UseofRationalandModified RationalMethodinTexas - - PowerPoint PPT Presentation

Use
of
Rational
and
Modified
 Rational
Method
in
Texas


T.G.
Cleveland,
Texas
Tech
University
 X.
Fang,
Auburn
University
 W.
H.
Asquith,
US
Geological
Survey
 D.
B.
Thompson,
R.O.
Anderson,
Inc.
 Project
0‐6070.

May
28,
2009


Outline


• Introduction

• Definitions

• What
We
Have
Done/What
We
Have
Learned


– Activities/Evidence


• Summary

• Where
We
are
Headed


Introduction


• Rational
Equation



 

 
 
Qp
=
C
i
A
 
 
Qp
=
peak
discharge
(L^3/T)
 
 
C
=
runoff
coefficient

 
 
i
=
rainfall
intensity
(L/T)
 
 
A
=
contributing
area
(L^2)


Introduction


• Rational
(and
Modified
Rational)
method


abridged
sequence.


– (A)
Choose
a
risk
level.
 – (B)
Determine
a
characteristic
time
for
drainage
 area.
(need
to
specify
area
and
time)
 – (C)
Rainfall
intensity
(Needs
(A)
and
(B)
a‐priori)
 – (D)
Choose
a
runoff
generation
coefficient
 – (E)
Compute
Qp
(or
a
hydrograph
if
using
modified
 rational
method
–
peak
value
is
Qp)


Introduction


• Rational
Method



http://onlinemanuals.txdot.gov/txdotmanuals/hyd/the_rational_method.htm


– Determine
drainage
area.
 – Determine
time
of
concentration.
 – Verify
appropriate
with
stated
assumptions.
 – Determine
e,b
and
d
values
for
desired
design
 frequency.
 – Compute
rainfall
intensity.
 – Select
runoff
coefficients.
 – Calculate
estimated
peak
discharge.


Introduction


• Modified
Rational


Method


– Similar
approach
but
 generates
a
triangular
or
 trapezoidal
hydrograph
 depending
on
rainfall
 duration
as
related
to
the
 watershed
characteristic
 time
(Tc).
 – Largest
possible
peak
 discharge
is
same
as
 rational
method
if:


• Td
>=
Tc


Introduction


• Modified
Rational


Method


– Similar
approach
but
 generates
a
triangular
or
 trapezoidal
hydrograph
 depending
on
rainfall
 duration
as
related
to
the
 watershed
characteristic
 time
(Tc).
 – Largest
possible
peak
 discharge
is
same
as
 rational
method
if:


• Td
>=
Tc


Introduction


• Assumptions
(current)


– The
rate
of
runoff
resulting
from
any
constant
 rainfall
intensity
is
maximum
when
the
duration
of
 rainfall
equals
the
time
of
concentration.


• This
assumption
is
a
statement
of
mass
conservation
in


a
system
where
loss
rates
after
some
time
become
 constant.

It
[the
assumption]
is
probably
valid
for
most
 hydrologic
scales
with
the
qualification
that
requisite
 inputs
may
not
ever
occur
in
nature,
and
constant
loss
 rates
may
never
be
realized.


Introduction


• Assumptions
(current)


– The
frequency
of
peak
discharge
is
the
same
as
 the
frequency
of
the
rainfall
intensity
for
the
given
 time
of
concentration.


• Known
non‐linearity
in
hydrologic
systems
such
that


response
does
not
scale
at
the
same
rate
as
input.




Introduction


• Assumptions
(current)


– The
rainfall
intensity
is
uniformly
distributed
over
 the
entire
drainage
area.



• In
reality,
rainfall
intensity
varies
spatially
and


temporally
during
a
storm.



• For
small
areas,
the
assumption
of
uniform
distribution


is
reasonable.



• For
larger
areas
partial
contribution
may
be
applicable


–
hard
to
quantify
for
design
use.


Introduction


• Assumptions
(current)


– The
fraction
(C)
of
rainfall
that
becomes
runoff
is
 independent
of
rainfall
intensity
or
volume.


• This
assumption
is
important
to
be
consistent
with
the


first
assumption.


