United States
Environmental Protection
Agency
Municipal Environmental Research ^
Laboratory
Cincinnati OH 45268
Research and Development
EPA-600/S2-83-087 Nov. 1983
SERA Project Summary
The Value of Flow Calibration for
Decision Making in Infiltration/
Inflow Studies
William C. Pisano, David S. Watson, and Gerald L Aronson
A study was conducted to determine
the value of flow calibration for decision
making in sewer system studies. The
results may be applied to any sewer
system flow measurement program.
Data indicated that the conventional
use of Manning's equation for relating
discharge from stage records collected
by manual and automated recorders
without primary flow calibration could
grossly misestimate discharge condi-
tions over all flow regimes. This
problem translates into erroneous cost
conclusions regarding treatment, trans-
port, and rehabilitation recommenda-
tions.
Primary data used in this investigation
were obtained from an earlier study
conducted for the Metropolitan District
Commission (MDC) in Boston, Massa-
chusetts, covering portions of West
Roxbury-Newton-Brookline-Dedham in
the Boston metropolitan area (Environ-
mental Design & Planning, Inc., "Infil-
tration/Inflow Study for West Roxbury-
Brookline-Newton-Dedham Area,"
Metropolitan District Commission).
This previous study included a number
of primary discharge measurements at
key manholes during both dry and wet
weather flow conditions.
The present study developed a variety
of scenarios for key manholes as to the
amount of information available to
develop stage/discharge calibration
curves. Alternative stage/discharge
curves were used to estimate costs for
the resulting rehabilitation and treat-
ment programs. The problem of in-
creased gaging costs was weighed
against possible reductions in erroneous
recommendations. Results indicate
that the additional care and cost
involved in primary flow calibration of
stage/discharge curves for sewer
system evaluations are well worth the
effort. The best method for developing
stage/discharge curves was to curve-
fit Manning's equation with a variable
roughness coefficient to calibration
data using least squares methods.
This Project Summary was developed
by EPA's Municipal Environmental
Research Laboratory. Cincinnati, Ohio.
to announce key findings of the research
project that is fully documented in a
separate report of the same title (see
Project Report ordering information at
back).
Introduction
Accurate determination of sewer
system flow characteristics in dry and
wet weather is vital to establishing
infiltration and/or inflow rehabilitation
requirements. Inaccurate gaging (parti-
cularly for borderline situations) may lead
to erroneous recommendations for
further actions and result in the inefficient
allocation of local or Federal resources.
Flow determinations are often per-
formed in infiltration/inflow studies
using a dipstick (instantaneous spot)
procedure to obtain stage levels. These
results are then converted into discharge
using any one of a variety of different
procedures such as Manning's, Chezy's,
and/or Kutter's equation together with
measurements of various hydraulic input
parameters (size, shape, length, and
slope) taken from plan and profile as-built
drawings. Roughness coefficients are
usually estimated by inspecting the
conditions of the test segment and its age
since construction. In other situations,
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automatic stage recording devices are
installed to obtain a long-time series of
stage records. Most often these records
are then converted to discharge quantities
using similar procedures for relating
stage to discharge.
Accurate measurement of either dry or
wet weather flowdischarge in a sewerage
system is a difficult process beset by
constantly changing problems. For exam-
ple, sediment accumulations erratically
increase the values of roughness coeffi-
cients nominally determined for clean-
pipe conditions, and also change the
cross-sectional area of flow. Grit and
debris tend to impair velocity determina-
tions and bias discharge computations.
Backwater problems elevate depths of
flow and change velocity profiles.
Two different measurement procedures
can be used: direct methods that meausre
both depth of flow and velocity, and
indirect methods that record only depth of
flow and then convert to discharge using
some predictive relationship. Direct
measurement approaches can provide
accurate determinations of flow for given
instants of time, but they are not routinely
used in gaging studies. Secondary or
indirect methods provide a less accurate
flow estimate, but they are commonly
used in infiltration/inflow studies because
of their simplicity and low cost. In many
studies, no direct measurements are
made to corroborate or calibrate indirect
methods.
One approach used in some studies is
to obtain primary measurements at
several flow depths (most often at dry
weather flow conditions) and then to use
Manning's equation (or an equivalent) in
dimensionless form to estimate full-pipe
discharge conditions so that the upper end
of the stage/discharge curve can be
established. This method is very reason-
able, provided that the calibration points
are made without measurement error
(which is often difficult). The presence of
sediment beds complicates the numerical
simplicity of this direct approach. This
approach may be acceptable when
measurements are made accurately. But
when measurement errors exist, and
when all the requisite assumptions
underlying Manning's equation are not
met, a least squares curve-fitting proce-
dure can be used with Manning's
equation (in dimensionless form) to fit
observed data. This report presents an
alternative and numerically equivalent
method consisting of simple, statistical,
least squares, curve-fitting techniques
that compensate for both the measure-
ment error problem and the inherent
limitation of using a simple model
(Manning's equation) to represent com-
plicated phenomena.
