U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2011–5107

In cooperation with the Montana Department of Transportation

Investigation of Pier Scour in Coarse-Bed Streams in 

Montana, 2001 through 2007
Front cover. Upper left: Two Medicine River near Browning, Mont., July 2006. Coarse-bed pier scour.

Upper right: Clark Fork near Superior, Mont., April 2007. Survey-grade echo sounder and 

global positioning system (GPS) being used to conduct coarse-bed pier-scour measurements.

Center right: Boulder River northwest of Boulder, Mont., June 2006. Scour rod attached to pier.

Lower right: North Fork Blackfoot River near Ovando, Mont., October 2006.

Back cover. Landers Fork east of Lincoln, Mont., August 2011. Coarse-bed pier scour. 

All photographs by Stephen R. Holnbeck, U.S. Geological Survey.
Investigation of Pier Scour in Coarse-Bed 

Streams in Montana, 2001 through 2007

By Stephen R. Holnbeck

In cooperation with the Montana Department of Transportation

Scientific Investigations Report 2011–5107

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2011

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Suggested citation:

Holnbeck, S.R., 2011, Investigation of pier scour in coarse-bed streams in Montana, 2001 through 2007: 

U.S. Geological Survey Scientific Investigations Report 2011–5107, 68 p.

ISBN 978-1-4113-3214-0
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................2

Purpose and Scope .............................................................................................................................3

Background............................................................................................................................................3

Hydraulic Conditions and Real-Time Scour Measurements ................................................3

Development of Pier-Scour Equations for Coarse-Bed Streams.........................................6

 Recent Research of Pier-Scour Equations.............................................................................8

Methods of Data Collection ...............................................................................................................9

Site Selection ...............................................................................................................................9

Pier-Scour Depth .......................................................................................................................15

Bed-Material Characteristics..................................................................................................23

Ancillary Data.............................................................................................................................25

Pier Scour in Coarse-Bed Streams in Montana .....................................................................................27

Hydrologic and Hydraulic Conditions..............................................................................................27

Scour Data Collected .........................................................................................................................28

Comparisons with Other Pier-Scour Data ......................................................................................29

Comparisons of Measured and Computed Scour ........................................................................37

Effects of Bed-Material Characteristics on Pier Scour ..............................................................39

Effects of Surface Bed-Material Characteristics.................................................................39

Effects of Shallow-Subsurface Bed-Material Characteristics ..........................................51

Combined Effects of Surface and Shallow-Subsurface Bed-Material 

Characteristics .............................................................................................................55

Limitations of the Study .....................................................................................................................62

Summary and Conclusions.........................................................................................................................62

References Cited..........................................................................................................................................65

Figures

1. Map showing location of bridge sites with pier-scour data used for this study ...............4

2. Diagram showing flow field around a bridge pier and principal hydraulic 

features ..........................................................................................................................................5

3. Graph showing surveyed cross section at upstream side of bridge showing 

pier-scour holes and reference bed surface used to determine pier-scour 

depth .............................................................................................................................................22

4. Photographs showing examples of differences in surface and shallowsubsurface bed material at sites where pier-scour measurements were 

collected for this study ..............................................................................................................24

5. Graph showing pier-scour depth and streamflow measured at site 33, 

Yellowstone River south of Livingston, Montana (station 06192500), 

May and June, 2006....................................................................................................................28

6. Graph showing comparison of pier-scour measurements collected for this 

study and from the Bridge Scour Data Management System All dataset........................31
iv

7. Graph showing comparison of approach velocities for pier-scour measurements collected for this study with the Bridge Scour Data Management 

System Coarse and All datasets ..............................................................................................32

8. Graph showing comparison of relative scour velocities for pier-scour 

measurements collected for this study and from the Bridge Scour Data 

Management System Coarse dataset.....................................................................................33

9. Graph showing comparison of bed-material particle size (D50 and D95) for 

pier-scour measurements collected for this study with the Bridge Scour 

Data Management System Coarse dataset ...........................................................................34

10. Graph showing comparison of idealized K4 to Mueller velocity-intensity 

ratios for pier-scour measurements collected for this study, historical pierscour measurements from Montana, and pier-scour measurements in the 

Bridge Scour Data Management System...............................................................................36

11. Graph showing comparison of idealized K4 to relative bed-material size for 

pier-scour measurements collected for this study, historical pier-scour 

measurements from Montana, and pier-scour measurements in the 

Bridge Scour Data Management System Coarse dataset...................................................36

12. Graphs showing comparison of measured with computed pier-scour depths 

for data collected for this study using selected pier-scour equations..............................38

13. Graph showing comparison of residual pier-scour depth for six pier-scour 

equations evaluated for this study...........................................................................................39

14. Graphs showing relation between relative pier scour and relative velocity 

based on critical velocity for incipient motion of bed material upstream from 

the bridge; and relative velocity based on critical velocity to initiate scour of 

bed material at the pier..............................................................................................................47

15. Graph showing comparison of gradation coefficient for surface bed material 

for pier-scour measurements collected for this study with the Bridge Scour 

Data Management System Coarse dataset ...........................................................................48

16. Graph showing relation between relative pier scour and gradation coefficient 

for pier-scour measurements collected for this study, historical pier-scour 

data from Montana, and pier-scour measurements in the Bridge Scour Data 

Management System All dataset in ripple-forming (D50 < 0.6 millimeter) and 

nonripple-forming (D50≥ 0.6 millimeter) sediments..............................................................49

17. Graphs showing relation between relative pier scour and relative velocity 

based on critical velocity for incipient motion of bed material upstream from 

the bridge; and relative velocity based on critical velocity to initiate scour of 

bed material at the pier..............................................................................................................50

18. Graph showing comparison of surface and shallow-subsurface bed-material 

particle size for pier-scour measurements collected for this study ..................................52

19. Graph showing comparison of surface with shallow-subsurface bed-material 

gradation coefficients for pier-scour measurements collected for this study ................53

20. Graph showing relations between corresponding surface and shallowsubsurface bed-material particle size (D50 and D95) for pier-scour measurements collected for this study ..................................................................................................53

21. Graph showing comparison of ratios of surface to shallow-subsurface bedmaterial particle size based on selected percentiles (Di

) from the particlesize distribution curves for each site where pier-scour measurements were 

collected for this study ..............................................................................................................54
v

Tables

1. Bridge sites where pier-scour measurements were made for this study ........................10

2. Pier-scour data collected for this study .................................................................................16

3. Bed-material size classification used for this study.............................................................25

4. Variation in surface bed-material particle size for multiple pebble counts made 

at bridge sites where pier-scour measurements were made for this study ....................26

5. Summary statistics for pier-scour measurement data collected for this study...............30

6. Comparison of the relative bed-material size associated with pier-scour 

measurements made in this study with the Bridge Scour Data Management 

System All dataset......................................................................................................................31

7. Comparison of bed-material size classification associated with pier-scour 

measurements made for this study and measurements from the Bridge 

Scour Data Measurement System Coarse and All datasets...............................................32

8. Selected historical pier-scour measurements (1996–97) used in this study ....................35

9. Particle-size distribution and gradation coefficient for surface and shallowsubsurface bed-material data collected for this study ........................................................40

10. Percentage of pier-scour measurements with a gradation coefficient greater 

than 2.5 .........................................................................................................................................51

11. Statistical summary of measured pier scour and computed pier scour based 

on surface and shallow-subsurface bed-material data.......................................................56

12. Groups of pier-scour measurements based on classification of particle sizes 

of the surface and shallow-subsurface bed material ..........................................................57

22. Graph showing comparison of residual pier-scour depth based on pier-scour 

measurements and bed-material data collected for this study with pier-scour 

depth computed by the HEC-18-K4Mu equation....................................................................57

23. Graph showing relation between relative pier scour and relative velocity for 

different groups of combinations of surface and shallow-subsurface bedmaterial size classifications for the pier-scour measurements collected for 

this study ......................................................................................................................................58

24. Graph showing comparison of relative pier scour to gradation coefficient for 

surface and shallow-subsurface bed-material data collected for this study ..................59

25. Graph showing relation between relative pier scour and relative velocity 

based on surface and shallow-subsurface bed-material gradation 

differences for pier-scour measurements collected for this study. ..................................60

26. Graph showing relation between relative pier scour and relative velocity 

based on surface and shallow-subsurface bed-material size and gradation 

differences for pier-scour measurements collected for this study....................................61
vi

Conversion Factors, Datum, Abbreviations, Symbols, and 

Variables

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

foot per second (ft/s) 0.3048 meter per second (m/s)

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

foot per second squared (ft/s2) 0.3048 meter per second squared (m/s2)

Hydraulic gradient

foot per foot (ft/ft) 1.0 meter per meter (m/m)

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F–32)/1.8

Vertical coordinate information for study data corresponds to the North American Vertical 

Datum of 1988 (NAVD 88).

Horizontal coordinate information for study data corresponds to the North American Datum 

of 1983 (NAD 83) except for figure 1, which corresponds to the North American Datum of 1927 

(NAD 27). 

Elevation, as used in this report, refers to distance above the vertical datum or local datum.

Acronyms used in this report:

ADCP acoustic Doppler current profiler

BSDMS Bridge Scour Data Management System, U.S. Geological Survey bridge scour 

database

BSDMS All All noncohesive BSDMS pier-scour data meeting initial screening criteria for 

the Montana study

BSDMS Coarse All BSDMS pier-scour data meeting further coarseness criteria of 

D50 ≥ 2 millimeters and D95 ≥ 20 millimeters 

CSU Colorado State University

FISP Federal Interagency Sedimentation Project

FHWA Federal Highway Administration
vii

GPS global positioning system

HEC-18 Hydraulic Engineering Circular No. 18

IR interquartile range

MDT Montana Department of Transportation

USGS U.S. Geological Survey

Symbols and variables used in this report:

= equal to

≈ approximately equal to

< less than

> greater than

≤ less than or equal to

≥ greater than or equal to

b effective width of the pier measured perpendicular to the approach flow, in feet

bn nominal pier width measured perpendicular to the pier length, in feet

b/D50 relative bed-material size, dimensionless

D bed-material particle size, in millimeters or feet

DCF particle size for the coarse size fraction of the bed material, in millimeters or feet

Di bed-material particle size, D, where i percent of particles are finer by weight, in 

millimeters or feet

Di,s surface bed-material particle size, D, where i percent of particles are finer by 

weight, in millimeters or feet

Di,ss shallow-subsurface bed-material particle size, D, where i percent of particles

are finer by weight, in millimeters or feet

Fr Froude number directly upstream from the pier, , dimensionless

g acceleration due to gravity, in feet per second squared

Iv bed-material variability index that expresses the difference between the 

surface- and shallow-subsurface bed-material, as measured by the variables 

D50, D95, and σg ; dimensionless

Idealized K4 relative error between pier-scour depth from field measurements and pier-scour 

depth computed by an equation, dimensionless

K1 correction factor for pier-nose shape, dimensionless

K2 correction factor for flow angle of attack on the pier, dimensionless

K3 correction factor for bed-form condition, dimensionless

K4 correction factor for the coarse bed-material armoring effect on pier-scour 

depth, dimensionless

Kc a constant equal to 12.0, in inch-pound units
viii

KI a constant equal to 4.80, in inch-pound units

Ki correction factor for the initiation of motion used by Molinas (2003), dimensionless

Ku a constant equal to 1.81, in inch-pound units

K′

U a constant equal to 1.21, in inch-pound units

n number of data values

L length of the pier measured perpendicular to the nominal pier width, in feet

p p-value is the probability of obtaining a result that is more extreme than a given 

value, where in this study p less than 0.05 indicates statistical significance 

(significance level) of the regression relation

r2 coefficient of determination, has a value that ranges from 0 to 1 and the strength of 

the relation increases as the value of r 2 approaches 1, dimensionless

Ri bed-material size ratio of the corresponding surface and shallow-subsurface bedmaterial particle size Di

, determined at percentile i, dimensionless

Rg relative gradation or the ratio of shallow-subsurface-to-surface bed-material 

gradation coefficient, dimensionless

S water-surface slope, dimensionless

V velocity in the experimental flume or stream channel, in feet per second

Vci critical velocity for incipient motion of bed material upstream from the bridge for 

the Di particle size, where i indicates the percentage of material finer by weight, in 

feet per second

V′

ci critical velocity to initiate scour of bed material at the pier for the Di

 particle size, 

where i indicates the percentage of material finer by weight, in feet per second

VcCF critical velocity for incipient motion of the DCF particle size, in feet per second

Vc50,s critical velocity for incipient motion of surface bed material upstream from the 

bridge for the D50 particle size, where 50 indicates the percentage of material finer 

by weight, in feet per second

Vi critical velocity to initiate scour of bed material at the pier, in feet per second

Vim critical velocity to initiate scour of bed material at the pier and equal to 0.4Vc35, in 

feet per second
ix

Vo approach velocity directly upstream from the pier, in feet per second

Vo / Vc50 relative velocity associated with incipient motion of bed material upstream from 

the bridge, dimensionless

Vo / V′

c50 relative velocity associated with initiation of scour of bed material at a pier, 

dimensionless

VR velocity-intensity ratio, dimensionless

VRM Mueller velocity-intensity ratio, dimensionless

yo approach depth of flow just upstream from the pier, in feet

yo /b relative flow depth, dimensionless

ys pier-scour depth computed by an equation, in feet

y′

s pier-scour depth from field measurements, in feet

y′

s

/b relative pier scour, dimensionless

y′

s

/ys idealized K4 or relative error in computed scour (ys) based on measured pier scour 

(y ′

s

), dimensionless

α flow angle of attack on the pier, in degrees

ψ velocity-intensity ratio used by Molinas (2003), dimensionless

σg gradation coefficient of the bed material, , dimensionless

σg,s gradation coefficient of the surface bed material, , dimensionless

σg,ss gradation coefficient of the shallow-subsurface bed material, , 

dimensionless

σg,c composite gradation coefficient or geometric mean of the corresponding surface 

gradation coefficient (σg,s) and shallow subsurface gradation coefficient (σg,ss), 

dimensionless

θ Shields parameter, dimensionless

Investigation of Pier Scour in Coarse-Bed Streams in 

Montana, 2001 through 2007

By Stephen R. Holnbeck

Abstract

Determination of pier-scour potential is an important 

consideration in the hydraulic analysis and design of highway 

bridges that cross streams, rivers, and other waterways in the 

United States. A primary goal of ongoing research in the field 

of bridge scour is to improve scour-prediction equations so 

that pier-scour depth is neither underpredicted nor excessively 

overpredicted. Scour depth for piers in noncohesive, nonuniform streambeds with a mixture of sand, gravel, cobbles, 

and boulders (coarse-bed streams) generally is less than the 

scour depth in finer-grained (mostly sand) streambeds under 

similar hydraulic conditions. The difference in scour depth 

is attributed to formation of an armor layer. Pier-scour data 

collected by the U.S. Geological Survey were used to develop 

a bed-material correction factor called K4. The equation 

recommended by the Federal Highway Administration for 

computing pier scour is a version of the HEC-18 pier-scour 

equation that includes K4, which is referred to in this report 

as the HEC-18-K4Mu equation. The Montana Department 

of Transportation was interested in pier-scour prediction in 

coarse-bed streams because coarse-bed streams are common 

in Montana. Consequently, the U.S. Geological Survey and the 

Montana Department of Transportation began a cooperative 

study in 2001 to investigate pier scour in coarse-bed streams 

in Montana. 