Definitions


• Time
Concepts


– Rainfall
duration


• How
long
a
constant
rainfall
rate
is
applied
‐‐‐


numerically
the
same
as
Tc,
conceptually
different.


Single
pulse
of
rainfall
over
T2
 5
pulses
of
rainfall,
T1
units
long,
in
a
row.
 T1
 T2


Definitions


• Time
Concepts


– Watershed
 characteristic
 times


• Concentration

• Peak

• Lag


Definitions


• Runoff
Coefficients


– Volumetric
 – Rational



CV = Q(t)dt

∫

p(t)dt

∫

CR = Qp i(td )A

Definitions


• Runoff
Coefficients

• Coefficients
are
related
to
basin
characteristics.


Definitions


• Watershed
Characteristics


– Area
(already
in
rational
equation)
 – Slope
 – Channel
Length
 – Basin
Development
Factor
 – Percent
Impervious
 – Land
Use
(Descriptive)


Definitions


• Basin
Development
Factor
(BDF)


– Reflects
improvement
in
drainage
network.
 – Correlates
with
development,
makes
no
 assumptions
about
land
use
surrounding
the
 network
per‐se


Definitions


• BDF


– Channel
 modification
 – Channel
lining
 – Storm
drain
 – Curb‐gutter


What
We
Have
Done


• Table
of
Cstd
values
from
an
extensive


literature
review.


– These
were
presented
in
TM1
 – They
will
be
repeated
with
further
commentary
in
 TM2
(in
progress,
due
soon)


What
We
Have
Done


• Table
of
Cstd
values
from
an
extensive


literature
review.


– These
were
presented
in
TM1
 – They
will
be
repeated
with
further
commentary
in
 TM2
(in
progress,
due
soon)


What
We
Have
Done


• Determined
(by
literature
review
and


numerical
experiments)
that
the
modified
 rational
method
is
a
special
case
of
the
unit
 hydrograph
method.


– A
valuable
consequence
of
this
determination
is
to
 remove
a
conceptual
decoupling
of
the
methods,
 instead
there
is
a
continuum
of
methodology.


 – Care
is
still
required
with
regards
to
the
scale
of
 applicability.


What
We
Have
Done


• Rational
Method
is
a


Unit
Hydrograph


– Watershed
is
 represented
as
a
 linear,
time‐invariant
 system

 – Assumptions
1
and
4


• f
current
method.


What
We
Have
Done


• Invested
in
clarifying
the
different
concepts
in


the
rational
method
especially
the
time
and
 the
runoff
coefficients.


– These
are
to
be
explained
in
detail
in
TM2.


What
We
Have
Done


• Forward
computation
of
Cv.


– Application
of
rational
method
as
if
ungaged
 watershed.
 – Assumed
that
rational
method
is
a
unit
hydrograph.
 – Substitute
the
rectangular
kernel
in
the
convolution
 integral.


What
We
Have
Done


Gamma
UH
 Modified
Rational
Method


What
We
Have
learned


• Forward
computation
of
Cv.


– Rational
method
is
a
unit
hydrograph.
 – Size
limitation
is
arbitrary
(from
standpoint
of
linear
 systems
theory).
 – Current
limitation
is
meaningful
for
other
reasons
 (uniform
rainfall,
storage,
etc.)


What
We
Have
Done


• Backward
computation
of
Cv
for
Texas


watersheds.


– Compute
best
Cv
to
explain
observations.
 – Same
kind
of
simulations
as
just
presented.
 – Again
assume
rational
method
is
a
unit
 hydrograph.


What
We
Have
Done


10
 100
 1000
 10000
 10
 100
 1000
 10000


Modeled
Qp
(cfs)
 Observed
Qp
(cfs)


1:1
line
 C
from
the
ratio
of
runoff
and
 rainfall


What
We
Have
Learned


• Backward
computation
of
Cv
for
Texas


watersheds.


Using
Cstd
 Using
Cv
–
backcomputed


What
We
Have
Learned


• Backward
computation
of
Cv
for
Texas


watersheds.