Study Design
In the earlier study, sewage flowdepths
at key manholes throughout a 3000-acre
sewerage system were measured under
both dry and wet weather conditions,
including instantaneous and continuous
measurements (see Figure 1). Discharge
estimates at a number of locations using
measured flow areas and velocities were
also obtained. A single stage/discharge
method was examined at a representative
number of key manholes.
Step A of this study derives alternative
stage/discharge relationships for all key
manholes using different assumptions.
Step B computes alternative estimates of
flow conditions throughout the sewerage
system for both dry and wet weather
conditions using the alternative stage/
discharge relationships derived in Step
A, the raw flow depth information, and
the discharge calibration points (computed
from velocity and flow area measure-
ments from the earlier study. Step C
proposes various sewer system rehabili-
tation programs that attain a similar
degree of extraneous flow reduction.
Earlier Study
• Depth of Flow Measurements
• Discharge Calibration Points
• Single Stage/Discharge Method
• Single Cost Effectiveness Analysis
I"
Current Study
Step A jt
Alternative Stage/ Discharge
Relationships
Step B i
Alternative Sets of Flow Conditions
Step C .r
Preparation of Alternative
Sewer System Rehabilitation Programs
and Costs
Step D
Overall Assessment of Measurement
Versus
Rehabilitation Programs
Figure 1. Schematic of project
research and development.
Complete analyses of the required sewer
system upgrading are prepared along
with cost estimates for each set of
assumed flow volumes. Finally, Step D
assesses the economic benefits of mea-
surements versus the implied costs of the
alternative rehabilitation programs.
The full report presents several statisti-
cal methods for developing stage/
discharge relationships in which Manning's
equation with a variable roughness
coefficient is curve-fitted by least squares
methods of calibration data obtained by
direct field measurements. A surrogate
value of energy slope is determined that
minimizes the sum of squares of field
discharge information about a curve
defined by Manning's equation. The
procedure makes the best use of limited
depth-of-flow/discharge measurements
in developing the overall rating curve.
Alternative fitting procedures are devel-
oped using ordinary least squares and
weighted least squares approaches. The
stage/discharge curve-fitting methodo-
logy is described in the following section.
In the case study examined in this
report, depth-of-flow data collected at 45
key manholes were translated into
discharge levels using three alternative
methods for developing stage/discharge
curves. The methods included: a) applica-
tion of weighted least squares procedures
to Manning's equation using a variable
roughness coefficient, b) application of
Manning's equation using plan and
profile slope and a variable roughness
coefficient formulation, and c) application
of Manning's equation using plan and
profile slope with a fixed roughness
coefficient. These methods are compared
to show the differences between the least
squares procedures and methods com-
monly used in actual practice. The
alternative stage/discharge methods
prepared for one of the key manholes are
described in the last section of this
summary.
The three different sets of peak dry-
weather flow, infiltration, and inflow for
the 45 key manholes were then used as
raw information to determine technically
equivalent, least-cost infiltration/inflow
control programs. The controls consisted
of (1) infiltration rehabilitation and inflow
remedial programs in areas tributary to
an interceptor undergoing backwater and
surcharge conditions, and (2)construction
of replacement (increased conveyance)
segments along the existing interceptor.
The aim of the effort was to investigate
how the mix of final recommended
controls and their associated costs would
change using flows computed by the
different flow estimation schemes.
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Table 1. Overview of Control Plans Developed Using Different Method of Flow Estimation
Flow Estimation Method Control Plan Overall Program Cost
Method 1
(Least squares estimation
of slope in Manning's
Equation with variable
roughness factor co-
efficient)
Method 2
(Manning's Equation with
plan & profile slope
and variable roughness
coefficient)
Method 3
(Manning's Equation with
plan & profile slope and
fixed roughness coefficient)
$3,160.000
• Infiltration control,
60%
• Inflow control, 71%
• New replacement segments
in portions of upper
westerly interceptor
> Infiltration control,
85%
i Inflow control, 85%
i New replacement segments
in portions of upper
westerly interceptor
system (about the same as
plan 1)
Infiltration control,
85%
Inflow control, 85%
New replacement segments
throughout the entire
system
$3,580,000
$6,130,000
Identical criteria were used to determine
the level and need for each type of control
for the three different methods. Present
value costs were computed for each plan,
and they included infiltration rehabilita-
tion, remedial inflow corrections, con-
struction of replacement segments along
the existing interceptor, and treatment
costs of extraneous wet weather dis-
charges. An overview of the rehabilitation
analysis and the associated costs for the
three methods appear in Table 1. Cost of
the overall control program (present
value) for the three methods of flow
estimation were $3.16 million, $3.58
million, and $6.13 million, respectively.