This report describes results of a study of pier scour in 

coarse-bed streams at 59 bridge sites during 2001–07 in the 

mountain and foothill regions of Montana. Drainage areas for 

the streams at bridge sites where measurements were collected 

ranged from about 3 square miles (mi2) to almost 20,000 mi2. 

Data collected and analyzed for this study include 103 coarsebed pier-scour measurements. The report also describes how 

coarse-bed pier-scour measurements were collected, shows 

the extent that the coarse portion of the national pier-scour 

database was expanded, discusses how these new data were 

used to evaluate the relative accuracy of various equations for 

predicting scour in coarse-bed streams, and demonstrates how 

differences in size and gradation between surface bed material and bed material underlying the surface layer (shallowsubsurface bed material) might relate to pier scour.

Ninety-six out of 103 pier-scour measurements were 

made under clear-water scour conditions, when the streambed 

upstream from the bridge is stable and there is no substantial 

incoming sediment supply to the bridge opening. Of the measurements made, 50 percent had an approach velocity (Vo ) that 

equaled or exceeded 70 percent of the critical velocity (Vc50 ) 

for incipient motion of bed material, which might indicate that 

scour was measured very near the threshold between clearwater and live-bed scour (Vo /Vc50 equal to 1.00) where maximum scour was shown in laboratory studies. 

Pier-scour data collected for this study were compared to 

selected pier-scour data from the Bridge Scour Data Management System (BSDMS), a database of scour measurements 

made nationwide, to show the effect of bed-material size 

and gradation on scour depth. The relation between relative 

pier scour (y′

s /.

b), or the ratio of measured pier scour (y′

s

) to 

effective pier width (b), and relative velocity (Vo /Vc50) defined 

by an envelope curve for data collected for this study and the 

BSDMS data were compared to relations observed in laboratory data. The envelope curve for data used for this study 

displays an earlier peak, with a decrease in relative pier scour 

followed by an asymptotic rise. Interestingly, the first peak 

in the curve occurs at a lower relative velocity (Vo /Vc50 about 

0.75) than the clear-water and live-bed threshold (Vo /Vc50

equal to 1.00). Also, the magnitude of this peak (maximum of 

1.25) is much less than the published maximum value of 2.4 

obtained from laboratory data. Furthermore, equilibrium livebed scour is about 18 percent shallower than the maximum 

pier scour under clear-water conditions, a larger reduction 

than the 10 percent indicated in published research. Armoring 

associated with nonuniform coarse-bed material and unsteadyflow conditions in the field probably accounts for differences 

between envelope curves developed for this study and those 

from previous studies conducted in the laboratory using finegrained material and steady-flow conditions. 

Scour depth was computed for the measurements collected for this study using the HEC-18 equation without the K4

correction factor and five pier-scour equations that use K4 for 

the armoring effect of coarse bed material. The HEC-18-K4Mu 

equation was the best equation for predicting pier-scour 

depth in coarse-bed streams because the equation generally 

predicted pier scour in closer agreement to measured scour 
2 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

than the other equations used for computing pier-scour depth. 

Furthermore, the magnitude of the residuals of underpredicted and overpredicted scour depth was the lowest using the 

HEC-18-K4Mu equation.

Gradation coefficients (σg) for the surface bed-material 

data collected for this study ranged from 1.48 to 4.14, with a 

median of 2.01. For data used in this study, maximum relative 

pier scour was greater when σg was less than or equal to about 

2.5. Pier-scour measurements collected for this study and 

BSDMS confirm a general lack of data for coarse-bed sites 

with σg greater than about 2.5. 

Paired samples of surface and shallow-subsurface bed 

material collected for this study were analyzed and the D50 and 

D95 particle sizes were compared. Median D50 particle sizes 

for 103 surface and shallow-subsurface bed-material samples 

were about 49 millimeters (mm) and 32 mm, respectively, and 

median D95 particle sizes for 103 surface and shallow-subsurface bed material samples were 122 mm and 86 mm, respectively. The median gradation coefficients for surface and shallow-subsurface bed material associated with measurements 

made for this study were 2.01 and 4.14, respectively. The 

frequency and magnitude of the residual pier-scour depth associated with underpredicted and overpredicted scour was about 

the same regardless of whether surface or shallow-subsurface 

particle-size data were used with the HEC-18-K4Mu equation. 

The combined effect that surface and shallow-subsurface 

bed-material characteristics might have on measured pier 

scour also was examined. The combined influence of relative 

differences in surface and shallow-subsurface bed-material 

size and gradation on pier scour was investigated by defining 

a bed-material variability index (Iv) based on ratios of particle 

sizes and gradation coefficients between layers for each of 

the 103 pier-scour measurements. Selected percentiles were 

determined and the relative pier scour and relative velocity for 

each measurement in the percentile were plotted. Envelope 

curves bounding the measurements in each percentile were 

drawn. Maximum relative pier scour associated with each 

envelope curve was concentrated at about the same relative 

velocity. Based on these curves, as differences in particle size 

and gradation between the surface and shallow-subsurface 

bed-material layers increase (higher values of Iv), pier scour 

decreases.

 Runoff conditions during the study mostly were limited to bankfull discharge approximating the 1.5- to 2-year 

recurrence-interval flood, which have a 67- and 50-percent 

chance, respectively, of happening in any year. Lack of higher 

flows having greater velocity intensities may indicate that 

scour depths were limited by hydrologic conditions. Pier-scour 

depths measured in this study also may have been limited by 

streambed armoring and unsteady flow conditions. Conclusions presented for this study are generally limited to the range 

of hydraulic conditions and bed-material characteristics demonstrated in the data collected for this study and the BSDMS 

data.

Introduction

Determination of pier-scour potential is an important 

consideration in the hydraulic analysis and design of highway 

bridges that cross streams, rivers, and other waterways in the 

United States. Existing bridges continue to undergo scour 

assessments as part of a national program, and pier scour is 

among the key scour components investigated. Bridge rehabilitation and the design of new bridges require that pier footings be located below the scour zone of the streambed during 

large floods. Pier-scour depth is commonly computed using 

empirical equations, with many different equations having 

been developed over the years. Such equations were traditionally based on studies conducted in hydraulic laboratories using 

flumes and scaled pier models. Scale effects associated with 

model studies restricted the maximum bed-material particle 

size to mostly noncohesive sand. A primary goal of ongoing 

bridge scour research is to improve laboratory-based scourprediction equations using field-scale data so that pier-scour 

depth is neither underpredicted nor excessively overpredicted. 

Underpredicted pier scour can result in pier footings not being 

located deep enough in the streambed to resist scour. Scour 

during large floods could, therefore, undermine pier footings 

and result in damage or instability to the bridge and possibly 

catastrophic failure. Overpredicted pier scour might indicate 

that more bridges are scour critical, or prone to failure due to 

scour, than is actually the case and thus may lead to expensive 

over-design or unnecessary retrofitting of pier foundations 

(Chase and Holnbeck, 2004). Research efforts, therefore, 

continue to evaluate and improve the accuracy of pier-scour 

equations. 

Since the early 1990s, research has expanded to include 

field measurements of pier scour during high-flow conditions. 

These scour measurements enable pier-scour equations to be 

evaluated beyond the theoretical or limited laboratory conditions frequently used in equation development. Mueller (1996) 

and Mueller and Jones (1999) studied bed-material size and 

gradation effects on pier-scour depth in noncohesive streambeds. Their research indicated that scour depth for piers in 

noncohesive, nonuniform streambeds with a mixture of sand, 

gravel, cobbles, and boulders (coarse-bed streams) generally is 

less than the scour depth in finer-grained (mostly sand) streambeds under similar hydraulic conditions. The difference in 

scour depth is attributed to formation of an armor layer (Shen 

and others, 1966). This finding, supported by field measurements collected by the U.S. Geological Survey (USGS), has 

led to modifications in the scour equations recommended by 

the Federal Highway Administration (FHWA). Pier-scour data 

collected by the USGS were used to develop a bed-material 

correction factor called K4. Collection of additional coarsebed pier-scour data was recommended to further validate the 

method for computing K4 and to provide additional data for 

continued evaluation of pier-scour equations (Mueller and 

Jones, 1999). As recently as 2005, Mueller and Wagner (2005) 

noted that relations and equations developed from laboratory research had not been adequately verified with field data. 
Introduction 3

Much advancement has been made since the mid–1990s in 

the research and development of equations that account for 

coarse-bed pier scour, but additional data were needed to 

evaluate whether these equations accurately predicted scour in 

coarse-bed streams. 

The Montana Department of Transportation (MDT) 

was interested in pier-scour prediction in coarse-bed streams 

because coarse-bed streams are common in Montana. Furthermore, the USGS has collected bridge-scour data and conducted bridge-scour investigations in Montana since the early 

1990s in cooperation with MDT. Consequently, the USGS and 

MDT began a cooperative study in 2001 to investigate pier 

scour in coarse-bed streams in Montana. 

Purpose and Scope 

This report presents the results of a study of pier scour in 

coarse-bed streams at 59 bridge sites during 2001–07 in the 

mountain and foothill regions of three major river basins in 

Montana: the Missouri, Yellowstone, and Clark Fork of the 

Columbia (fig. 1). Additionally, the report:

1. Describes how coarse-bed pier-scour measurements 

were collected.

2. Shows the extent to which the coarse portion of the 

national pier-scour database used for conducting 

scour research was expanded. 

3. Discusses how these new data were used to evaluate the relative accuracy of various equations for 

predicting scour in coarse-bed streams.

4. Demonstrates how differences in size and gradation between surface bed material and bed material 

underlying the surface layer, referred to in this report 

as shallow-subsurface bed material, might relate to 

pier scour. 

Drainage areas for the streams at bridge sites where 

measurements were made ranged from about 3 mi2

 to almost 

20,000 mi2

. Data collected and analyzed for this study include 

103 coarse-bed pier-scour measurements and represent a substantial sample of coarse-bed streams in Montana. When these 

data are added to the USGS Bridge Scour Data Management 

System (BSDMS) database at the conclusion of the study, the 

number of coarse-bed pier-scour measurements in the current 

(2011) database will increase substantially. 

Field methods (Jarrett and Boyle, 1986; Landers and 

Mueller, 1996; and Mueller and Landers, 2000), hereinafter 

referred to as limited-detail field procedures, and equipment 

ranging from simple scour rods to complex hydroacoustic 

instrumentation were used to obtain pier-scour measurements. Pier-scour measurements were made during spring 

snowmelt runoff when streamflow and hydraulic conditions 

generally approximated bankfull discharge with recurrence 

intervals of 1.5 to 2 years, which have a 67- and 50-percent 

chance, respectively, of happening in any year. Pier-scour 

measurements also were made during low-flow conditions. 

Because the low-flow measurements are considered ancillary, 

only the high-flow measurements are presented and discussed 

in the report. Selected Montana pier-scour measurements for 

peak streamflows (5– to about 100–year recurrence interval) 

obtained prior to this study also were used as an independent 

check on envelope curves developed similar to those from 

previous studies (Ettema, 1980; Melville, 1984; Chiew, 1984; 

and Mueller and Wagner, 2005). The envelope curves have 

either linear or curvilinear shape and define an upper boundary 

to the data. 

In this report, gradation and gradation coefficient (σg

 ) are 

terms frequently used to describe and measure the variation in 

particle-size distribution of the bed material. Bed material is 

considered to be poorly graded or has a smaller gradation (also 

referred to as well sorted) when the material is more uniform, 

or has a narrow range of grain sizes. Bed material is considered to be well graded or has a larger gradation (also referred 

to as poorly sorted) when the material is nonuniform, or has 

a wider range of grain sizes. The magnitude of the gradation 

coefficient increases as gradation increases or as the range in 

particle sizes increases. 

Because bed armoring relates to surface and subsurface 

particle-size differences, surface and shallow-subsurface bedmaterial data were collected at each site. Particle-size gradation was determined from pebble counts and sieve analyses. 

Surface and shallow-subsurface bed material sampled during 

this study were used to evaluate the sensitivity of computed 

scour to size and gradation differences between the two layers. 

Data for each layer also were used to assess the effects of bedmaterial characteristics on measured pier scour.

Mathematical symbols (=, ≈, <, >, ≤, and ≥) used in this 

report are defined near the front of the report under the section 

“Symbols and variables used in this report,” and are either 

spelled out or appear as the symbol thereafter in the body of 

the report. Variable symbols used in this report (for example, 

yo

, Di, and Vo) are defined near the front of the report under 

the section “Symbols and variables used in this report,” and at 

first occurrence and occasionally thereafter in the body of the 

report.

Background

Hydraulic Conditions and Real-Time Scour 

Measurements

The relative stability of a streambed is related to hydraulic conditions in the stream and the particle sizes of the 

streambed material. Scour at a bridge pier depends on the 

stability of the streambed near the bridge, the extent of bedmaterial transport from upstream, and forces acting on the 

streambed near the pier that result from the interaction of the 

stream and the pier. Pier scour can occur when the streambed 

upstream from the bridge is stable and there is no substantial 

incoming sediment supply to the bridge opening (clear-water 
Figure 1. Location of bridge sites with pier-scour data used for this study.

Boxelder

Creek

Musselshell

River

Bighorn

 Lake Clarks

Fork

108°

Billings

Dodson

Edgar

Absarokee

Shelby

Cr

Fork

Two

South

Fork

Yaak River

Kootenai

River

Flathead

 Lake

Flathead

River

Middle

Cut Bank Creek

Medicine River Marias

River

Birch

Milk

River

Lodge

Creek

Battle

Creek

River

Judith

Sun River

Smith

River

River

Teton

River Swan

Flathead

R

Bitterroot

River

Blackfoot

River

Creek

Rock

Boulder

River

Gallatin

River

Madison

River

Ruby

River

River

Jefferson

R.

Beaverhead

Big

Hole River

N.

Frk.

0 20 40 60 MILES

0 20 40 60 KILOMETERS

111° 110° 109° 113° 112° 114° 115° 116° 49°

48°

47°

46°

45°

Peoples

Holter

 Lake

Hauser Lake

Canyon

 Ferry

 Lake

Lake

Koocanusa

Kalispell

Missoula

Butte

Helena

Great

Falls

Havre

Bozeman

Dillon

Lewistown

Milk River

M Fork

British

Alberta CANADA

UNITED STATES

Montana

Wyoming

Idaho

Base modified from U.S. Geological Survey 

State base map, 1968

North American Datum of 1927 (NAD 27)

Park

Glacier

CONTINENTAL DIVIDE

DIVIDE

CONTINENTAL

National

Hudson Bay Basin

Dearborn

Big

Timber

Stillwater River

Livingston

Shields

River

Three

Forks

Twin

Bridges

Garrison

Little

Blackfoot

River

Clinton

Bonner

Perma St.

Regis

Troy Libby

Fisher

River

Bigfork

West

 Glacier

Whitefish

River

Columbia

Hungry

Horse

Reservoir

Saskatchewan

Creek

Ulm

Basin

Belt

Augusta

Creek

Belt

Creek

River

Wisconsin

Boulder

R. Boulder

S.

R. S. Wil

Cr.

low

Mill

St. Regis R.