– Evidence
over‐prediction
potential
of
peak
 discharge.
 – Evidence
suggests
bias
towards
smaller
areas
‐‐‐
 exactly
where
the
method
is
intended
to
be
 applied.
 – Median
C
value
from
literature
based
approach
is
 nearly
two
times
larger
than
for
back
computed.


What
We
Have
Done


• Back
compute
Cr
values
from
Texas
+
ARS.


– Used
Tr
as
the
characteristic
time
value,
removes
 ambiguity
for
specification
of
Tc


• Not
dimensionally
homogeneous.

• Consistent
with
a
watershed
integrated
“velocity”
of
1


mile
per
hour.
(~1.46
ft/sec)


TRhrs = Asq.mi.

What
We
Have
Done


• Use
iPAR
tool


developed
early
 in
the
project.


– Analyst
selects
 time
windows.
 – Software
returns
 various
C
values
 – Allows
isolation


• f
interesting


parts
of
time
 series.


What
We
Have
Done


• Use
iPAR
tool


developed
early
 in
the
project.


– Cv
 – Cr
 – Cw
 – Cx


Evidence:
Granularity


• Stem‐leaf
plots
Will
(14000
storms)


What
We
Learned


• Regionalized


Tr.


– Estimates
 most
 sensitive
to
 timing.
 – Cr
medians
 for
each
 station


What
We
Learned


• Runoff
Coefficients
from
Tr
analysis


– Rural,
non‐highway
(ARS)


Cars
=
0.09;
BDF=0


– Houston
(coastal
plains?):




Chou
=
0.13
;
BDF<6
 Chou
=
0.28
;
BDF>=6


– Other
Texas
(Not
Houston)



Ctex
=
0.23
;
BDF<6
 Ctex
=
0.35;
BDF>=6


• Huge
differences
in
timing
concepts
are
imbedded
in


these
groups:


– huge
differences
in
applied
intensity
 – huge
differences
in
Qp


What
We
Learned


• Runoff
Coefficients


(from
Kuchling
1889)


– “vacant
land”
~
0.3
 – “developed”
~
0.5‐0.7


• Runoff
Coefficients


(our
work)


– “vacant
land”
~
0.15
 – “developed”
~
0.32


Summary


• Rational
method
is
a
special
case
of
the
unit


hydrograph.



– Thus
it
represents
a
continuum
of
technique
and
 should
be
viewed
as
such.


 – Limitations
as
to
upper
area
of
applicability
and
 minimum
rainfall
duration
are
sound
to
prevent
 improper
use,
but
are
not
specifically
related
to
the
 rational
concept.



• As
currently
applied
a
potential
for
over‐

estimation
exists,
in
part
from
the
“C”
value,
and
 in
part
from
the
“Tc”
value.


Summary


• Granularity
in
the
literature
tabulated
values


implies
a
greater
level
of
precision
than
is
 reasonable.


– Our
work
to
date
suggests
that
the
literature
 values
are
too
large,
and
too
granular.
 – The
researchers
have
found
very
little
actual
data
 to
support
the
literature
C
values.


 – We
speculate
that
the
current
values
are
the
 result
of
a
“Delphi”
process.


Summary


• Using
literature
values
and
reasonable


application
of
current
methodology,
can
 expect
to
overestimate
Qp
in
about
1/3
of
 applications.


Where
We
Are
Headed


• Rational
Method
has
utility
and
need
not
be


abandoned.


• Tables
of
“C”
values
can
be
reduced
in
scope
and


still
provide
meaningful
estimates
of
Qp


– Excess
Modified
Rational
Method


• Focused
on
the
time
parameter
and
a
reduced
set
of
“C”


values.


• Careful
documentation
of
the
linkage
to
the
unit


hydrograph
method,
as
well
as
when
to
switch
 the
kernel
function.


Commentary


• Kuchling’s
original
concept
was
that
the
“C”


value
was
the
fraction
of
impervious
area,
and
 assumed
that
vacant
land
produced
negligible
 runoff
(for
his
work).


• The
“time
of
concentration”
concept
is


mentioned
in
his
1889
paper
–
the
definition
 in
that
paper
is
unchanged
today.