The mix of recommended controls was
substantially different for each of the
three plans. The engineering costs
involved in performing the field calibration
discharge measurements together with
performing the least squares calculations
were about $13,500 (as determined in
the earlier study). Field calibration
accounted for about 90 percent of the
additional field effort. The least squares
computations can be quickly performed
using typical engineering-office desk
calculators.
Significant benefit can be derived from
carefully preparing stage/discharge
rating curves in infiltration/inflow
studies by performing primary measure-
ments and making the best use of this
information with curve-fitting procedures.
This conclusion holds true for any
sewerage system management study.
Least Squares
Stage/Discharge Methodology
A number of the hydraulic parameters
in Manning's equation can be expressed
as a function of the ratio of observed flow
depths, hi, to pipe diameter, D, for
measured values of discharge, Q,. Mann-
ing's equation is as follows:
n,=1.49r,2/3S1/2a,
n,
d)
where:
Qi = fi(hi/D), measured flow rate (cfs)
r, = faJh/D), hydraulic radius (ft)
S = constant, energy slope (ft/ft)
a, = f3(h,/D), wetted area (ft2)
n, = f4(h,/D), Manning's roughness
coefficient
The value of n at full pipe, nf, and its
variations with depth was assumed to be
known, and the slope, S, was taken as a
surrogate for all the uncertainties with
respect to S itself and the roughness
coefficient, n. We preferred to assume
that n (or its variation with depth) is
known and to let a surrogate slope, S,
constant for all water depths, be the free
parameter to be determined from the
least squares procedure.
A number of least squares approaches
are possible, depending on how we pose
the minimization problem of fitting the
observed values of (hi, Q,) about Manning's
equation. Three approaches are possible.
The first approach is the standard,
ordinary least squares formulation in
which the standard deviation of the
residuals (observed-predicted) is inde-
pendent of the magnitude of discharge.
The resulting expression for the sum of
the squares of the residuals, O(S), can be
written as:
m
Min 0(S) = I [Q, - S1/2 Ki(h/D)]2
S i=1 (2)
where: Q, = observed values of flow from
the calibration
Ki(hi/D)=1-49a,r,2/3 (3)
n,
m = number of observed Qi
all other variables = as defined before
To minimize O(S), its derivative is taken
with respect to S and set equal to zero.
The resulting expression for the best-fit
S is:
m
(4)
m
I K2
In the second approach, the variance of
the measured value, Qi, from its computed
value is assumed to be proportional to the
magnitude of Q. The objective function to
be minimized under this assumption is
as follows:
m
Min 0(S) = I * J [Qi - K, S1/2]2 (5)
S S i=1 K,
The resulting estimate of slope, S, is
given by:
m
I Q,
s1/2 = if]
m (6)
I K,
i=1
Another feasible model would be to
assume that the standard deviation of Q,
is proportional to Q,. This assumption is
the same as a constant coefficient of
variation. The resulting objective function,
O(S), is given by:
Min O(S) =
S
m
1/S2 I 1/K2(Qi-K,S1/2)2 (7)
i=1
The resulting estimate of slope, S, is
given by:
m
S1/2=J_2 Q/Ki (8)
mi=1
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The choice of a particular model to use in
any given application is somewhat
arbitrary. The ordinary least squares
approach given by Equation 4 weights
large observations of (Q,, K,) too heavily,
whereas the estimate of S, resulting from
the third approach given by Equation 8,
weights large observations too little. The
second approach, which assumes that
the variance of the observed Q about its
true value is proportional to its magnitude,
seems to provide a reasonable com-
promise.
Pertinent data illustrating the least
squares fitting procedures are presented
in Table 2. Depth of flow and velocity
measurements in a 27-in., vitrified clay
pipe (vcp) sanitary trunk sewer in the
Boston area were determined on four
separate occasions. Depth of flow and
velocities were measured using a Marsh-
McBirney* meter. The tabulation of
variable Manning's roughness coefficient
for nt = 0.015 was used to prepare the
estimates shown in Table 2.