Yellowstone

National

Park

Missouri

Yellowstone

River

Fork

Clark

River

Clark

Fork

9

8

7

6

5

4

3

2

1

62

61 60

59

48

52

53

43

42

45

40

37

58

55

41

51

44

57

47 46

50

49

36

35

34

33

32

29

27

26

25

24 23

22

21

20

1917 16

15 14

13

12

10

54 56

39

30

28

31

18

11

38

Cr.

Study area 

Montana

19

EXPLANATION

Missouri River Basin

Yellowstone River Basin

Columbia River Basin

Sampling site and site number—Pier-scour measurements collected for this study (table 1)

Sampling site and site number—Pier-scour measurements collected in 1996 and 1997 (table 8) 

 associated with peak streamflows of greater recurrence interval (up to about 100 years) 

 than pier-scour measurements collected for this study 

60

Hudson Bay Basin 

Little Missouri River Basin 

4 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Pier

Pier

Flowacceleration 

zone

Flowacceleration 

zone

Detailed

perspective

hole Pier-scour

Streambed

Plan view

Wake vortex

Surface roller

Horseshoe vortex Reference bed 

surface

Reference bed surface

Downflow

Pier

Flow-acceleration 

zone

Flow-acceleration 

zone

Figure 2. Flow field around a bridge pier and principal hydraulic features (modified from 

Melville, 1995).

Introduction 5

scour) and when the streambed is unstable and bed material is 

actively being transported by the stream into the bridge opening from upstream (live-bed scour). Documenting the stream 

stability and sediment-transport conditions associated with 

each pier-scour measurement is necessary to improve understanding of the processes contributing to pier scour. In order 

to develop equations relating scour and hydraulic conditions, 

pier scour measured in the field must be the result of hydraulic 

conditions at the time of measurement (real-time scour). 

The critical velocity for incipient motion of bed material 

(Vci ) is used to determine if clear-water or live-bed conditions 

predominate upstream from a bridge site (Richardson and 

Davis, 1995, p. 12–13), with observed velocity less than Vci

indicating clear-water scour and observed velocity greater than 

Vci indicating live-bed scour:

 , (1)

where

θ is the Shields parameter, dimensionless, based 

on particle size and hydraulic conditions;

Ku is a constant equal to 1.81 for inch-pound 

units, dimensionless;

yo is the approach depth of flow just upstream 

from the pier, in feet for this equation; and

Di is the bed-material particle size D, in feet for 

this equation, where i percent of particles 

are finer by weight. 

The equation by Gao and others (1993) is used to compute the critical velocity to initiate scour of bed material at the 

pier (V′

ci ) based on the critical velocity for incipient motion of 

bed material upstream from the bridge (Vci , equation 1):

 , (2)

where

bn is the nominal pier width measured 

perpendicular to the pier length, which 

takes on the units of Di in this equation 

(feet); 

and where all other terms are as previously defined. By this 

equation, the velocity to initiate scour of bed material at the 

pier is about one-half the velocity for incipient motion of bed 

material in the channel upstream from the bridge.

Equation 2 is used to determine if a valid real-time measurement of pier-scour was made in the field. Measurements 

(scour depth, flow velocity, and flow depth) made at remnant 

scour holes under base or moderate flows do not reflect the 

true hydraulic conditions accounting for the geometry of the 

scour hole. The measured approach velocity (Vo) is compared 

Equation 1 is commonly applied to 

determine the critical velocity for any 

particle size in the streambed. For 

incipient motion, the median or D50

particle size and corresponding θ typically are used to determine stream stability. Clear-water scour often occurs 

in streams with coarse bed material. 

 Pier scour is a local scour 

process resulting from vortex action 

and flow acceleration when flow is 

obstructed by the pier. The vortex 

systems produced are complex threedimensional flow fields that can act 

as continuous or intermittent energy 

systems that vary with hydraulic 

conditions. The two primary vortices 

involved are the horseshoe vortex and 

the wake vortex (fig. 2). The horseshoe 

vortex forms near the upstream base of 

the pier as flow impinges and moves 

down the pier face and along the sides 

of the pier. The wake vortex forms as 

shearing layers of flow detach from 

the sides of the pier and move downstream away from the pier, which 

results in bed material being carried or 

scoured away (Melville, 1995).
6 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

to the critical velocity (V′

c50) to initiate scour of the D50 particle 

size computed by equation 2, to determine if the hydraulic 

conditions at the time of measurement are sufficient to cause 

pier scour, according to the following criteria:

1. If Vo > V′

c50 , a valid real-time scour measurement was 

made because bed material can scour at the pier under the 

observed conditions, or

2. If Vo ≤ V′

c50 , a valid real-time scour measurement was not 

made because bed material cannot scour at the pier under 

the observed conditions. 

Under clear-water conditions, scour continues to develop 

until the scour hole becomes armored, and the resulting scour 

is referred to as maximum-local-pier scour. Under live-bed 

conditions, scour continues to develop until the rate of sediment transport out of the scour hole equals the rate of sediment transport into the scour hole, or the scour hole becomes 

armored and the stream velocity is insufficient to transport 

the size of bed material in contact with flow (Richardson and 

Davis, 2001). Scour depth under live-bed conditions tends to 

fluctuate around a mean-depth value over time, partly due to 

the transient nature of bedload transport in coarse-bed streams 

and passage of dunes in sand-bed streams. This mean-depth 

value is referred to as the equilibrium local live-bed pier scour. 

According to Richardson and Davis (1995, p. 16), the equilibrium local live-bed pier scour is about 10 percent shallower 

than the maximum-local-pier scour under clear-water conditions. Maximum-local-pier scour may be attained under clearwater conditions before the threshold for incipient motion is 

reached in the upstream channel and before live-bed scour 

conditions result in transport of bed material into the scour 

hole. Exceptionally high flood discharges that induce high 

bed-material transport upstream from the bridge, therefore, 

may not produce the deepest scour at a pier. Local pier scour 

is frequently used to describe scour caused by local hydraulic 

conditions at the pier, where the effects of any contraction 

scour, abutment scour, general scour, or long-term scour at the 

bridge are excluded. Hereinafter, local pier scour is referred to 

as pier scour.

Clear-water and live-bed distinctions have particular 

importance in laboratory research where either condition can 

be maintained throughout the hydraulic experiment. However, 

these distinctions become less clear under field conditions. 

During floods, piers in coarse-bed streams can be subjected 

to clear-water scour at lower flows, live-bed scour at higher 

flows, and clear-water scour again during the receding flood 

stages (Mueller, 1996). Furthermore, Mueller (1996) indicated 

that only a relatively short interval of live-bed scour is needed 

for pier scour to attain an equilibrium scour depth typical of 

live-bed conditions. Finally, Mueller and Wagner (2005, p. 19) 

noted that differences in armoring rates between the upstream 

channel and the pier-scour hole can complicate scour-hole 

development, overall scour depth, and the actual condition 

(clear-water or live-bed). They concluded that the traditional 

clear-water and live-bed designations are insufficient to fully 

describe the actual conditions for streams with nonuniform 

bed material. However, clear-water and live-bed scour designations continue to be important concepts describing the 

predominant scour condition at a site.

Development of Pier-Scour Equations for 

Coarse-Bed Streams

Pier-scour research in the mid–1970s resulted in a pierscour equation, which was recommended by FHWA (Richardson and others, 1975) and widely adopted for conducting pierscour analysis and design in the United States. This equation 

was largely based on laboratory flume studies with no verification using actual field data. Additional research based on field 

data in later years indicated that pier scour in nonuniform, 

coarse, bed material was not well predicted by the recommended equation. The equation has evolved to include correction factors for pier shape, flow alignment, bed-form conditions, and bed-material size and gradation. The latest (2011) 

version of the equation incorporates a bed-material correction 

factor (K4 ) to compute pier scour in coarse-bed streams. 

The pier-scour equation recommended by the FHWA for 

conducting detailed bridge-scour investigations in the mid–

1970s was the Colorado State University (CSU) equation 

(Richardson and others, 1975). A modified version of the CSU 

equation was later published by FHWA in Hydraulic Engineering Circular No. 18 (HEC-18) (Richardson and others, 1991) 

and is referred to in this report as the basic HEC-18 equation. 

The basic HEC-18 equation is:

 , (3)

where

ys is the computed pier-scour depth that assumes 

the units of yo (in feet for this study);

K1 is the correction factor for pier-nose shape, 

dimensionless;

K2 is the correction factor for flow angle of attack 

on the pier (α), dimensionless; and

Fr is the Froude number directly upstream from 

the pier, dimensionless;

where the Froude number is computed according to: 

 , (3a)

where

g is the acceleration due to gravity, in feet per 

second squared for this study; 

and where all other terms are as previously defined. Further 

research produced a second version of the basic HEC-18 equation recommended by FHWA (Richardson and others, 1993, 

p. 39) that included a correction factor (K3) to account for the 

bed-form condition (dunes, plane bed, and other variations): 
Introduction 7

 , (4)

where

K3 is the correction factor for bed-form 

condition, dimensionless; 

and where all other terms are as previously defined.

The value of K3 is equal to 1.1 for clear-water scour and planebed conditions—typical of coarse-bed streams analyzed in this 

study. Other values of K3 range from 1.1 to 1.2 for beds with 

medium dunes to 1.3 for beds with large dunes (Richardson 

and others, 1993, p. 39). These first two versions of the basic 

HEC-18 equation did not account for bed-material size and 

gradation effects related to armoring of the streambed. 

Most of the pier-scour equations that have been developed were based on laboratory research that used bed material 

of uniform sand or fine gravel. Consequently, these equations 

do not adequately describe the effect of bed-material size and 

gradation on scour depth under actual field conditions. McIntosh (1989) reported that more than 35 equations have been 

proposed to compute pier scour. Mueller (1996) found that of 

these 35 equations, only 8 included an explanatory variable 

(median particle size or D50) to account for bed-material size.

The effect of nonuniform bed material on scour depth 

was observed by Shen and others (1966), where the presence 

of larger particle sizes appeared to reduce scour depth due to 

armoring. Research by Nicollet and Ramette (1971) compared scour for three separate sand sizes and a sand mixture 

of equal parts of each size. They concluded that the maximum 

scour occurred at the critical velocity of each sand size used 

and that scour depth was less for experiments with the sand 

mixture. The bed-material gradation coefficient (σg) was used 

by Ettema (1976, 1980) to show that nonuniform bed material 

affected the rate of development and final depth of scour under 

clear-water conditions. These findings were extended to livebed scour conditions by research conducted by Chiew (1984) 

and Baker (1986). Furthermore, research by Abdou (1993) 

substantiated that an increase in σg resulted in a decrease in 

clear-water scour. In all of these studies, findings were based 

only on laboratory investigations with no validation with field 

data.

The need to account for bed-material size and gradation 

effects on pier scour led to research sponsored by FHWA, 

where laboratory data were used by J. Sterling Jones to 

develop a correction factor (K4) that was first presented by 

Richardson and Davis (1995) and described later by Mueller 

and Jones (1999). K4 accounts for armoring of the scour hole 

and generally reduces computed scour in coarse-bed materials 

compared to what would be computed if the factor were not 

applied. 

In response to the work of Jones, the FHWA recommended a new version of the HEC-18 equation for use for 

both clear-water and live-bed scour conditions that incorporated the K4

 correction factor (Richardson and Davis, 1995):

 , (5)

where all terms are as previously defined. Richardson and 

Davis (1995) defined K4 as: 

 , (6)

where 

VR is the velocity-intensity ratio, dimensionless, 

computed as:

 , (7)

where all other terms are as previously described.

The laboratory research by Molinas and Abdou (1999) 

and Molinas and Noshi (1999) proposed the following equations for computing K4 :

 , (8)

DCF is the particle size that represents the coarse fraction of the 

bed material, in feet, computed as:

 , (9)

Vim is the critical velocity that initiates scour 

of bed material at the pier, and equal to 

0.4Vc35 , in feet per second; 

VcCF is the critical velocity for incipient motion of 

the DCF particle size, in feet per second; 

and where all other terms are as previously defined.

Mueller (1996) used field data collected by the USGS 

throughout the United States to develop a K4 factor. Partial 

residual analysis and multiple linear regression were used to 

identify the important variables affecting pier-scour depth. 

The investigation showed that the D95 particle size was the 

dominant bed-material variable affecting pier scour under both 

clear-water and live-bed scour conditions. The D50 particle 

size also was considered important based on Gao and others 

(1993), where the velocity-intensity ratio based on only the 

D50 particle size was shown to have an influence on pier-scour 

depth. Critical velocities associated with these two important 

bed-material variables resulted in the Mueller velocity-intensity ratio (VRM ,dimensionless):

 , (10)

and where all other terms are as previously defined.
8 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

The following equation incorporating the variables that define 

VRM was then proposed by Mueller (1996) for computing K4: 

 , (11)

Equations 10 and 11 apply for the condition when:

. (12)

In a review of equations describing pier scour, Mueller 

and Jones (1999) concluded that equations 10–12 proposed by 

Mueller (1996) for quantifying K4 provided the best explanation of the differences between observed and computed 

pier-scour depth. However, the authors recognized the need 

that additional field data be obtained and further research be 

conducted to improve the accuracy of pier-scour equations.

Chase and Holnbeck (2004) compared 42 pier-scour 

measurements made by the USGS in 5 States at 11 bridge sites 

with scour depth computed by 5 of the more commonly cited 

equations. The version of the HEC-18 equation that includes

K4 as proposed by Mueller (equations 5 and 11) was statistically determined to be the best equation for predicting pier 

scour in coarse-bed materials for that data set.

Mueller (1996, p. 115) found that the HEC-18 equation 

(equation 5) underestimated very few instances of observed 

scour but substantially overestimated observed scour depths 

because of the restrictions placed on K4 . When first published 

as part of the HEC-18 equation (Richardson and Davis, 1995, 

p. 36), K4 could only be applied when the D50 bed-material 

size was greater than or equal to 60 millimeters (mm). Furthermore, the minimum value of K4

 that could be used in any 

instance was 0.70. 

The current (2011) HEC-18 equation with K4 as recommended by the FHWA still has the form of equation 5, but the 

Mueller K4 is computed by equation 11 for bed-material size 

conforming to current (2011) HEC-18 criteria (Richardson and 

Davis, 2001, p. 6.6). The criteria for applying K4 require that 

the D50 particle size be greater than or equal to 2 mm and the 

D95 particle size be greater than or equal to 20 mm. The minimum value of K4

 under current (2011) criteria is 0.40. 

 Recent Research of Pier-Scour Equations

Pier-scour research has continued and new equations 

have been proposed that might lead to more reliable scour 

prediction than currently provided by conventional FHWA 

and HEC-18 equations. Recent (2003–05) pier-scour research 

in nonuniform sediment mixtures includes results reported 

by Molinas (2003) and Mueller and Wagner (2005). Molinas 

(2003) proposed two new methods that include equations 

similar in principle to those currently (2011) recommended 

by FHWA (equations 5 and 11). The first method proposed by 

Molinas (2003) includes the pier-scour equation:

, (13)

where

K’U is a constant equal to 1.21, in inch-pound 

units;

ψ is the velocity-intensity ratio, dimensionless

 , (14)

where 0 ≤ ψ ≤ 1 (this is the same ratio contained in equation 8); and where all other terms are as previously defined. 

Also, velocity relations associated with equation 14 are:

 , (15)

which is essentially the same as Vci in equation 1, and

 , (16)

which is the same as Vim in equation 8 (where Vim = 0.4Vc35)

where 

Kc is a constant equal to 12.0, in inch-pound 

units;

KI is a constant equal to 4.8, in inch-pound units.

