International Journal of Geosciences, 2012, 3, 297-302 Published Online May 2012 (
Developing a Family of C urves for the H EC-18
Scour Equation
Timothy Calappi1, Carol Miller1, Donald Carpenter2, Travis Dahl3
1Department of Civil and Environmental Engineering, Wayne State University,
Detroit, USA
2Department of Civil Engineering, Lawrence Technological University,
Southfield, USA
3United States Army Corps of Engineers, Detroit, USA
Received October 21, 2011; revised January 20, 2012; accepted February 25, 2012
Accurate pier scour predictions are essential to the safe and efficient design of bridge crossings. Current practice uses
empirical formulas largely derived from laboratory experiments to predict local scour depth around single-bridge piers.
The resulting formulas are hindered by insufficient consideration of scaling effects and hydrodynamic forces. When
applied to full-scale designs, these formula deficiencies lead to excessive over prediction of scour depths and increased
construction costs. In an effort to improve the predictive capabilities of the HEC-18 scour model, this work uses
field-scale data and nonlinear regression to develop a family of equations optimized for various non-cohesive soil con-
ditions. Improving the predictive capabilities of well-accepted equations saves scarce project dollars without sacrificing
safety. To help improve acceptance of modified equations, this work strives to maintain the familiar form of the HEC-
18 equation. When compared to the HEC-18 local pier scour equation, this process reduced the mean square error of a
validation data set while maintaining over prediction.
Keywords: Scour; Piers; Bridges; Erosion; Estimation; Failures; Bridge Foundations
1. Introduction
Riverbed scour is a continuous process with natural and
anthropomorphic causes. Local accelerations in river ve-
locity increase the ability for a river to erode sediment.
Bridge support structures at river crossings create local
acceleration. Removing enough sediment from the river
bottom near bridge piers or abutments can cause the bri-
dge to become unstable, increasing the risk of failure.
According to the Federal Highway Administration, the
United States has approximately 600,000 bridges; about
80 percent require some sort of scour mitigation [1]. Due
to uncertainty in current scour prediction equations, ul-
timate scour depth is typically overestimated to ensure
safety. While the incremental cost for deeper foundations
may be reasonable for a small bridge with a single pier, it
can be exorbitant for larger bridges with several large-
diameter piers. Decreasing uncertainty associated with
scour-prediction models can lead to cheaper construction
costs without sacrificing safety.
Over the last few decades, statistical and physical mo-
deling dominated scour research with the goal of relating
hydrodynamics, geometry and sediment data to scour
depth. Empirically derived scour prediction equations,
largely based on experimental flume data using cohesion-
less sediment, currently dominate the state of the practice.
Although laboratory data are the most typical source of
data to define relationships affecting pier scour [2], it does
not capture the complexity of bridge scour due to diffi-
culties in scaling effects [3,4]. Scaled physical models
often use sediment of similar size as the field condition
they represent. Sediment is difficult to scale due to cohe-
sive effects and the presence of bed forms are determined
by particle size relative to the height of the viscous sub-
layer [4]. Uncertainty also stems from the fact that the
ranges of the various parameters over which the equations
are valid are typically unknown [5]. Considerable uncer-
tainty is also associated with these equations due to diffi-
culties in measuring complex velocity fields and bathy-
metry in the field.
The Federal Highway Administration issued Hydraulic
Engineering Circular (HEC) 18 [6], HEC-20 [7] and HEC-
23 [8] to provide guidance for local scour determinations.
HEC-18 provides specific guidance regarding the predic-
tion of local pier scour depth primarily through the em-
pirically derived Equation (1) [6]:
opyright © 2012 SciRes. IJG
 (1)
where ys is the scour depth, a is the pier width, K1 is the
correction factor for pier nose shape, K2 is the correction
factor for the angle of attack (the angle at which the flow
impinges upon the pier, K3 is the correction factor for bed
condition (plane bed, dune, ripple), K4 is the correction
factor for armoring by bed material size, y1 is the flow
depth directly upstream of the pier and Fr is the Froude
number. The remainder of this work refers to (a/y1) as the
normalized pier width (NPW). Equation (1) represents the
state of the practice and is included in one-dimensional
hydraulic models such as the Hydraulic Engineering
Center-River Analysis System [9]. Equation (1) is based
on work performed at Colorado State University and is
frequently referred to as the CSU equation.