The ordinary least squares estimate of
the surrogate slope parameter S, using
Equation 4, is computed as follows:
S = [ 562.557 p =0.00323
9902.724
The weighted least squares estimate of
the surrogate slope parameter using
Equation 6 is computed as follows:
S = [JLZ46_]2 = 0.00292
161.925
A sensitivity analysis was performed
(Table 3) using two alternative least
squares approaches on different sets of
data derived by various representative
assumptions of measurement error. Four
different cases of measurement error for
the field data in Table 2 are considered.
The first case entails a positive velocity
meter bias of 10 percent for all four
observations. An erroneous positive
depth-of-flow error of 1 in. for all
observations in considered in the second
case. An erroneous positive depth-of-
flow error of 1 in. for only the high-flow
measurements is depicted for the third
case. And the last case considers both
depth of flow error and positive velocity
meter bias for the high-flow measure-
ments. Many other combinations of
measurement error are also possible.
The square root of the ratio of the fitted
ordinary least squares (ols) and the
weighted least squares (wls) slopes for
each case (Table 3) reflect the increase of
hydraulic conveyance about the nominal
'Mention of trade names or commercial products
does not constitute endorsement or recommendation
for use
estimate, assuming that the derived
relationships are thereafter used without
error (i.e., correct depths of flow are
entered into the computations). The
results show that for these data sets,
either method will yield comparable rates
of error. The greatest degree of error
derives from the first case, involving
velocity measurement bias.
Estimates provided by the wls approach
are nevertheless preferred, particularly
when very high (full pipe) flow conditions
are gaged with the possibility of both
velocity and depth-of-flow measurement
errors. The wls approach tends to reduce
the impact of large (Qk K,) values. A good
compromise in practice would be to
calculate the slopes by both methods and
then compute the geometric average of
the two estimates.
Two types of discharge estimates were
compared for the 27-in. trunk sewer —
those computed using plan and profile
slopes of 0.003 with fixed roughness
coefficients of 0.013 and 0.015, and
those computed using both least squares
approaches for various flowdepths (Table
4). The calculations show that discharges
computed using plan and profile slope
with fixed roughness coefficients (cases
A and B) greatly exceedthose computed by
either of the least squares procedures.
Thus the differences in Table 4 between
cases A and B and Cases C and D are
considerably decreased if the variable
roughness coefficient formulation is
used for cases A and B.
Comparison of
Stage/Discharge Curves from
the Earlier Study
Alternative stage/discharge curves for
a key man hole in area 25 in the study area
is presented in Figure 2. Field discharge
measurements are also shown. The
sewer segment is a 12-in. vcp with IViin.
of sediment and was constructed at a
slope of 0.00208. Four alternative curves
are plotted for the following conditions:
(1) minimum square fit (wls) for S
assuming variable n (nf = 0.015); (2)
minimum square fit (wls) for S assuming a
constant n = 0.015 and considering
sediment, if present; (3) clean pipe, with
slope from the plans and a constant n =
0.013; and (4) clean pipe with slope from
the plans and a constant n = 0.015.
Common practice in flow measurement
studies is to use the pipe slope from as-
built drawings in Manning's equation and
to disregard the presence of sediments in
the estimation process. Implicit assump-
tions are that the pipe slope is known with
certainty and that the hydraulic flow
profile equals the pipe slope (i.e., the -
water slope is parallel to the pipe slope), m
These important assumptions underlie ^
the computational process used to
derive curves 3 and 4. The least squares
fitting process used to establish curyes 1
and 2 makes no such limiting assump-
tions; instead, it uses actual flow
measurements affected by sediment
beds and by pipe and water slope
irregularities to develop realistic rating
curves.
Comparisons of curves 1 and 2 in
Figure 2 show that though both curves
comply with the mathematical condition
that the sum of squares of the deviations
from the observed values is minimum,
curve 1 seems to adhere better to the
observed points. This fact can be inter-
preted as additional evidence that
Manning's equation is more capable of
reproducing the physical phenomena of
flow in pipes if n is considered a function
of flow depth.
The presence of sediment beds can be
considered in the least squares curve-
fitting procedure by appropriate adjust-
ment of the hydraulic parameters defined
in Equation 1. The cross sectional area, a,,
and the hydraulic radius, r,, are directly
computed using the top of the sediment
bed as the reference level. The variable
roughness factor tabulation is based on a
circular cross section. Estimates of the
variable roughness factor can be gener-
ated for noncircular cross sections
(circular or noncircular, with or without
sediment) using equivalent circular
estimates. First, an equivalent diameter
is computed for the total cross section
free from sediments. Next, an equivalent
depth of flow (circular section) is computed
using the wetted area of the noncircular
section for each flow observation. Last,
the appropriate estimate of variable n is
determined using the equivalent circular
depth of flow and the equivalent diameter
estimates.