K4 is then given by:

 , (17)

for 0.0 ≤ K4 ≤ 1.0, and 0.0 ≤ ψ ≤ 1.0; and where all other terms 

are as previously defined. 

The second method proposed by Molinas (2003) incorporates K4 computed by equation 17 into the HEC-18 equation 

(equation 5), with the addition of Ki given by:

for Vo > Vim ,

 , (18)

for Vo ≤ Vim ,

 Ki = 0.0 , (18a)

where Ki is the correction factor for initiation of motion. 

K4 from equation 17 and Ki factor from equation 18 are incorporated to produce the form of the HEC-18 equation proposed 

by Molinas (2003):

, (19)
Introduction 9

and where all other terms are as previously defined. The 

research by Molinas that resulted in equation 19 showed 

improvements among the laboratory data over the most 

revised version of the HEC-18 equation (equation 5 with K4 by 

equation 6); however, that research was not supported by field 

data.

Finally, Mueller and Wagner (2005) used field data from 

266 pier-scour measurements in the sand and coarser size 

range to show that the magnitude of overprediction by the 

HEC-18 equation without the inclusion of K4 increased as the 

median size of the bed material increased. They proposed the 

development of a new methodology for determination of K4

based on numerous combinations of variables analyzed. The 

best correlation resulted when K4 was related to the pier width 

and median bed-material size according to:

, (20)

where all terms are as previously defined. Mueller and Wagner 

(2005) determined that HEC-18 equation results using K4 

determined by equation 20 did not perform as well (statistically) as the HEC-18 equation using K4 determined by equation 11 proposed by Mueller (1996). Still, the basis for the 

approach using equation 20 was considered justified because 

of previous research (Sheppard, 2004) that showed the ratio of 

pier width to sediment particle size to be an important scour 

mechanism.

Methods of Data Collection 

Data collected in the field for this study included information needed to measure pier scour along with bed material 

and ancillary data needed to compute scour using equations 

like those presented in the “Background” section of this report. 

Measured pier scour is a scour-depth estimate derived from 

various hydraulic measurements made at a bridge. Computed 

scour is a scour-depth prediction derived from variables measured in the field and equations developed in previous studies. 

During 2001–07, pier-scour and related data were collected at 

bridges in Montana with a variety of stream sizes, streamflow 

conditions, pier configurations, and bed-material sizes and 

gradations. 

Limited-detail field procedures were used to investigate 

pier scour during this study. Measurements made at each site 

(tables 1, 2) included the following:

1. Pier-scour depth derived from cross-section survey data.

2. Bed-material characteristics.

3. Approach velocity (Vo) and approach depth of flow (yo ) at 

each pier.

4. Water-surface elevation and slope at the bridge during 

high-flow conditions.

5. Pier geometry and flow angle of attack on the pier (α).

6. Background notes describing site-specific conditions.

Upon returning from the field, equations 1 and 2 were 

used to compute the critical velocity to initiate scour of 

bed material at the pier (V′

c50), which was compared to the 

measured approach velocity (Vo ) to determine whether or not 

a valid real-time pier-scour measurement was obtained for a 

site. If a valid real-time pier-scour measurement was indicated, 

the information collected was used to determine measured 

scour and computed scour using equation 5. Where measured 

scour exceeded scour computed by equation 5, follow up data 

were collected to help explain the inconsistency. Follow up 

data-collection efforts included (1) measuring approach depth 

of flow at different magnitudes of streamflow to confirm the 

elevation of the reference bed surface (fig. 2), also called the 

concurrent ambient-bed surface (Landers and Mueller, 1996, 

p. 36–37; Mueller and Wagner, 2005, p. 8–9); (2) determining 

the potential effects of the thalweg on scour-depth estimation, 

(3) identifying the presence of substantial amounts of large, 

woody debris; (4) collecting additional bed-material samples; 

and (5) reassessing the flow angle of attack.

Site Selection

Bridge sites were selected to include a range of coarse 

bed-material characteristics, bridge-pier configurations, and 

stream characteristics such as bed-material size and gradation, 

pier width, and stream size that can affect pier scour. Several 

bed-material, pier, and site criteria were used to select suitable 

bridge sites for measurement of pier scour. 

Foremost was the bed-material criteria used to select 

sites. Only sites with noncohesive, nonuniform, coarse-bed 

material were considered, which is consistent with HEC-18 

criteria (Richardson and Davis, 2001), where the D50 particle 

size is ≥ 2 mm and the D95 particle size is ≥ 20 mm. Most of 

the bridge sites investigated were located in stream reaches 

that would likely be classified as pool-riffle or plane-bed 

(Montgomery and Buffington, 1997). 

Criteria also were used to determine if piers associated 

with a site were suitable for scour measurement. When piers 

are aligned parallel with the flow, maximum scour typically 

occurs at the upstream nose of the pier (Mueller and Wagner, 

2005, p. 18). Bridge sites selected for measurement had piers 

with little or no flow angle of attack (α generally less than 

about 5 degrees). A few sites where α ranged from 5 to a maximum of 10 degrees were selected because remnant scour holes 

with no substantial infilling indicated maximum scour depth 

was located near the upstream nose of the pier, and a few sites 

also were selected because flow became more aligned with the 

pier at higher stages of flow. Thus, scour measurements were 

made at the upstream end of the pier in all instances. 

Other criteria also were considered in the selection 

of sites. Sites selected had no unique hydraulic conditions 

that could produce pier scour by processes other than those 

accounted for in the pier-scour equations evaluated in this 

report. Furthermore, piers that had substantial accumulation 
Table 1. Bridge sites where pier-scour measurements were made for this study.—Continued

[All sites in Montana. Horizontal coordinate information referenced to North American Datum of 1983 (NAD 83). Abbreviations: DMS, degrees, minutes, seconds; MDT, Montana Department of Transportation; NA, not applicable; USGS, U.S. Geological Survey. Symbol: -- not provided]

Site 

number 

(fig. 1)

Site name Site description

Drainage area 

above bridge, 

in square miles

Bridge-structure 

number1

Total length 

of bridge, 

in feet

Latitude, 

in DMS

Longitude, 

in DMS

Missouri River Basin

1 Big Hole River east 

of Wise River

County road just off State Route 43, at Jerry Creek bridge 

fishing access, about 2 miles east of Wise River. Steel continuous girder. Construction date unknown.

1,993 Not in MDT system 200 45°47′08.5′′ 112°54′52.9′′

2 Big Hole River west 

of Divide

State Route 43, about 2 miles west of Divide. Prestressed 

concrete girder bridge built in 1968.

2,029 P00046 075+00661 304 45°45′27.2′′ 112°46′52.6′′

3 Beaverhead River 

north of Dillon

Old State Route 91, about 1 mile north of Dillon. Steel 

two-girder continuous system bridge built in 1946. 

3,339 L01311 000+06001 151 45°13′59.8′′ 112°37′55′′

4 Big Hole River near 

Melrose

County road west of I-15; near Salmonfly fishing access 

at Melrose; secondary channel 0.38 mile or about 2,000 feet east of main channel. Prestressed concrete girder bridge built in 2001.

2,476 L47026 000+02001 216 45°37′36.3′′ 112°41′13.9′′

5 Boulder River near 

Basin

I-15 southbound, about 2 miles southwest of Basin. Prestressed concrete girder bridge built in 1985. 217 I00015 155+00551 168 46°15′45.4′′ 112°17′17.7′′

6 Wisconsin Creek 

near Sheridan

Abandoned railroad grade, just east of State Route 287, 

about 4 miles northwest of Sheridan. Construction date unknown.

27.7 NA - abandoned 

railroad bridge

59 45°29′30′′ 112°14′58′′

7 Gallatin River 

northwest of West Yellowstone

U.S. Highway 191 about 7 miles northwest of Yellowstone 

National Park boundary and 25 miles north of West Yellowstone. Concrete continuous tee-beam bridge built in 1955.

99.3 P00050 023+06711 120 44°58′15.4′′ 111°04′43.3′′

8 Big Hole River 

southwest of Twin Bridges

County road at Pennington Bridge, west of State Route 41, 

about 6 miles southwest of Twin Bridges. Continuous steel multi-girder bridge built in 1987.

2,718 L29141 000+03001 208 45°29′52.3′′ 112°26′16′′

9 Jefferson River north 

of Twin Bridges

State Route 41, about 7 miles north of Twin Bridges. Continuous steel girder bridge built in 1992. 7,685 P00029 049+08261 458 45°38′32.9′′ 112°18′37′′

10 Madison River south 

of Cameron

County road at McAtee Bridge, about 7 miles south of 

Cameron. Prestressed concrete girder bridge built in 1967.

1,545 L29092 000+07001 280 45°05′47′′ 111°39′40′′

11 Boulder River 

at I-90, near Cardwell

I-90 westbound, about 1.75 miles north of Cardwell. Prestressed concrete girder bridge built in 1968. 758 I00090 256+08672 112 45°52′14′′ 111°56′34.1′′

12 Boulder River near 

Cardwell

State Route 2, about 1.5 miles north of Cardwell. Prestressed concrete girder bridge built in 1968. 758 L22249 000+08001 111 45°52′12.2′′ 111°56′33.7′′

10 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Table 1. Bridge sites where pier-scour measurements were made for this study.—Continued

[All sites in Montana. Horizontal coordinate information referenced to North American Datum of 1983 (NAD 83). Abbreviations: DMS, degrees, minutes, seconds; MDT, Montana Department of Transportation; NA, not applicable; USGS, U.S. Geological Survey. Symbol: -- not provided]

Site 

number 

(fig. 1)

Site name Site description

Drainage area 

above bridge, 

in square miles

Bridge-structure 

number1

Total length 

of bridge, 

in feet

Latitude, 

in DMS

Longitude, 

in DMS

Missouri River Basin—Continued

13 South Boulder River 

near Cardwell

State Route 359, about 3 miles south of Cardwell. Timber 

stringer bridge built in 1958.

70.3 S00359 006+02091 50 45°48′17.7′′ 111°55′37.1′′

14 South Willow Creek 

near Harrison

U.S. Highway 287, about 0.5 mile south of Harrison. Timber stringer bridge built in 1934. 3.13 P00013 075+01321 39 45°41′42.3′′ 111°47′02′′

15 Gallatin River west 

of Bozeman

State Route 84, about 7 miles west of Bozeman. Prestressed concrete girder bridge built in 1987. 947 P00084 027+09401 152 45°40′18.5′′ 111°12′32′′

16 Jefferson River west 

of Three Forks

Old U.S. Highway 10, 2.5 miles northwest of Three Forks, 

at USGS streamflow-gaging station 06036650. Steel continuous girder bridge built in 1949.

9,532 P00013 093+06931 248 45°53′48.1′′ 111°35′45.9′′

17 Gallatin River near 

Logan

County road just north off State Route 205, 0.7 mile 

northwest of Logan, 0.2 mile downstream from USGS streamflow-gaging station 06052500. Prestressed concrete girder bridge built in 1964.

1,795 L16200 000+01001 200 45°53′10.2′′ 111°26′30′′

18 Jefferson River north 

of Three Forks

I-90 eastbound about 1 mile north of Three Forks. Prestressed concrete girder bridge built in 1968. 9,627 I00090 277+04471 274 45°54′36′′ 111°32′48′′

19 Madison River near 

Three Forks

State Route 205, about 1 mile northeast of Three Forks. 

Steel continuous girder bridge built in 1948.

2,551 S00205 000+04911 224 45°54′03.6′′ 111°31′31′′

20 Missouri River near 

Townsend

U.S. Highway 287 about 1 mile north of Townsend. Steel 

continuous girder bridge built in 1935.

15,318 P00008 076+03431 500 46°20′00.6′′ 111°31′48′′

21 Sun River north of 

Augusta

U.S. Highway 89 about 4 miles northeast of Augusta. Steel 

two-girder continuous system bridge built in 1936. 

822 P00009 043+01451 320 47°32′53.9′′ 112°21′53.7′′

22 Marias River near 

Shelby

Old U.S. Highway 91 just east of I-15, about 5 miles 

south of Shelby, at USGS streamflow-gaging station 06099500. Steel two-girder continuous system bridge built in 1936. 

3,242 L51306 003+02001 545 48°35′37.9′′ 111°53′20′′

23 Smith River southeast of Ulm State Route 330, about 13 miles southeast of Ulm. Pre- stressed concrete girder bridge built in 1962. 1,881 S00330 015+02001 144 47°15′41′′ 111°25′28.6′′24 Smith River at old 

truss bridge, southeast of Ulm

County road about 6 miles southeast of Ulm. Steel truss 

bridge built in 1908.

1,992 L07410 000+01001 150 47°21′24′′ 111°26′24′′

25 Belt Creek 7 miles 

south of Belt

State Route 89 about 7 miles south of Belt. Prestressed 

concrete girder bridge built in 1968.

374 P00060 066+03551 160 47°16′54.5′′ 110°54′15.4′′

Introduction 11
Table 1. Bridge sites where pier-scour measurements were made for this study.—Continued

[All sites in Montana. Horizontal coordinate information referenced to North American Datum of 1983 (NAD 83). Abbreviations: DMS, degrees, minutes, seconds; MDT, Montana Department of Transportation; NA, not applicable; USGS, U.S. Geological Survey. Symbol: -- not provided]

Site 

number 

(fig. 1)

Site name Site description

Drainage area 

above bridge, 

in square miles

Bridge-structure 

number1

Total length 

of bridge, 

in feet

Latitude, 

in DMS

Longitude, 

in DMS

Missouri River Basin—Continued

26 Belt Creek 6 miles 

south of Belt

State Route 89 about 6 miles south of Belt. Prestressed 

concrete girder bridge built in 1968.

376 P00060 067+02311 182 47°17′40.8′′ 110°53′59.6′′

27 Belt Creek 2 miles 

southeast of Belt

State Route 200 about 2 miles southeast of Belt. Continuous concrete tee-beam bridge built in 1954. 585 P00060 071+02931 156 47°21′08.1′′ 110°53′55.9′′

Yellowstone River Basin

28 Yellowstone River at 

Emigrant

County road about 0.1 mile east of U.S. Highway 89 and 

Emigrant. Steel continuous girder bridge built in 1949.

3,073 L34206 000+04001 302 45°22′03.9′′ 110°43′40′′

29 Shields River near 

Wilsall

County road about 0.2 mile east of U.S. Highway 89, about 

1 mile northeast of Wilsall. Steel girder bridge built in 1960.

143 L34006 000+07001 63 46°01′34.1′′ 110°38′26.5′′

30 Mill Creek near Pray State Route 540 about 3 miles northeast of Pray. Steel 

girder bridge built in 1925.

164 S00540 017+03701 67 45°24′46.6′′ 110°38′56.8′′

31 Yellowstone River 

near Pray 

County road about 4 miles northeast of Pray. Continuous 

steel multi-girder bridge built in 1960.

3,301 L34204 000+05001 374 45°25′09.3′′ 110°38′32.6′′

32 Yellowstone River 

near Pine Creek

County road about 1 mile west of Pine Creek. Steel continuous girder bridge built in 1990. 3,467 L34203 001+07001 364 45°30′38.8′′ 110°34′53′′

33 Yellowstone 

River south of Livingston

State Route 540 at Carter Bridge, about 4 miles south 

of Livingston, about 50 feet upstream from USGS streamflow-gaging station 06192500. Concrete arch bridge built in 1921.

