones) for Pond Storage
Figure 8.14  Schematic
of wedge storage
To assist in
estimating the available storage on parking lots, MIDUSS provides a
wedge storage procedure that calculates the volume of a sector of a
flat, inverted cone as illustrated in Figure 8.14. The angle
subtended by the segment is defined as an angle(in
radians) for generality. In practice, this will often be 90 degrees
with four such segments describing the storage around a
catchbasin draining the parking lot with
grades g1
and g2
mutually at right angles.
Figure 8.15  Calculation
of surface area of a segment of inverted, ovoid cone.
The radius R
and grade g are assumed
to vary linearly with the angle as shown in Figure 8.15. Then a small
element of the surface area is described as:
:
Integrating between the limits 0 and gives
the surface area as
The volume V
is then calculated as:
MIDUSS assumes that the invert of the tail pipe
or the Inflow Control Device (ICD) in the catch basin is approximately
3 ft (0.92 m) below the rim elevation and that the maximum depth of
ponding will probably be less than 1 ft
(0.3m) above rim elevation as illustrated in Figure 8.14. To provide
an initial estimate for design purposes MIDUSS assumes that the last
defined impervious area has a catch basin density of 1 per 2500
sq.m or 2989 sq.yd.
It is further assumed that each catch basin has a drainage area with
orthogonal grades in a ratio of 2:1 (e.g. 40H:1V in one direction and
80H:1V at right angles). Based on a depth of 1 ft (0.3m) above rim
elevation, the necessary grades to provide the estimated required
volume are calculated and displayed together with the total number of
elliptical quadrants (four such quadrants per catch basin).
In setting up parking
lot storage the depth range should be slightly more than 4 ft (1.22m)
to comply with the assumptions made above.
Rooftop Flow
Control for Pond Storage
For developments
involving large commercial buildings with flat roofs, onsite storage
can be provided by installing roof drain controls. Typically these
devices contain one or more "notches" which take the form of a linear
proportional weir in which discharge is directly proportional to the
head or depth of storage, for example 24 litres
per minute per 25 mm of head or 6 US gallons per minute per inch of
head. The actual value is defined along with other relevant
parameters.
If the roof is dead
level then the volume of storage is calculated simply as roof area
times head where the roof area available for storage is smaller (e.g.
75%) than the building footprint to allow for service structures
(access, elevator, HVAC) on the roof.
When a finite grade is used to promote drainage
the calculation of available storage depends on whether the head H is
less than or greater than the fall or difference in elevation between
ridge and valley in the roof profile.
for
for
where
and L
= flow length from ridge to drain
S0 = roof grade