Attempts to improve fit and reduce uncertainty in com-
monly used scour prediction equations appeared in the
1990s when researchers, such as [5] tried using field data
to determine valid ranges for typical parameters. Johnson
[5] also compared several competing models based on
computed bias in predictions. Johnson concluded some
equations were not fit for design purposes because they
often under predict scour. Conversely, equations used for
design purposes over predict with a large, positive bias
leading to an improved design from a safety perspective,
while unnecessarily increasing construction costs [5].
For this effort, the National Bridge Scour Database
(NBSD) provided field-scale data for an attempt to im-
prove the scour prediction capabilities of the HEC-18 lo-
cal pier scour equation. The NBSD, last updated in 2004
and maintained by the US Geologic Survey (USGS),
provides data from 20 sites in eight states [10]. For selec-
tion, a record must contain enough data to apply the cur-
rent version of the HEC-18 scour equation.
2. Methods
No single equation reliably predicts scour in all scenarios,
Ettema et al. [11]. The goal of the present effort is to
reduce mean square error of scour prediction through the
development and application of a family of scour predic-
tion equations. Each member of the family is similar in
form to HEC-18, but with various exponents applied to
the normalized pier width and Froude number. Currently,
these exponents are fixed in HEC-18 and apply for all
conditions. Maintaining the form of HEC-18 as the basis
for the non-linear regression ensures previously identified
parameters important in describing the scour process are
included. That is, no attempt is made to link important
scour parameters to a new functional form. Grouping si-
milar data and splitting the domain into multiple regions
provides a mechanism to develop multiple equations
termed a family of equations. Each member is tailored to
specific conditions.
Generally, the proposed model will require develop-
ment of several pairs of exponents, each pair developed
for a specific set of conditions. Collectively, the equations
generated from each exponent pair, apply to the same
broad range of conditions as the current HEC-18 equation.
Specifically, this effort will develop two pairs of expo-
nents (Case 1 and Case 2) applicable to live-bed scour
where the median particle size is in the sand fraction,
Equation (2). The value of the normalized pier width, as
defined by the geometry and flow conditions at the study
site, delineates the choice of exponent pairs for Case 1 and
Case 2. Case 1 is defined as live-bed scour, median par-
ticle size in the sand fraction and a normalized pier width
less than 0.3. Case 2 is defined the same as Case 1 but the
normalized pier width ranges from 0.3 to 1.25. Figure 1
illustrates the decision process used to choose exponents
for the current work as well as exponents for future deve-
 (2)
The parameters in Equation (2) are defined the same as
in Equation (1) where K is the collection of K1 through K4
and b1 and b2 are regression coefficients to be determined.
2.1. Data Description
The National Bridge Scour Database contains 148 re-
cords meeting all of the conditions described above
(complete for HEC-18 application, cohesionless, live-bed
scour and D50 < 2 mm). These records represent 20 uni-
que sites from eight states (Alaska, Colorado, Georgia,
Indiana, Louisiana, Missouri, Mississippi and Ohio). Due
to limited representation in the database, eleven records
with a normalized pier width greater than or equal to 1.25
were removed. Table 1 provides descriptive statistics
Figure 1. Flow chart depicting currently derived equations
and conditions where equations still need to be derived.
Copyright © 2012 SciRes. IJG
Table 1. Combined descriptive statistics for Case 1 and Case
2 data.
Variable Mean MedianStandard
Deviation Min Max
pier width 0.35 0.29 0.22 0.0431.18
Froude 0.25 0.24 0.12 0.04 0.55
Median grain
size (mm) 0.81 0.90 0.45 0.15 1.82
from the remaining queried data.
This analysis requires two datasets from the queried
records: one set to derive and validate exponents for Case
1 (described above), and the other dataset to derive and
validate exponents for Case 2. The median normalized
pier width was determined and the values used to split
the 137 records into two datasets. From Table 1, the me-
dian normalized pier width is 0.29 and rounded to 0.3 for
this analysis. Currently, HEC-18 uses a special correction
factor for wide piers (i.e. normalized pier widths greater
than 1.25). This criterion provides a natural upper bound
for the normalized pier widths for Case 2. Analyses for
Case 1 and Case 2 were performed with 71 and 66 re-
cords, respectively. Data for each analysis is described in
Table 2.
Multiple visits to the same bridge or multiple piers
from a single bridge generate multiple records in the da-
tabase. The datasets used in Case 1 and Case 2 model de-
velopment were further parsed into derivation and vali-
dation datasets. However, records from a single site were
prevented from simultaneously contributing to both the
derivation and validation datasets. This prevented site-
specific processes from artificially increasing perform-
ance statistics on the validation dataset. For example, if a
specific location contributes five records to a dataset and
that site is chosen to contribute to the derivation dataset,
then all five records will be in the derivation dataset.