The full report was submitted in
fulfillment of Grant No. R-804578 by
Environmental Design & Planning, Inc.,
under the sponsorship of the U.S.
Environmental Protection Agency.
4
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Table 2. Sample Calculations of Least Squares Procedures
Measured Information Physical Data
Calculations
Depth of Flow*
(in.)
9.5
11.5
4.75
4.25
Sum
Measured Velocity
(fps)
2.50
2.90
1.15
1.10
(ft)
.437
.504
.242
.219
(ft2)
1.249
1.614
.471
.401
n?
.0192
.0190
.0190
.0189
a
(cfsri
3.123
4.680
0.542
0.401
8.746
Kfi
55.868
8O.282
14.322
11.453
161.925
Kf
3121.233
6445.200
205. 120
131.171
3302. 724
AfiQ,
174.464
375.743
7.756
4.594
562.557
*27-in. vcp trunk sewer with no sediment, plan and profile slope = 0.0036.
* Variable Manning's roughness coefficient with n> = 0.015.
n\
JCubic meters/second = 0.0283 cfs.
Table 3. Sensitivity Analysis of Least Squares Procedures
Case
listing data (nominal
stimate)
'elocity measurement
tcrease 110%) for all
observations
lepth of flow increase
1 in.) for all obser-
ations
tepth of flow increase
1 in.) for 1st and
:nd observations
Ordinary Least A. *
.00323
.0039?
.00286
.OO294
Squares-Eq. 5 B.
1.00
1.10
0.94
0.95
Weighted C.
.00292
.00353
.00253
.00270
Least Squares-Eq. 7 D.
1.OO
1.10
0.93
0.96
Depth of flow increase
(1 in.) and velocity
increase (10%) for 1st
and 2nd observations
.00354
1.05
.00321
1.05
"A. Ordinary feast squares estimate of slope.
B. Square root (estimate slope for stated data/estimated nominal slope). Slopes computed by ordinary least squares formulation — Equation 5.
C. Weighted least squares estimate of slope.
D. Square root (estimated slope for stated data/estimated nominal slope). Slopes computed by weighted least squares formulation — Equation 7.
Table 4. Comparison of Discharges Computed Using Various Stage/Discharge Formulations*
Assumed
Depth of
Flow (in.J+
10
18
25
Discharges (cfs)+
At
4.98
13.98
18.27
B
4.22
11.43
15.86
C
3.50
9.98
15.64
D
3.33
9.55
14.80
A/D
1.495
1.372
1.234
Ratio of
B/D
1.267
1.197
1.O72
C/D
1.051
1.045
1.057
*27-in. trunk sewer.
+1 in. = 2.54 cm.
Cubic meters/second = 0.0283 cfs.
A. Manning's equation, plan and profile slope = O.O03, fixed n = 0.013
B. Manning's equation, plan and profile slope = O.O03, fixed n = 0.015
C. Manning's equation, fitted slope (ols), variable n with n, = 0.075
D. Manning's equation, fitted slope (wls), variable n with n, = 0.01 S
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Legend
Curve 1: Best Fit Slope = .000982, N Variable
Curve 2: Best Fit Slope = .000786. N = 0.015
7.8 Curve 3: As Built Slope = .003, N = 0.015
1.7-
1.6-
1.5-
1.4-
1.3-
1.2-
1.1-
•s
„.- 1.0-
0.8-
0.7-
0.6-
0.5-
0.4-
0.3-
0.2-
0.1
Curve 4: As Built Slope = .003, N = 0.013
Measured Points
Depth above sediment
for curves 1 and 2.
Note: (1 in. = 2.54 cm)
(1 cubic meter/second
= 0.0283 cfs)
70 11 12 13 14 15 16
Depth, in.
Figure 2. Alternative stage/discharge curves for Area 25.
•&U. S. GOVERNMENT PRINTING OFFICE: 1983/759-102/0796
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William C. Pisano, David S. Watson, and Gerald L. Aronson are with
Environmental Design & Planning, Inc., Boston, MA 02134
Richard Field and Robert Turkeltaub are the EPA Project Officers (see below).
The complete report, entitled "The Value of Flow Calibration for Decision Making
in Infiltration/Inflow Studies," (Order No. PB 83-259 739; Cost: $10.00, subject
to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, v'A 22161
Telephone: 703-487-4650
EPA Project Officer Richard Field can be contacted at:
Storm and Combined Sewer Program
Municipal Environmental Research Laboratory—Cincinnati
U.S. Environmental Protection Agency
Edison, NJ 08837
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
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