3,551 S00540 031+06621 285 45°35′48′′ 110°34′00′′

34 Boulder River near 

McLeod 

State Route 298, about 14 miles southwest of Big Timber, 

about 3 miles northeast of McLeod. Steel continuous girder bridge built in 1950.

483 S00298 013+00001 150 45°41′21.2′′ 110°03′54.9′′

35 Boulder River south 

of Big Timber

State Route 298 about 11 miles southwest of Big Timber. 

Prestressed concrete girder bridge built in 1975. 

504 S00298 008+05001 206 45°43′23.6′′ 109°59′48.7′′

36 Boulder River at I-90 

near Big Timber 

I-90 eastbound just south of Big Timber. Prestressed concrete girder bridge built in 1980. 523 I00090 369+00271 523 45°49′34.6′′ 109°56′42.3′′

Columbia River Basin

37 Little Blackfoot 

River near Avon

U.S. Highway 12 about 0.5 mile east of Avon. Prestressed 

concrete girder bridge built in 1982.

269 P00008 011+08791 118 46°35′46.1′′ 112°35′16′′

38 Blackfoot River (old 

bridge) west of Lincoln

State Route 141 near junction with State Route 200, about 

14 miles west of Lincoln. Steel truss bridge built in 1936 was replaced with new bridge (site 39).

483 P00041 032+04711 186 46°56′28′′ 112°56′50′′

12 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Table 1. Bridge sites where pier-scour measurements were made for this study.—Continued

[All sites in Montana. Horizontal coordinate information referenced to North American Datum of 1983 (NAD 83). Abbreviations: DMS, degrees, minutes, seconds; MDT, Montana Department of Transportation; NA, not applicable; USGS, U.S. Geological Survey. Symbol: -- not provided]

Site 

number 

(fig. 1)

Site name Site description

Drainage area 

above bridge, 

in square miles

Bridge-structure 

number1

Total length 

of bridge, 

in feet

Latitude, 

in DMS

Longitude, 

in DMS

Columbia River Basin—Continued

39 Blackfoot River 

(new bridge) west of Lincoln

State Route 141 near junction with State Route 200, about 

14 miles west of Lincoln, MT. Steel girder bridge built in 2004 to replace old bridge (site 38). 

483 P00041 032+04711 205 46°56′28′′ 112°56′50′′

40 Little Blackfoot 

River near Garrison

I-90 west bound near Garrison. Prestressed concrete girder 

bridge built in 1979.

407 I00090 175+02152 131 46°31′10.7′′ 112°47′36′′

41 Clark Fork near 

Gold Creek

County road just south of I-90, 0.4 mile north of town of 

Goldcreek, 1.1 miles downstream from Gold Creek, at USGS streamflow-gaging station 12324680. Prestressed concrete tee-beam bridge built in 1982.

1,760 L39248 000+02001 200 46°35′25.7′′ 112°55′43′′

42 North Fork 

Blackfoot River west of Lincoln

State Route 200 about 20 miles west of Lincoln. Concrete 

continuous tee-beam bridge built in 1956.

241 P00024 049+06571 182 47°00′01′′ 113°02′10′′

43 North Fork 

Blackfoot River near Ovando

Off-system road located about 3.5 miles southeast of 

Ovando. Prestressed concrete girder bridge built in 2000.

314 L39001 004+09001 143 46°58′47.1′′ 113°053′2.5′′

44 Clark Fork south of 

Drummond

State Route 1 about 0.75 mile south of Drummond. Steel 

multi-girder bridge built in 1934.

2,404 L20016 000+05001 270 46°39′42′′ 113°09′02′′

45 Lost Horse Creek 

north of Darby

U.S Highway 93 about 7 miles north of Darby. Prestressed 

concrete girder bridge built in 1990.

74 P00007 037+05961 163 46°06′47.8′′ 114°10′48′′

46 Clark Fork near 

Drummond

County road just north of I-90, about 9 miles northwest 

of Drummond, at USGS streamflow-gaging station 12331800. Prestressed concrete tee-beam bridge built in 1986.

2,501 L20009000+01001 170 46°42′43.7′′ 113°19′51′′

47 Clark Fork at 

Turah Bridge

County road 4 miles southeast of Bonner, about 0.8 mile 

southeast of Turah, at USGS streamflow-gaging station 12334550. Prestressed concrete girder bridge built in 1962.

3,641 L32210 001+00801 287 46°49′34′′ 113°48′48′′

48 Swan River east of 

Ferndale 

State Route 209 about 5 miles east of Big Fork. Prestressed 

concrete girder bridge built in 1964.

676 S00209 004+09861 226 48°03′06′′ 113°58′59′′

49 Clark Fork at 

Missoula

Clark Fork at Madison Street bridge, in Missoula. Steel 

two-girder continuous system bridge built in 1958.

6,099 P00007 095+00581 552 46°52′02.3′′ 113°59′15.3′′

Introduction 13
Table 1. Bridge sites where pier-scour measurements were made for this study.—Continued

[All sites in Montana. Horizontal coordinate information referenced to North American Datum of 1983 (NAD 83). Abbreviations: DMS, degrees, minutes, seconds; MDT, Montana Department of Transportation; NA, not applicable; USGS, U.S. Geological Survey. Symbol: -- not provided]

Site 

number 

(fig. 1)

Site name Site description

Drainage area 

above bridge, 

in square miles

Bridge-structure 

number1

Total length 

of bridge, 

in feet

Latitude, 

in DMS

Longitude, 

in DMS

Columbia River Basin—Continued

50 Clark Fork near 

Alberton

County road about 2.5 miles southeast of Alberton, just 

upstream from Petty Creek. Continuous steel two-girder system bridge built in 1964.

9,279 L32090 000+01001 457 46°59′32.4′′ 114°26′46′′

51 Clark Fork near 

Superior

Frontage road at Dry Creek Exit 43 north of I-90 about 

4.5 miles northwest of Superior. Steel continuous girder bridge built in 1982.

10,295 L31169 000+01001 425 47°13′43′′ 114°57′58.8′′

52 St. Regis River at 

Ward Creek

Access bridge to off-system road, just south of I-90 near 

east bound exit ramp. Prestressed concrete beam bridge. Construction date unknown.

276 Not in MDT system 146 47°18′43.4′′ 115°13′59.3′′

53 St. Regis River west 

of St. Regis

Bridge on exit ramp of I-90 west bound, about 3 miles 

west of St. Regis. Concrete tee-beam bridge built in 1981.

300 I00090 030+02024 150 47°17′45.4′′ 115°09′49.9′′

54 St. Regis River at 

St. Regis 

Frontage road on east end of St. Regis. Steel continuous 

girder bridge built in 1937.

363 L31179 013+02001 192 47°17′46′′ 115°05′46′′

55 Flathead River near 

Perma

State Route 382 on Flathead Indian Reservation, 0.3 mile 

north of Perma. Steel continuous girder bridge built in 1965.

9,008 S00382 000+01001 654 47°22′02.6′′ 114°34′38′′

56 Clark Fork above 

Flathead River, near Paradise

State Route 200 about 0.6 mile upstream from confluence 

with Flathead River, about 2.5 miles southeast of Paradise. Steel continuous girder bridge built in 1996.

10,838 P00006 084+07531 470 47°21′20.3′′ 114°46′56.3′′

57 Clark Fork below 

Flathead River, near Paradise

State Route 200 about 0.5 mile southeast of Paradise and 

1.75 miles downstream from the confluence with Flathead River. Steel continuous girder bridge built in 1997.

19,930 P00006 082+07591 1,110 47°22′56.2′′ 114°47′53′′

58 Kootenai River near 

Libby

State Route 37 just north of Libby. Steel two-girder continuous system bridge built in 1959. 10,168 P00033 000+07121 698 48°23′49.9′′ 115°32′51′′

59 Yaak River near Troy U.S. Highway 2 about 8 miles northwest of Troy. Steel 

continuous girder bridge built in 1934.

766 P00001 006+03521 213 48°33′36′′ 115°58′12′′

1Bridge-structure number from Montana Department of Transportation (1994).

14 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Introduction 15

of large, woody debris or a tendency for debris accumulation 

based on historical information were not considered for scour 

measurements. Finally, sites were selected where contraction 

and abutment scour did not contribute to pier scour because 

(1) all piers were beyond the influence of abutments that may 

have projected into the main-channel flow, (2) the site had 

an incised channel with little or no flood plain or lacked any 

flow contraction at the bridge, (3) bridge abutments did not 

encroach within the bankfull width of the stream, or (4) runoff 

conditions at the site were insufficient to cause measurable 

contraction or abutment scour effects.

Pier-Scour Depth

Measured pier-scour depth (y′

s

) is the difference between 

a measured reference surface of the streambed and the lowest 

measured location within a pier-scour hole. Scour measurements primarily consist of cross-section surveys, but also may 

include longitudinal profile surveys and continuous measurement of scour using fixed instrumentation. At all sites, scour 

measurements were conducted during intermittent site visits 

using portable instrumentation. At some sites, scour measurements also were made using instrumentation fixed at a 

particular location (Mueller and Landers, 2000 and Mueller 

and Wagner, 2005). 

Scour measurements obtained during high-flow conditions at each bridge fell into one of four categories:

1. One measurement at a single pier.

2. One measurement at multiple piers.

3. Multiple measurements at a single pier.

4. Multiple measurements at multiple piers.

Most measurements were in category 1 or 2. 

The most common method to measure scour is to 

conduct cross-section surveys. For this study most crosssection surveys involved sounding with a cable-suspended 

Columbus-type (or C-type) weight (Jarrett and Boyle, 1986) 

deployed from the upstream side of the bridge deck. A few 

scour measurements were obtained by probing the streambed 

with a graduated metal rod when the bridge deck was close to 

the streambed surface and the streamflow depth was less than 

about 3 or 4 ft. 

In addition to these more-conventional methods, 

hydroacoustic devices also were used. A portable chartrecording echo sounder was used at several sites to survey 

cross sections. Two devices used with the echo sounder 

included a knee board with a transom-mounted transducer fitted through the board and a transducer mounted to the bottom 

of a Columbus weight (Mueller and Landers, 2000, p. 37–51). 

The echo sounder and transducer worked well in the stream 

channel, some distance away from the pier, but scour measurements at piers during high-flow conditions generally 

were problematic because (1) turbulence and drawdown near 

the pier made the knee board pitch, heave, or roll, so the 

transducer was not submerged beneath the water surface or 

was out of vertical alignment at times; (2) air entrainment 

interfered with acoustic transmission; or (3) high surface 

velocities swept the kneeboard away from the pier. 

A boat equipped with a portable survey-grade echo 

sounder and global positioning system (GPS) was used in 

several of the large, deep rivers during moderate streamflow 

conditions to obtain channel-bottom data near the pier and 

longitudinally upstream and downstream from the bridge cross 

section. These data were used with other high-flow crosssection surveys to assess scour depth and to evaluate remnant 

scour. The prevalence of clear-water scour conditions resulting in no substantial infilling allowed for measurements of 

remnant scour that would not have been valid under live-bed 

conditions. The boat and echo sounder were used at two large 

river sites exceeding 300 ft in flow width (sites 51 and 57, 

table 1), where the channel bottom was surveyed for several 

hundred feet upstream and downstream from the piers to aid in 

determining the reference bed surface longitudinally. The boat 

and echo sounder also were used at site 57 to map the pierscour hole, where the flow angle of attack on three out of four 

piers was 10 degrees. At this site, soundings were conducted 

at depths in excess of 10 ft along the sides and along the 

upstream and downstream ends of the piers under moderate 

streamflow conditions to confirm that the deepest scour was 

located at the upstream end of the piers. 

Fixed instrumentation used to measure scour included 

acoustic transducers and scour rods installed at 10 sites. 

Transducers were used at three sites (sites 21, 33, and 41) and 

consisted of one or more 6-degree beamwidth, 235-kilohertz 

devices, fastened to the pier near the upstream end to measure 

scour depth on a near-continuous basis (15-minute time step). 

Scour rods were used at seven sites (sites 4, 5, 30, 37, 40, 43, 

and 45) and typically consisted of 3-ft- to 6-ft-long smooth 

steel rods 1-inch in diameter that were encased most of their 

lengths in a 2-inch-diameter steel pipe fixed to the pier near 

the upstream end to measure only maximum scour during 

spring runoff. Elevations of the top of the rod and casing were 

surveyed and tied to a local vertical datum before and after 

runoff and were used with other information to determine 

scour depth. The top of the scour rod and outer casing projected above the water surface, or the top of casing was sealed 

to prevent intrusion of sand and finer suspended sediment during submergence. Streambed material supporting scour rods 

generally exceeded the diameter of the rod. Scour rods were 

observed in operation during very high approach velocities 

(velocity at site 45 was 11.1 ft/s, table 2) with no indication 

of erratic rod movement or vibration that might cause drilling 

action that would induce additional scour. 

Cross sections, piers, and other features were described 

using a USGS convention, in which features are referenced in 

a downstream view from left to right. Therefore, the left bank 

was considered to be on the left-hand side of the observer 

when looking downstream in the direction of streamflow. The 

left pier at a site, therefore, was designated pier number 1 and 

piers to the right were assigned pier numbers 2, 3, and so on. 
Table 2. Pier-scour data collected for this study.—Continued

[All pier-scour data are for sites with nonuniform, coarse-bed material (predominantly gravel and cobble). Pier number: 1, 2, 3 are piers left to right looking 

downstream. The term “single” is used when bridges had one pier. Pier type: S, single; G, group; Debris effects: I, insigificant; M, moderate. Pier-scour 

depth from field measurements. Abbreviations: ft, feet; ft/s, feet per second; σg , gradation coefficient, dimensionless; mm, millimeters; Di, surface bed-material 

particle size where i percent of particles are finer by weight, in millimeters]

Site 

number

(fig.1)

Site name

Measurement

number

Date Time Pier 

number

Pier

type

Pier 

nose 

shape

Nominal

pier width, bn

(ft)

Pier 

length, L

(ft)