The process of splitting the data into derivation and
validation data was repeated four times. Each time, the
site, or combination of sites contributing records to the
validation dataset changed. Resampling continued until
each site contributed to both the derivation and validation
datasets. This technique ensured the equations developed
with this process did not rely on the records chosen to be
in the derivation and validation datasets.
2.2. Regression Types
The HEC-18 pier scour equation was re-derived with
nonlinear regression analysis which will both under-and
over predicts scour. Therefore, an adjustment factor is
applied to the best-fit equation to minimize the number
of under predictions. Two adjustment factors were con-
sidered in this study, a multiplicative adjustment as in the
current HEC-18 equation and an additive adjustment as
Table 2. Descriptive statistics for national bridge scour da-
tabase data.
National Bridge Scour Database—Froude
Mean Std. DevMin Max
Number of
Case 10.18 0.09 0.04 0.37 71
Case 20.32 0.10 0.16 0.55 66
National Bridge Scour Database—D50 (mm)
Case 10.74 0.33 0.16 1.82 71
Case 20.89 0.54 0.15 1.80 66
in the Froehlich Design Equation [12]. Equations (3a)
and (3b) provide the two forms of the adjusted equations
examined in this study.
 (3a)
 (3b)
The adjustment factors are computed by examining the
maximum under-prediction of scour from the deriving
data set. The multiplier required to increase the most
under-predicted value in the deriving data set to the ob-
served value was determined. Relative scour depth ratios
in the validation data set were predicted using the best-fit
equation and increased by the multiplicative adjustment.
Similarly, the additive adjustment was determined and
added to each best-fit prediction in the validation set.
This study applied four different regression techniques
to Equation (2) and investigated the ability of Equations
(3a) and (3b) (for both Case 1 and Case 2) to over predict
observed scour but by a lesser margin than the current
HEC-18 local pier scour equation. Regression techniques
Unrestricted, ordinary least-squares;
Unrestricted, weighted least-squares;
Restricted, ordinary least-squares;
Restricted, weighted least-squares.
The National Bridge Scour Database includes informa-
tion describing the accuracy for each scour measurement.
Accuracy ranged from ±0.08 meters to ±0.61 meters. The
weighted regression schemes considered the measure-
ment accuracy for each record to determine the regres-
sion parameters. For example, records with high accu-
racy had more influence in the fitting process than inac-
curately measured records. Each record in the ordinary
regression models were considered equally precise and
assigned equal weight. Restricted regression helped main-
tain intuitive ranges on regression parameters.
3. Results
This process results in a series of equations based on
Copyright © 2012 SciRes. IJG
various regression forms and types. The mean-square
error and number of over predictions were determined
for each case and for each resampling. Not all regression
types or forms resulted in over-predicted scour depths or
a reduced mean square error compared to the original
HEC-18 equation. However, the restricted, ordinary, least-
squares (OLS) regression applied to Equation (3b) con-
sistently over-predicted scour depth (at least as often as
the current HEC-18 model), but with a smaller mean
square error than the current HEC-18 implementation.
Table 3 summarizes the mean square error and number
of over predictions from each of the resampled validation
data sets. The remainder of this manuscript focuses on
comparing OLS Equation (3b) and the current HEC-18
local pier scour equation.
In every sampling for Case 1 and Case 2 records, the
modified version over predicted scour as often as the
current HEC-18 approach (Table 4). The mean square
error for each sample was also determined for both Case
1 and Case 2. Mean square errors for the original HEC-
18 ranged from 0.06 to 1.55 and from 0.01 to 0.38 for the
modified version and were generally higher for Case 2 in
both the original and modified models (Table 5).
In order to maximize the number of records used in
equation development, all available data was used to de-
rive a final pair of equations but only after a regression
type (restricted OLS) and model form (Equation (3b))
were determined through the four re-sampled trials. The
first case with a/y1 < 0.3 is predicted with Equation (4a)
and Case 2 with 0.3 a/y1 < 1.25 predicted by Equation
(4b). The 95-percent confidence interval around the re-
gression parameters for each trial are shown in Table 6.
The exponents of the final equations fit within the bounds
of the exponents based on the four resampled cases.
Table 3. Average MSE and number of over predictions fr om
resampled validation data sets.