Missouri River Basin 

1 Big Hole River east of Wise River 1 06/04/02 1100 2 S Sharp 2.00 15.6 

1 Big Hole River east of Wise River 2 06/04/02 1045 3 S Sharp 2.00 15.6 

1 Big Hole River east of Wise River 3 05/24/06 1810 3 S Sharp 2.00 15.6 

2 Big Hole River west of Divide 4 06/04/02 1245 1 S Sharp 4.00 36 

2 Big Hole River west of Divide 5 06/04/02 1230 2 S Sharp 4.00 36 

2 Big Hole River west of Divide 6 05/25/06 1100 2 S Sharp 4.00 36 

3 Beaverhead River north of Dillon 7 05/31/06 1130 2 S Sharp 2.67 28 

4 Big Hole near Melrose 8 05/24/06 1040 1 S Round 7.00 24 

5 Boulder River near Basin 9 06/10/06 1845 1 S Round 1.69 84 

6 Wisconsin Creek near Sheridan1 10 06/29/05 1350 1 G Round 1.02 14 

7 Gallatin River northwest of West Yellowstone 11 06/28/06 1700 Single S Sharp 2.00 32 

8 Big Hole River southwest of Twin Bridges 12 05/31/06 1440 1 S Sharp 3.67 26 

9 Jefferson River north of Twin Bridges 13 06/13/06 1505 1 S Sharp 3.00 46 

9 Jefferson River north of Twin Bridges 14 06/13/06 1520 2 S Sharp 3.00 46 

9 Jefferson River north of Twin Bridges 15 06/13/06 1535 3 S Sharp 3.00 46 

10 Madison River south of Cameron 16 06/01/03 1245 1 S Sharp 2.80 33 

10 Madison River south of Cameron 17 06/16/05 1730 1 S Sharp 2.80 33 

10 Madison River south of Cameron 18 06/16/05 1745 2 S Sharp 2.80 33 

11 Boulder River at I-90, near Cardwell 19 05/19/05 1445 1 S Sharp 3.00 40 

11 Boulder River at I-90, near Cardwell 20 06/06/05 1220 1 S Sharp 3.00 40 

12 Boulder River near Cardwell 21 06/06/05 1045 1 S Sharp 3.00 37 

13 South Boulder River near Cardwell 22 06/29/05 1015 Single S Square 1.60 26 

13 South Boulder River near Cardwell 23 06/11/06 1650 Single S Square 1.60 26 

14 South Willow Creek near Harrison 24 06/11/06 1840 Single G Round 1.00 37 

15 Gallatin River west of Bozeman 25 05/30/06 1820 Single S Sharp 3.00 41 

15 Gallatin River west of Bozeman 26 06/07/06 2025 Single S Sharp 3.00 41 

16 Jefferson River west of Three Forks 27 06/13/06 2005 2 S Sharp 3.30 32 

17 Gallatin River near Logan 28 05/24/05 1715 2 S Sharp 2.80 24 

18 Jefferson River north of Three Forks 29 06/28/05 1215 2 S Sharp 3.33 41 

19 Madison River near Three Forks 30 05/23/05 1315 Single S Sharp 3.00 32 

19 Madison River near Three Forks 31 06/15/05 1345 Single S Sharp 3.00 32 

20 Missouri River near Townsend 32 06/06/05 1630 1 S Sharp 3.00 46 

20 Missouri River near Townsend 33 06/06/05 1700 2 S Sharp 3.00 46 

20 Missouri River near Townsend 34 06/12/06 1920 1 S Sharp 3.00 46 

20 Missouri River near Townsend 35 06/12/06 1935 2 S Sharp 3.00 46 

21 Sun River north of Augusta 36 06/12/02 1345 1 S Sharp 3.50 28 

21 Sun River north of Augusta 37 06/12/02 1415 2 S Sharp 3.50 28 

21 Sun River north of Augusta 38 05/25/06 1545 1 S Sharp 3.50 28 

22 Marias River near Shelby 39 06/11/02 1015 1 S Sharp 6.002 27 

16 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Flow angle 

of attack, α

(degrees)

Approach 

velocity, Vo

(ft/s)

Approach depth 

of flow, yo

(ft)

Debris 

effects D16 D50 D84 D95 σg

Pier-scour

depth, y′

s

(ft)

Accuracy 

of y′

s

(ft)

Missouri River Basin

0 7.50 7.51 I 40.5 102 176 263 2.09 1.14 0.30

0 7.89 7.41 I 40.5 102 176 263 2.09 1.44 .30

0 7.79 7.09 I 40.5 102 176 263 2.09 1.12 .30

0 6.09 4.54 I 23.7 71.3 155 190 2.56 1.36 .30

0 7.21 4.94 I 23.7 71.3 155 190 2.56 .76 .30

0 6.65 4.76 I 23.7 71.3 155 190 2.56 1.35 .30

0 3.62 1.62 I 20.3 35.8 59.2 80.3 1.71 .98 .30

0 5.15 4.10 I 28.2 59.5 104 157 1.92 2.23 .20

0 4.82 2.60 I 29.1 58.0 111 163 1.96 .32 .10

0 5.27 1.40 I 23.2 47.8 78.8 104 1.84 .40 .20

0 7.63 1.82 I 32.7 64.7 117 167 1.89 1.58 .20

0 4.58 5.25 M 20.8 38.5 60.9 80.1 1.71 1.27 .40

10 5.98 7.27 I 28.8 60.7 114 148 1.99 1.35 .40

10 7.21 6.02 I 28.8 60.7 114 148 1.99 2.33 .30

8 6.80 7.83 I 28.8 60.7 114 148 1.99 2.06 .30

0 5.53 3.01 I 29.9 79.8 149 253 2.23 .86 .30

0 5.64 3.22 I 29.9 79.8 149 253 2.23 .78 .30

0 6.09 3.22 I 29.9 79.8 149 253 2.23 .79 .30

0 4.00 4.73 I 2.58 17.1 44.1 82.9 4.14 1.37 .30

0 5.02 5.67 I 2.58 17.1 44.1 82.9 4.14 2.03 .30

0 4.45 4.71 I 2.58 17.1 44.1 82.9 4.14 .71 .30

0 5.78 .80 I 29.3 55.2 111 155 1.95 .40 .20

0 6.38 2.24 I 29.3 55.2 111 155 1.95 .64 .30

0 4.57 1.85 I 8.76 38.3 77.3 109 2.97 .60 .20

0 6.45 3.61 I 45.5 70.4 109 143 1.55 .68 .30

0 8.33 4.49 I 45.5 70.4 109 143 1.55 1.25 .30

0 6.25 15.9 I 5.91 22.3 57.0 89.6 3.11 2.08 .40

0 6.24 5.80 I 17.8 33.3 57.4 76.0 1.80 1.70 .30

0 3.87 8.47 I 8.85 19.1 31.1 47.5 1.87 .90 .30

0 4.67 2.73 I 19.7 29.8 43.2 56.8 1.48 1.47 .20

0 5.22 1.76 I 19.7 29.8 43.2 56.8 1.48 1.10 .20

0 5.83 8.47 I 11.9 29.7 64.1 87.4 2.32 1.23 .50

0 4.74 8.18 M 11.9 29.7 64.1 87.4 2.32 1.43 .50

0 7.84 10.3 I 11.9 29.7 64.1 87.4 2.32 2.19 .50

0 6.83 10.0 M 11.9 29.7 64.1 87.4 2.32 2.99 .50

0 5.74 3.60 I 26.8 57.4 111 156 2.03 1.50 .20

0 4.77 2.83 I 26.8 57.4 111 156 2.03 1.17 .20

0 5.18 3.12 I 26.8 57.4 111 156 2.03 1.38 .20

0 9.56 14.7 I 7.36 22.7 44.0 59.3 2.44 2.97 .50

Introduction 17
Table 2. Pier-scour data collected for this study.—Continued

[All pier-scour data are for sites with nonuniform, coarse-bed material (predominantly gravel and cobble). Pier number: 1, 2, 3 are piers left to right looking 

downstream. The term “single” is used when bridges had one pier. Pier type: S, single; G, group; Debris effects: I, insigificant; M, moderate. Pier-scour 

depth from field measurements. Abbreviations: ft, feet; ft/s, feet per second; σg , gradation coefficient, dimensionless; mm, millimeters; Di, surface bed-material 

particle size where i percent of particles are finer by weight, in millimeters]

Site 

number

(fig.1)

Site name

Measurement

number

Date Time Pier 

number

Pier

type

Pier 

nose 

shape

Nominal

pier width, bn

(ft)

Pier 

length, L

(ft)

Missouri River Basin—Continued 

23 Smith River southeast of Ulm 40 06/08/05 0820 1 S Sharp 3.20 31 

23 Smith River southeast of Ulm 41 06/14/05 1115 1 S Sharp 3.20 31 

24 Smith River at old truss bridge, southeast of 

Ulm

42 06/08/05 1415 2 S Round 3.53 16 

25 Belt Creek 7 miles south of Belt 43 06/13/05 1330 1 S Sharp 3.50 42.5 

26 Belt Creek 6 miles south of Belt 44 06/10/05 1445 1 S Sharp 3.50 42.5 

27 Belt Creek 2 miles southeast of Belt 45 06/13/05 1430 Single S Sharp 2.00 33.5 

Yellowstone River Basin 

28 Yellowstone River at Emigrant 46 06/22/05 1630 2 S Sharp 4.00 25 

29 Shields River near Wilsall 47 06/30/05 1430 1 S Square 2.50 25 

30 Mill Creek near Pray 48 06/22/05 1230 Single S Square 2.00 21 

30 Mill Creek near Pray 49 06/14/06 1600 Single S Square 2.00 21 

31 Yellowstone River near Pray 50 06/23/05 1000 1 S Sharp 3.00 27 

31 Yellowstone River near Pray 51 06/23/05 0930 3 S Sharp 3.00 27 

31 Yellowstone River near Pray 52 06/08/06 1400 1 S Sharp 3.00 27 

31 Yellowstone River near Pray 53 06/08/06 1420 3 S Sharp 3.00 27 

32 Yellowstone River near Pine Creek 54 06/05/02 1145 1 S Round 3.50 26 

32 Yellowstone River near Pine Creek 55 06/05/02 1230 2 S Round 3.50 26 

33 Yellowstone River south of Livingston 56 06/10/06 1445 1 S Round 10.0 25 

33 Yellowstone River south of Livingston 57 06/14/06 1145 2 S Round 10.0 25 

34 Boulder River near McLeod 58 06/08/06 2015 1 S Sharp 3.00 22 

35 Boulder River south of Big Timber 59 06/08/06 1745 2 S Sharp 3.60 34 

36 Boulder River at I-90 near Big Timber 60 06/09/06 1045 2 S Sharp 3.60 44 

Columbia River Basin 

37 Little Blackfoot River near Avon 61 06/10/02 1500 1 G Round 1.69 46 

37 Little Blackfoot River near Avon 62 06/07/05 1135 1 G Round 1.69 46 

37 Little Blackfoot River near Avon 63 06/08/07 1100 1 G Round 1.69 46 

38 Blackfoot River (old bridge) west of Lincoln 64 06/10/02 1215 Single S Sharp 3.00 21 

38 Blackfoot River (old bridge) west of Lincoln 65 06/19/02 1445 Single S Sharp 3.00 21 

39 Blackfoot River (new bridge) west of Lincoln 66 05/13/05 1745 Single S Sharp 2.00 42 

39 Blackfoot River (new bridge) west of Lincoln 67 06/05/06 1845 Single S Sharp 2.00 42 

39 Blackfoot River (new bridge) west of Lincoln 68 06/08/07 1745 Single S Sharp 2.00 42 

40 Little Blackfoot River near Garrison 69 06/11/06 1330 Single S Sharp 2.00 46 

40 Little Blackfoot River near Garrison 70 06/08/07 1320 Single S Sharp 2.00 46 

41 Clark Fork near Gold Creek 71 06/07/07 0745 2 S Round 5.00 31 

42 North Fork Blackfoot River west of Lincoln 72 05/30/02 1115 1 S Sharp 3.00 42 

42 North Fork Blackfoot River west of Lincoln 73 05/30/02 1130 2 S Sharp 3.00 42 

42 North Fork Blackfoot River west of Lincoln 74 05/17/05 1400 2 S Sharp 3.00 42 

43 North Fork Blackfoot River near Ovando 75 05/20/05 1330 1 G Round 1.69 34 

18 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Flow angle 

of attack, α

(degrees)

Approach 

velocity, Vo

(ft/s)

Approach depth 

of flow, yo

(ft)

Debris 

effects D16 D50 D84 D95 σg

Pier-scour

depth, y′

s

(ft)

Accuracy 

of y′

s

(ft)

Missouri River Basin—Continued

5 6.18 5.08 I 6.67 13.6 26.7 41.5 2.00 2.92 0.30

5 5.89 4.57 I 6.67 13.6 26.7 41.5 2.00 2.24 .30

0 6.59 7.88 I 19.8 39.2 86.0 123 2.08 .85 .30

0 5.97 6.88 M 15.1 44.6 88.8 125 2.42 2.13 .30

0 6.45 2.80 I 14.5 38.0 75.1 105 2.28 1.85 .50

0 6.11 2.86 I 23.1 39.5 76.3 109 1.82 1.14 .20

Yellowstone River Basin

0 6.84 9.69 I 14.6 33.2 67.6 94.4 2.15 2.81 0.50

0 5.04 1.37 I 24.8 50.6 88.0 115 1.88 .45 .20

0 6.10 2.52 I 32.2 67.4 130 164 2.01 .75 .20

0 5.73 2.12 I 32.2 67.4 130 164 2.01 .53 .20

0 6.48 6.32 I 41.5 92.1 133 165 1.79 1.67 .30

0 7.61 8.97 M 41.5 92.1 133 165 1.79 1.40 .40

0 6.66 7.10 I 41.5 92.1 133 165 1.79 1.20 .30

0 8.13 9.77 M 41.5 92.1 133 165 1.79 1.63 .40

0 6.85 10.3 M 10.9 54.5 114 174 3.23 2.95 .50

0 8.25 11.5 I 10.9 54.5 114 174 3.23 3.05 .40

0 10.5 8.75 I 24.1 55.5 88.8 117 1.92 6.60 .20

0 5.44 6.00 I 24.1 55.5 88.8 117 1.92 3.13 .40

0 7.64 4.72 I 31.9 55.6 108 128 1.84 .80 .40

0 7.17 8.71 I 43.2 76.6 125 171 1.71 .99 .30

0 7.22 2.44 I 46.9 84.6 136 196 1.70 .90 .30

Columbia River Basin

0 3.62 2.48 I 13.2 29.2 55.2 77.9 2.06 .52 0.20

0 6.00 1.72 I 13.2 29.2 55.2 77.9 2.06 1.08 .20

0 5.06 3.18 I 13.2 29.2 55.2 77.9 2.06 1.47 .20

0 3.03 7.80 I 1.68 9.63 26.8 38.2 4.00 1.00 .30

0 3.12 8.60 I 1.68 9.63 26.8 38.2 4.00 1.41 .30

0 3.04 5.24 I 1.68 9.63 26.8 38.2 4.00 .76 .30

0 3.78 4.76 I 1.68 9.63 26.8 38.2 4.00 .84 .30

0 4.08 5.04 I 1.68 9.63 26.8 38.2 4.00 1.16 .30

0 4.87 2.41 I 14.6 57.4 98.1 127 2.59 .60 .20

0 5.98 2.70 I 14.6 57.4 98.1 127 2.59 .75 .20

0 4.22 5.58 I 8.62 30.9 79.8 159 3.04 2.09 .20

0 6.77 3.94 I 22.4 36.6 60.6 82.2 1.65 1.46 .20

0 8.30 3.98 I 22.4 36.6 60.6 82.2 1.65 1.71 .30

0 5.81 2.84 I 22.4 36.6 60.6 82.2 1.65 .56 .30

0 5.33 2.06 I 25.9 50.1 93.7 128 1.90 .45 .20

Introduction 19
Table 2. Pier-scour data collected for this study.—Continued

[All pier-scour data are for sites with nonuniform, coarse-bed material (predominantly gravel and cobble). Pier number: 1, 2, 3 are piers left to right looking 

downstream. The term “single” is used when bridges had one pier. Pier type: S, single; G, group; Debris effects: I, insigificant; M, moderate. Pier-scour 

depth from field measurements. Abbreviations: ft, feet; ft/s, feet per second; σg , gradation coefficient, dimensionless; mm, millimeters; Di, surface bed-material 

particle size where i percent of particles are finer by weight, in millimeters]