NPW < 0.30 0.30 NPW <1.25
Original Modified P-value Original ModifiedP-Value
MSE 0.23 0.03 0.0001 1.05 0.30 0.001
Prediction 70/71 70/71 65/66 65/66
Table 4. Number of over predictions for original and modi-
fied models.
trials 1 to 4
Case 1 Case 2 Case 1 Case 2 Case 1Case 2
Trial 1 17 19 14 19 17 18
Trial 2 19 15 18 15 19 15
Trial 3 17 13 17 10 17 14
Trial 4 17 18 10 16 17 18
Table 5. Mean square error for trials 1 to 4 from models
developed with restricted, or dinary least-squares regression
for both Case one and Case two.
Trials 1 to 4
Case 1Case 2Case 1 Case 2 Case 1Case 2
Trial 1 0.46 1.55 0.18 7.26 0.02 0.14
Trial 2 0.15 1.09 0.09 0.50 0.04 0.39
Trial 3 0.25 0.96 0.36 1.04 0.03 0.39
Trial 4 0.06 0.14 0.03 0.04 0.01 0.29
Table 6. Modified exponents b1 and b2 with corresponding
95% confidence limits for each trial.
Case 1
Case 1
Case 1
Best Fit
Case 1
b1 b2 b1 b2 b1 b2
Trial 1 1.13 –0.551.86 0 2.59 0.55
Trial 2 1.27 –0.401.72 0 2.18 0.40
Trial 3 0.69 0.031.22 0.5 1.75 1.03
Trial 4 1.37 0.391.81 0 2.26 0.39
Case 2
Case 2
Case 2
Best Fit
Case 2
b1 b2 b1 b2 b1 b2
Trial 1 0.24 0.930.80 1.27 1.36 1.64
Trial 2 –0.080.810.50 1.26 1.08 1.71
Trial 3 –0.481.140 1.46 0.48 1.78
Trial 4 – 1.45 0.92 1.84
 (4a)
 (4b)
Figure 2 presents residuals from both the original
HEC-18 model and the final modified version. These re-
siduals show some records were better predicted with the
original HEC-18 model, but other records show the mo-
dified version improves the fit. Overall, the family of
equations better predicts the observed field-scale scour
measurements based on mean-square error shown in Ta-
ble 3.
The results of the Case 1 analysis shows that the modi-
fied HEC-18 equation with the multiplicative adjustment
under predicted relative scour depths in 11 instances,
compared to the original equation (Table 4). The modi-
fied equation with the additive adjustment over predicts
scour in the same number of instances as the original
HEC-18 model (Table 4). Similarly, the application of
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Copyright © 2012 SciRes. IJG
Figure 2. Residual comparisons for the final version of the modified HEC-18 family of equations Case 1 (top) and Case 2
an attempt to simplify a complex physical system into
predictive empirical equations, some empirical models
exclude variables. The Mississippi scour equation, which
is functionally dependent on pier width and flow depth
only, was ranked in the top six performers in a study by
Mueller and Wagner [2]. Equation (4a) is independent of
approach velocity. However, as seen with the Mississippi
equation, an empirical model need not contain every pa-
rameter associated with pier scour to perform well [11].
Ettema et al. [11] echo a similar sentiment and state ve-
locity is not a primary parameter to determine maximum
scour depth.
the modified model with the multiplicative adjustment
results in five more occurrences of under predicted scour
when compared to the original HEC-18 model (Table 4).
Whereas the modified model with the additive adjust-
ment under predicted scour once (in trial two) when com-
pared to the original model, it over predicted scour once
(in trial three) compared to the original model (Table 4).
This regression type and model form was chosen for use
in the final model due to a significant decrease in mean-
square error (Table 2) and the number of over predic-
tions (Table 4).