Site 

number

(fig.1)

Site name

Measurement

number

Date Time Pier 

number

Pier

type

Pier 

nose 

shape

Nominal

pier width, bn

(ft)

Pier 

length, L

(ft)

Columbia River Basin—Continued 

44 Clark Fork south of Drummond 76 06/02/05 1210 2 S Sharp 3.00 25 

44 Clark Fork south of Drummond 77 06/04/05 0945 2 S Sharp 3.00 25 

45 Lost Horse Creek north of Darby 78 05/16/06 1015 1 G Round 1.37 46 

46 Clark Fork near Drummond 79 06/15/06 1045 2 S Round 1.67 26 

47 Clark Fork at Turah Bridge 80 06/05/01 1230 3 S Sharp 3.00 26 

48 Swan River east of Ferndale 81 05/27/06 1035 2 S Sharp 3.00 32 

48 Swan River east of Ferndale 82 05/27/06 1055 3 S Sharp 3.00 32 

49 Clark Fork at Missoula 83 05/23/02 0730 1 S Sharp 3.25 68 

50 Clark Fork near Alberton 84 05/31/02 0830 2 S Sharp 3.00 26 

51 Clark Fork near Superior 85 06/01/06 1010 1 S Sharp 5.00 34 

51 Clark Fork near Superior 86 06/01/06 1055 2 S Sharp 5.00 34 

52 St. Regis River at Ward Creek 87 05/30/02 1930 2 S Square 1.35 25 

53 St. Regis River west of St. Regis 88 05/30/02 1745 Single S Sharp 3.00 29 

54 St. Regis River at St. Regis 89 05/30/02 1515 Single S Sharp 2.50 33 

55 Flathead River near Perma 90 06/22/06 1550 1 S Sharp 3.80 28 

55 Flathead River near Perma 91 06/22/06 1615 2 S Sharp 3.60 28 

55 Flathead River near Perma 92 06/22/06 1650 3 S Sharp 3.50 28 

56 Clark Fork above Flathead River, near 

Paradise

93 06/06/06 1215 2 S Sharp 3.50 34 

57 Clark Fork below Flathead River, near 

Paradise

94 06/02/06 1630 1 S Sharp 4.33 38 

57 Clark Fork below Flathead River, near 

Paradise

95 06/02/06 1615 2 S Sharp 4.33 38 

57 Clark Fork below Flathead River, near 

Paradise

96 06/02/06 1530 3 S Sharp 4.33 38 

57 Clark Fork below Flathead River, near 

Paradise

97 06/02/06 1500 4 S Sharp 4.33 38 

57 Clark Fork below Flathead River, near 

Paradise

98 06/22/06 1245 2 S Sharp 4.33 38 

57 Clark Fork below Flathead River, near 

Paradise

99 06/22/06 1315 3 S Sharp 4.33 38 

58 Kootenai River near Libby 100 05/26/06 1430 3 S Sharp 4.00 34 

58 Kootenai River near Libby 101 06/21/06 1150 3 S Sharp 4.00 34 

59 Yaak River near Troy 102 05/26/06 1130 1 S Sharp 3.50 48 

59 Yaak River near Troy 103 05/26/06 1045 2 S Sharp 3.50 48 

1

Site is abandoned railroad bridge on abandoned railroad grade.

2

Pier width based on exposed footing width of 6 ft.

20 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Flow angle 

of attack, α

(degrees)

Approach 

velocity, Vo

(ft/s)

Approach depth 

of flow, yo

(ft)

Debris 

effects D16 D50 D84 D95 σg

Pier-scour

depth, y′

s

(ft)

Accuracy 

of y′

s

(ft)

Columbia River Basin—Continued

0 7.46 6.63 I 5.46 34.4 86.4 118 3.98 2.07 0.30

0 5.79 5.77 I 5.46 34.4 86.4 118 3.98 1.53 .30

0 11.1 2.73 I 24.1 73.0 122 168 2.26 .75 .20

0 4.12 3.80 I 15.8 27.8 43.6 62.0 1.66 .70 .30

0 6.97 6.30 I 20.9 49.3 87.1 122 2.04 1.94 .30

0 7.09 7.36 I 34.8 72.9 115 146 1.82 1.15 .30

0 5.84 5.00 I 34.8 72.9 115 146 1.82 .80 .30

0 8.14 15.4 I 28.5 67.3 123 168 2.08 1.95 .40

0 7.02 12.4 M 18.1 46.1 95.4 140 2.30 2.11 .50

0 5.98 9.76 I 20.0 53.4 105 144 2.29 2.10 .50

0 6.43 8.96 M 20.0 53.4 105 144 2.29 3.54 .50

0 9.37 7.30 I 22.7 50.7 88.6 121 1.97 .90 .30

0 8.35 6.60 I 27.2 41.0 69.6 86.4 1.60 1.58 .30

0 8.04 6.53 I 12.7 24.5 50.0 84.8 1.99 1.37 .40

0 5.42 17.8 M 16.4 27.6 54.7 81.6 1.82 2.81 .50

0 4.90 19.4 M 16.4 27.6 54.7 81.6 1.82 1.92 .50

0 4.51 10.9 M 16.4 27.6 54.7 81.6 1.82 1.90 .50

0 6.34 10.7 I 16.2 54.9 115 159 2.67 2.29 .30

5 5.65 14.9 I 18.7 31.3 51.1 69.4 1.65 2.66 .50

10 6.73 13.5 M 18.7 31.3 51.1 69.4 1.65 7.00 .50

10 7.38 12.5 I 18.7 31.3 51.1 69.4 1.65 4.24 .50

10 6.03 17.6 M 18.7 31.3 51.1 69.4 1.65 6.43 .50

10 7.39 14.8 M 18.7 31.3 51.1 69.4 1.65 6.50 .50

10 7.59 13.9 I 18.7 31.3 51.1 69.4 1.65 5.06 .50

0 6.91 6.93 I 27.4 58.0 93.0 126 1.84 1.87 .30

0 9.78 8.03 I 27.4 58.0 93.0 126 1.84 2.47 .50

0 6.97 4.50 I 46.1 108 189 265 2.02 1.00 .30

0 6.51 3.95 I 46.1 108 189 265 2.02 1.05 .50

Introduction 21
03/07/2006 low-flow survey

63

64

65

66

67

68

69

70

50 100 150 200 250 300 350 400

Reference 

bed 

surface

Reference bed surface

Pier-scour

depth = 2.10 ft

Pier-scour

depth = 3.54 ft 

DISTANCE FROM LEFT EDGE OF CROSS SECTION, IN FEET

ELEVATION, IN FEET (LOCAL DATUM)

Centerline of pier

Site 51 - Clark Fork near Superior, Montana

06/01/2006 high-flow survey

Pier 1

Pier 2

EXPLANATION

Figure 3. Surveyed cross section at upstream side of bridge showing pier-scour holes and reference bed surface 

used to determine pier-scour depth.

22 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

The pier-numbering system used in the study differs from 

MDT pier numbers indicated on design drawings because 

bents or abutments at each end of the bridge structure included 

in the MDT pier-numbering system were excluded from the 

USGS pier-numbering system. The USGS pier-numbering 

system also was applied to sites operated and maintained by 

county and local entities that frequently had no engineering 

drawings with pier-numbering information. 

Cross-section data were used to determine the reference 

bed surface (fig. 2) from which the lowest measured elevation of the scour hole was subtracted to determine measured 

or observed scour. The reference bed surface was sketched 

(Mueller and Wagner, 2005, p. 8) using cross-section data 

obtained during high-flow conditions and used to determine 

pier-scour depth (fig. 3). Surveys to establish the reference 

bed surface mostly were made along the upstream bridge face 

by using a manually operated four-wheel-base bridge crane 

and B-reel setup with a cable-suspended sounding weight 

deployed from the bridge deck (Rantz and others, 1982; 

Turnipseed and Sauer, 2010). Cross-section surveys varied in 

detail; complete cross sections were surveyed at a majority 

of sites. At a few sites, cross-section data were only obtained 

close to the pier, but far enough away (laterally) from the pier 

to accurately define the reference bed surface (figs. 2 and 3). 

In all instances, detailed soundings were conducted near the 

pier to define the scour hole. Measurements occasionally were 

obtained longitudinally along the streambed for some distance 

upstream and downstream along the major axis of the pier to 

help define the reference bed surface. 

Surveys at most bridges were conducted on at least two 

separate dates—once during high-flow conditions (typically 

in May or June) to obtain pier-scour measurements, and once 

during low flow to assess channel-geometry changes, if any, 

attributed to general scour (Richardson and Davis, 2001, 

p. 5.1–5.18) or channel deposition relative to the high-flow 

measurement. Additionally, low-flow measurements were analyzed in combination with high-flow measurements to detect 

the extent of scour-hole infilling, to indicate whether clearwater or live-bed scour conditions prevailed at a site, and to 

assess the relative channel stability at the bridge cross section 

over a range of flow conditions.

Whether the total clear-water scour for a particular measurement resulted from a single scour event or multiple scour 

events was not addressed in this study. Cross-section measurements made during low-flow periods indicated remnant scour 

holes at most sites, so the total scour depth measured under 

high-flow conditions at a site could be the result of a single 

scour event or multiple scour events perhaps over consecutive below-average runoff years. As with similar studies using 

limited-detail field procedures to measure pier-scour depth, the 

assumption was made that scour depth produced by a single 

clear-water scour event could be the same as scour depth produced by multiple clear-water events of similar flow magnitude. Therefore, pier-scour depths reported in this study were 

attributed to high-flow conditions of similar magnitude leading 

up to and during the time scour measurements were made.
Introduction 23

Bed-Material Characteristics

Bed-material sizes used in scour-prediction equations 

were determined from the surface bed material that generally 

reflected the armor layer. The potential interaction between 

the surface and shallow-subsurface bed-material layers (fig. 4) 

with respect to pier scour and armoring also was of interest 

for this study. Therefore, shallow-subsurface bed material was 

sampled and data were compared to surface bed-material data 

for each site. Surface and shallow-subsurface bed-material 

measurements were obtained by pebble counts and sieve 

analyses. Bed-material data were used to construct particlesize distribution curves, which were used to conduct various 

coarse-bed pier-scour analyses.

Bed material was sampled at each site to characterize the 

average streambed material near the bridge, with an emphasis 

on the channel area leading into the bridge opening. Field 

observations indicated that coarse bed-material particle size 

and gradation varied at least as much vertically as areally near 

the bridge. Thus, surface bed-material samples were collected 

and analyzed to define bed-material characteristics in the same 

manner as traditional pier-scour studies. However, subsurface bed material typically was sampled at relatively shallow 

depths less than 1 ft beneath the streambed surface. Both samples generally were collected before or after the runoff season 

to more easily access and visually inspect the main-channel 

streambed. Samples were collected with square-hole templates 

and sieves to define the intermediate or b-axis (Bunte and Abt, 

2001, p. 20–27). 

The surface layer of the bed material was sampled by 

pebble counts (Wolman, 1954) where material was predominantly fine gravel and coarser particles (4 mm and larger, 

table 3), and samples were sieved when the composition 

included sand and very fine gravel. Particles were measured in 

the field by pebble counts using the US SAH-97 gravelometer 

developed by the Federal Interagency Sedimentation Project 

(FISP) and described by Davis (2005). Pebble counts typically 

were limited to a single sample set of at least 100 particles 

obtained near the bridge opening. Although only a single sample was obtained, considerable time was spent determining the 

optimal location to be sampled near the bridge. Samples were 

collected from the streambed in the main channel whenever 

possible. Locations were avoided where loose deposits of sediment or the presence of finer material might indicate greater 

sediment mobility than was likely for the average bed-material 

composition. For flood-bar deposits, the mid-bar region was 

sampled thus avoiding the coarser and finer material typically 

deposited at the upstream and downstream ends of the bar 

(respectively). Multiple sets of pebble counts were conducted 

for about 10 percent of the bridge sites (table 4) used in the 

study. Some variation in particle size between the datasets at 

each site is indicated (standard deviation for Di, table 4), but 

the relative magnitude of each standard deviation is small in 

most instances when compared to the corresponding mean 

particle size at each percentile (Di) shown.

Shallow-subsurface bed material was sampled within 

the zone typically considered part of the active layer or the 

upper layer of the streambed susceptible to sediment transport depending on hydraulic conditions (Molinas, 2000, p. 

33–37). Shallow-subsurface bed material was excavated by 

first removing the surface material or armor layer. Overall depth of excavation from which the subsurface sample 

was collected was either equal to at least two times the D90

particle size, which generally describes the active-layer depth 

in coarse-bed streams (DeVries, 2002) or was about an 8- to 

10-inch minimum depth (whichever was greater). Bed material 

representative of the overall streambed was sampled whenever 

possible. Some samples were obtained along channel fringes 

or flood bars where the composition appeared similar to bed 

material located farther out in the stream, but where the combined effects of flow depth and velocity prevented sampling. 

A single sampling location typically was chosen, with other 

locations excavated and visually inspected to gain an indication of average composition of the subsurface material. At a 

few sites, surface material was removed by merely scraping 

the surface material out of the way by hand. The majority of 

sites, however, required the use of a shovel to pry and excavate subsurface material. Many sites required the use of both a 

shovel and pry bar to obtain the sample. 

Surface material samples were used to characterize the 

subsurface material at four sites (site 31, site 37, site 38, and 

site 39; table 1) where excavation indicated that the size and 

gradation of the two layers were similar. Shallow-subsurface 

samples were sieved because sediment could not be characterized by pebble counts alone. Sieved samples were air dried for 

several weeks or longer, hand sieved using standard 8-inchdiameter metal sieves for 11 screen sizes ranging from 2 to 

128 mm, and weighed. Samples were analyzed at the USGS 

Montana Water Science Center office in Helena, Mont (Guy, 

1969). Particle-size distribution curves were determined for 

the surface- and shallow-subsurface layers of each sample, and 

important bed-material size percentiles were interpolated from 

the curves. Percentiles included the D16, D35, D50, D84, D85, D90, 

D95, and D99 particle sizes, where the subscript i denotes the 

percentage of material finer (by weight) than the indicated particle size (D). The D50 and D95 particle sizes were used to test 

the sensitivity of scour computations to vertical differences in 

bed-material size and to investigate how possible interactions 

between surface and shallow-subsurface streambed material 

are related to pier scour.

Although the gradation and stratification of coarse 

streambed material can vary (Bunte and Abt, 2001, p. 129), 

all but three sites where pier-scour data were collected for this 

study had a higher fraction of coarse material on the surface 

than in the shallow subsurface. Such conditions are typical of 

armored streambeds, where enough coarse particles can accumulate to shield the entire bed surface (Lagasse and others, 

1991, p. 78–79). Although general channel stability was indicated by clear-water conditions and bed armoring upstream 

from the bridge at most sites, shallow-subsurface bed material was exposed at the piers when local scour took place. 
 Surface

bed material

 Shallow-

 subsurface

bed material

 Surface

bed material

 Shallow-subsurface

 bed material

A

B

Figure 4. Examples of differences in surface and shallow-subsurface bed material at sites 

where pier-scour measurements were collected for this study. A, Near-vertical perspective 

of the bed-material variation at site 15 (Gallatin River west of Bozeman). B, Plan view of bedmaterial variation at site 36 (Boulder River at I–90 near Big Timber).