Another anomaly is the discontinuity that occurs in
scour prediction between use of Equations (9.4a) and
(9.4b) as is evidenced by the considerable change in re-
gression parameters and the additive adjustment. This
discontinuity is a product of the statistical formulation of
the equations, and engineering judgment is required for
cases near the transition point (NPW = 0.30). As addi-
tional field data becomes available, the regression proc-
esses may lead to a smoother and more continuous func-
4. Discussion and Conclusions
Pier scour is a complex phenomenon and is difficult to
predict. While many improvements have been made in
the field over the last 20 years, this complex behavior
prevents the use of a single design relationship or method
[11]. Like the current array of scour equations, this one is
not without its share of difficulties, some of which are
discussed below. The equations developed in this work
are far from comprehensive; they are merely part of a
larger framework of undeveloped equations. The Froehlich equation helps address two potential
concerns with the proposed equation: the additive ad-
justment and its reliance on field data. Use of the additive
adjustment term in the equation results in a “pseudo-
scour” even in the case of no flow (Fr = 0). While this is
not physically possible, it is not unique from other exist-
Physically, pier scour depends on various factors in-
cluding pier geometry, flow depth, approach velocity and
bed material characteristics [2]. Many empirical equa-
tions exist to predict scour, and compared to physical or
numerical models, offer expedience of bridge design. In
ing design equations. Specifically, the Froehlich Design
equation is both based on field data and uses an additive
adjustment. It is also considered among the top perfor-
mers in the Mueller and Wagner [2] study. Additionally,
laboratory-based equations are not without problems
(idealized conditions and scale effects). In fact, deficien-
cies in the leading equations stem from their reliance on
laboratory data [11].
This analysis shows that developing a family of equa-
tions in a similar format to the current HEC-18 equation
(Equation (1)) reduces the mean square error of predic-
tion and reduces the overall amount of over-prediction.
This study and others show the current HEC-18 equation
significantly over predicts scour in most cases, resulting
in increased construction costs. As shown in this study,
using field-scale data, partitioning the data set and defin-
ing regression parameters for specific conditions leads to
significant reductions in estimated scour depths while
maintaining scour over prediction.
HEC-18 was chosen for the starting point for this mo-
del development because it is the current scour model
approved by the FHWA and therefore widely used. Many
additions were made to the field of pier scour prediction
since the FHWA implemented HEC-18. The FHWA is in
the process of evaluating these new models. While the
CSU-based equation may not always be the recommended
pier scour equation in HEC-18, the authors feel the
framework developed in this study can be applied to
wide array of base equations and datasets.
5. Acknowledgements
The authors would like to acknowledge the Michigan
Department of Transportation for support of Project
#108493-Contract 2007-0436 and Project #85106-Con-
tract 2006-0413. The authors also recognize the com-
ments of all reviewers including the MDOT Bridge Scour
Technical Advisory Group and Dr. Peggy Johnson.
[1] H. Nassif, A. O. Ertekin and J. Davis, “Evaluation of
Bridge Scour Monitoring Methods, F,” United States
Department of Transportation, Federal Highway Admini-
stration, Trenton, 2002.
[2] D. Mueller and C. R. Wagner, “Field Observations and
Evaluations of Streambed Scour at Bridges,” United
States Department of Transportation, Federal Highway
Administration, Mclean, 2005.
[3] G. R. Hopkins and R. W. Vance, “Scour around Bridge
Piers,” Washington, 1980.
[4] R. Ettema, B. W. Melville and B. Barkdoll, “Scale Effect
in Pier-Scour Experiments,” Journal of Hydraulic Engi-
neering, Vol. 124, No. 6, 1998, pp. 639-642.
[5] P. Johnson, “Comparison of Pier-Scour Equations Using
Field Data,” Journal of Hydraulic Engineering, Vol. 121,
No. 8, 1995, pp. 626-629.
[6] E. V. Richardson and S. R. Davis, “Evaluating Scour at
Bridges,” 4th Edition, United States Department of Trans-
portation, Federal Highway Administration, Washington,
[7] P. F. Lagasse, J. D. Schall and E. V. Richardson, “Stream
Stability at Highway Structures HEC-20,” FHWA, 2001,
p. 260.
[8] P. F. Lagasse, et al., “Comprehensive Bridge Scour Eva-
luation Methodology,” 5th International Bridge Engi-
neering Conference, Bridges, Other Structures, and Hy-
draulics and Hydrology, Transportation Research Board
Natl Research Council, Washington, Vol. 1-2, 2000, pp.
[9] G. Brunner, “River Analysis System Hydraulic Reference
Manual,” D.o. Defense, Davis, 2008.
[10] M. Landers, D. Mueller and G. Martin, “Bridge-Scour
Data Managment System User’s Manual,” United States
Geologic Survey, Reston, 1996.
[11] R. Ettema, G. Constantinescu and B. Melville, “Evalua-
tion of Bridge Scour Research: Pier Scour Processes and
Predictions,” N.C.H.R. Program, 2011.
[12] D. Froehlich, “Analysis of Onsite Measurements of Scour
at Piers,” National Hydraulic Engineering Conference,
New York, 1988.
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