24 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Table 3. Bed-material size classification used for this study

(modified from Lane and others, 1947).

[Symbols: ≥, greater than or equal to; <, less than]

Particle-size range, 

in millimeters Classification

≥256 Boulders.

128 to <256 Large cobbles.

64 to <128 Small cobbles.

32 to <64 Very coarse gravel.

16 to <32 Coarse gravel.

8 to <16 Medium gravel.

4 to <8 Fine gravel.

2 to <4 Very fine gravel.

0.062 to <2 Very fine to very coarse sand.

<0.062 Silt and clay.

Introduction 25

The shallow subsurface layer may, therefore, be important in 

determining appropriate bed-material variables to be used in 

the scour equations and was investigated accordingly.

Based on the D50 particle size at each site, a Shields 

parameter (θ) equal to 0.047 (Meyer-Peter and Müller, 1948; 

Gessler, 1971) was used with equations 1 and 2. This value 

is consistent with a recent study (Mueller and others, 2005), 

in which a median value of θ equal to 0.040 was reported for 

45 gravel-bed streams and rivers.

Ancillary Data

 Other data also were collected that were necessary to 

determine measured pier-scour depth and to compute scour. 

These data included approach velocity (Vo) and approach 

depth of flow (yo) for all sites, water-surface elevation and 

slope, pier geometry, and flow angle of attack on the pier (α). 

Most of the ancillary data were collected during site visits, but 

some data at sites with fixed instrumentation were measured 

continuously. 

Approach velocity ideally represents the velocity just 

upstream from a pier, beyond the extent of the pier-scour 

hole and flow-acceleration zone (fig. 2). In the field, approach 

velocity typically is measured from a bridge deck during highflow conditions. A vertical-axis mechanical current meter and 

USGS standard streamgaging procedures (Rantz and others, 

1982 ; Turnipseed and Sauer, 2010) were used to measure the 

velocity on either side of the piers. Velocity measurements 

adjacent to piers but outside the scour hole were made to 

approximate the velocity just upstream from the pier where 

velocity was difficult to measure. Approach velocity measurements made using a current meter were checked at one site 

(site 55, table 1) using a portable acoustic Doppler current 

profiler (ADCP). The specific ways in which approach velocity was determined varied with the method used to measure 

pier scour at a site.

At sites where pier scour was measured using crosssection surveys during high-flow conditions, approach 

velocities were measured during the same site visit. Velocity 

was measured at several locations on either side of a pier 

to determine the approach velocity reported for the study. 

Multiple measurements of velocity were made to describe the 

variability of the approach velocities and any anomalies at a 

particular location. 

At sites where pier scour was measured with scour rods, 

approach velocities were determined using stage-velocity 

rating curves. At these sites, approach velocity and associated 

water stage were measured from the bridge deck during periodic site visits that included a range of moderate- to high-flow 

conditions. This limited number of measurements was used to 

develop a stage-velocity rating curve. The maximum seasonal 

water stage surveyed at the bridge, based on high-water marks, 

was then used in the stage-velocity rating curve to estimate 

maximum approach velocity by rating-curve extension. This 

method is consistent with findings that indicate maximum 

scour depth is closely correlated with maximum approach 

velocity (Müeller, 1996, p. 91).

At sites where pier scour was measured with acoustic 

transducers mounted to piers, approach velocities were determined based on periodic measurements of velocity, continuous 

measurement of stage, and development of stage-velocity rating curves. The recorded water stage at the bridge corresponding to maximum scour determined from transducer soundings 

was used with the stage-velocity rating curve to estimate, by 

rating-curve extension, approach velocity associated with 

maximum pier scour. 

Approach depth of flow ideally represents the depth to 

stream bottom just upstream from the pier beyond the extent 

of the pier-scour hole and flow-acceleration zone (fig. 2). In 

the field, approach depth of flow typically is measured by 

sounding from a bridge deck during high-flow conditions. 

Bottom depth was sounded at several locations on either side 

of a pier to determine the approach depth of flow reported 

for the study. Multiple measurements of depth were made to 

describe the variability of the approach depth of flow and any 

anomalies at a particular location. Bottom depth was referenced to a water-surface elevation to determine the approach 

depth of flow. For high-flow scour measurements using crosssection surveys, the referenced water-surface elevation was the 

stage noted on the day and time when the bottom depth was 

sounded. For sites where pier scour was measured with scour 

rods, the referenced water-surface elevation was the maximum 

stage based on high-water marks surveyed after runoff. For 

sites where pier scour was measured with acoustic transducers, the referenced water-surface elevation was the recorded 

stage at the time that maximum scour was indicated by the 

transducer. 

Water-surface elevation and slope during soundings 

were surveyed by trigonometric leveling with a total station 

and prism (Ghilani and Wolf, 2008) or by measuring from a 

reference mark down to the water surface. Vertical control for 

elevations and water-surface elevation measurements were 
Table 4. Variation in surface bed-material particle size for multiple pebble counts made at bridge sites where pier-scour 

measurements were made for this study.

[Abbreviations: Di, surface bed-material particle size where i percent of particles are finer by weight; mm, millimeters; n=, number of pebble counts in the 

dataset. Symbols: σg, gradation coefficient]

Site 1

n=3

Site 15

n=3

Site 17

n=3

Particle 

size

Mean value 

for Di, 

in mm

Standard 

deviation for Di, 

in mm

Particle 

size

Mean value 

for Di, 

in mm

Standard 

deviation for Di, 

in mm

Particle 

size

Mean value 

for Di, 

in mm

Standard 

deviation for Di, 

in mm

D16 41 3.4 D16 46 4.8 D16 18 0.7

D35 73 6.8 D35 58 3.0 D35 27 1.3

D50 102 7.3 D50 70 5.8 D50 33 2.2

D84 176 15 D84 109 19 D84 57 2.4

D85 179 14 D85 112 19 D85 58 2.5

D90 209 8.4 D90 127 22 D90 64 3.9

D95 263 4.9 D95 143 24 D95 76 12

D99 309 29 D99 179 13 D99 102 19

Mean σg = 2.09

Standard deviation = 0.01

Mean σg = 1.55

Standard deviation = 0.09

Mean σg = 1.80

Standard deviation = 0.01

Site 28

n=4

Site 36

n=3

Site 51

n=3

Particle 

size

Mean value 

for Di, 

in mm

Standard 

deviation for Di, 

in mm

Particle 

size

Mean value 

for Di, 

in mm

Standard 

deviation for Di, 

in mm

Particle 

size

Mean value 

for Di, 

in mm

Standard 

deviation for Di, 

in mm

D16 15 3.0 D16 47 18 D16 20 4.0

D35 25 2.9 D35 68 17 D35 37 5.8

D50 33 4.7 D50 85 15 D50 53 5.1

D84 68 21 D84 136 8.7 D84 105 4.1

D85 69 22 D85 140 7.7 D85 108 3.3

D90 79 26 D90 164 6.9 D90 121 .9

D95 94 32 D95 196 26 D95 144 9.1

D99 110 42 D99 266 31 D99 174 4.2

Mean σg = 2.16

Standard deviation = 0.40

Mean σg = 1.75

Standard deviation = 0.25

Mean σg = 2.31

Standard deviation = 0.18

26 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

referenced to a local arbitrary datum at some sites, whereas 

bench marks and other USGS gage reference marks were used 

where available. Water-surface slope is not an explanatory 

variable typically used in scour-prediction equations, but is 

needed to compute the shear stress on the streambed and to 

establish initial boundary conditions for site-specific hydraulic 

modeling studies. Therefore, water-surface slope was surveyed 

when the high-flow scour measurements were made to enable 

future research. 

Pier geometry was measured or obtained from engineering drawings, and flow angle of attack on the pier (α) was 

determined visually in the field. The shape of the pier nose 

was recorded. At sites with tapered piers, nominal pier width 

(bn) was computed as the average width in contact with flow at 

each surveyed water-surface elevation or measured stage. The 

flow angle of attack on the pier was determined during highflow measurements. At some sites, the flow angle of attack 

also was assessed during low-flow conditions by observing the 

relative symmetry of the scour hole and the relative alignment 

of sediment deposited alongside and downstream from a pier.

 Relative pier scour (y′

s / b) and relative bed-material size 

(b/D50) were calculated for each measurement and used to 

relate measured pier-scour depth (y′

s

), effective pier width (b), 

and median bed-material size (D50). These variables normalize 

the data and provide dimensionless plots that indicate a relation with pier-scour depth. Relative pier scour for study data 
Pier Scour in Coarse-Bed Streams in Montana 27

was computed by using the effective pier width (b) according 

to:

 , (21)

where

L is the length of the pier measured 

perpendicular to the nominal pier width 

(L ≥ bn), in feet; 

α is the flow angle of attack on the pier, in 

degrees; 

and where all other terms are as previously defined. The effective pier-width adjustment (where b > bn

, when α > 0 degrees) 

accounts for increased scour when flow is not parallel to the 

major axis of the pier. 

Collection of limited-detail measurements included 

recording background field notes and documenting hydraulic and scour conditions at each site. Typical observations 

included weather conditions, streamflow conditions, extent 

of any woody debris accumulated on piers, description of 

remnant scour holes, and bed-material characteristics and bed 

form in the stream reach near the bridge. For example, the 

bed-form correction factor (K3) was assigned a value of 1.1 

for all measurements, based in part on the coarse-bed material 

and plane-bed conditions recorded in the field. The quality 

rating of velocity measurements and soundings also was often 

described, for later use in assessing the suitability of measurements for determination of approach velocity and approach 

depth of flow. 

Water temperature can affect sediment transport and was 

therefore determined. Water temperature was either measured 

during a scour measurement with a calibrated thermometer, 

or estimated by analyzing real-time water temperature records 

reported for USGS streamflow-gaging stations near the measurement site. Based on an analysis of site measurements and 

water temperature records, water temperature was estimated to 

average about 50°F over the runoff periods for all sites. 

Pier Scour in Coarse-Bed Streams in 

Montana

A total of 103 pier-scour measurements were made at 

59 bridges on coarse-bed streams in Montana during 2001–07 

for this study. Runoff conditions during the study mostly were 

limited to bankfull discharge approximating the 1.5- to 2-year 

recurrence-interval flood, which have a 67- and 50-percent 

chance, respectively, of happening in any year. Data collected 

for this study were compared to BSDMS scour measurements 

for coarse-bed streams and envelope curves based on previous studies (Ettema, 1980; Melville, 1984; Chiew, 1984; and 

Mueller and Wagner, 2005). Data from this study also were 

compared with selected historical pier-scour measurements 

from Montana associated with peak streamflows of greater 

recurrence interval (up to about 100 years) than pier-scour 

measurements collected for this study. Surface and shallowsubsurface bed-material data were collected for each site 

where pier-scour measurements were made to characterize the 

particle size and gradation. Bed-material characteristics in the 

two layers were used to compute scour by using the current 

(2011) version of the HEC-18 equation (equation 5), and the 

computed scour was compared to the measured scour. Relations between surface and shallow-subsurface bed-material 

characteristics and pier scour also were analyzed. 

Hydrologic and Hydraulic Conditions

Seasonal runoff in the mountain and foothill regions of 

the western one-half of Montana (fig. 1) typically results from 

spring snowmelt supplemented with May and June rainfall. 

In that region, summary data (McCarthy, 2005) indicate that 

runoff can begin in April, typically peaks in May or June, and 

recedes through the summer months to base-flow conditions 

in the fall. Isolated thunderstorms and broader-based general 

storms can cause occasional periods of high flow with varying durations in the summer and fall. Streamflow conditions 

during this study generally were below average for the three 

major river basins in the study. Maximum daily mean flow 

during the spring runoff seasons was generally below average, and spring-runoff volumes were below to near average 

(http://waterdata.usgs.gov/mt/nwis). Annual flood-frequency 

data (Parrett and Johnson, 2004) indicate that instantaneous 

peak discharges at most gaging stations near bridge sites 

where scour was measured approximated the 1.5- to 2-year 

recurrence interval during 2001–07 runoff. Lawlor (2004) 

found that the recurrence interval for bankfull discharge at 

41 streams in western Montana generally ranged from 1.0 to 

4.4 years with a median value of 1.5 years. Stream velocities 

and seasonal high-water marks measured at some ungaged 

sites in smaller drainage basins indicated that more intense 

runoff conditions, and hence, higher recurrence-interval peak 

streamflows occurred in these smaller basins compared to peak 

flows in larger drainage basins. Despite general below-average 

runoff conditions during this study, streamflow at most sites 

probably reached bankfull to near-bankfull conditions, which 

can produce notable sediment transport (Andrews, 1983) and 

scour. For example, the rate of increase in bedload discharge, 

based on 1999–2001 data at one study site (site 32, fig. 1 

and table 1), increased as streamflow approached bankfull 

discharge and then bedload discharge diminished beyond the 

bankfull stage (Holnbeck, 2005, p. 12–16). Real-time pier 

scour recorded at a downstream site (site 33, fig. 1 and table 1) 

by a fixed acoustic transducer between May–June 2006 was 

closely linked to changes in streamflow (fig. 5) for flows up 

to bankfull. The response of scour-hole depth to changes in 

streamflow at this site is complicated, however, by the sediment supply from upstream that also increases during higher 

streamflows. For example, the scour hole is shown to be 

infilling (fig. 5) at times during the rising limb in early June, 
May 2006 June 2006

–8

–7

–6

–5

–4

–3

–2

–1

0

5,000

10,000

15,000

20,000

25,000

STREAMFLOW, IN CUBIC FEET 

PER SECOND

PIER-SCOUR DEPTH, IN FEET

Reference bed surface Maximum measured

pier-scour 

depth = 6.60 feet

Runoff

hydrograph

(15-minute

time step)

Transducer readings

(15-minute time step)

2-year recurrence interval peak streamflow

approximately bankfull discharge

Plateau likely due to 

debris or temporary 

scour-hole armoring 

Figure 5. Pier-scour depth and 

streamflow measured at site 33, 

Yellowstone River south of Livingston, 

Montana (station 06192500), May and 

June, 2006.

28 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

most likely as a result of bedload transport from upstream. The 

plateau that persisted for 3 days in early June (fig. 5) likely is 

due to woody debris deposited in the scour hole and later dislodged or to temporary armoring when small cobbles that can 

be transported (Holnbeck, 2005, p. 8–20) partially infilled the 

scour hole. Transducers at three other sites recorded similar 

responses of scour to changes in streamflow during the study 

period. 

The persistence of below- to near-average streamflow 

conditions during the study period likely produced repeated 

clear-water scour conditions at most study sites over successive years. Therefore, infilling of scour holes from one runoff 

season to the next was probably minimal or only partial, 

allowing for the continued scouring of holes during successive clear-water periods. Overall, runoff conditions during the 

study probably limited the opportunities to measure maximum scour, but sites where measurements were successfully 

made reflect real-time scour conditions at piers in coarse-bed 

streams. 

Scour Data Collected

Pier-scour measurements were made on various dates at 

59 bridge sites on coarse-bed streams. Of those measurements, 

103 satisfied criteria requiring the measured approach velocity 

(Vo, table 2) exceed the critical velocity (V′

c50) to initiate scour 

of bed material at the piers. Incipient motion computations 

based on the D50 particle size indicated that 96 out of 103 pierscour measurements were made under clear-water scour conditions. Two of the measurements were made under hydraulic 

conditions of supercritical flow, one measurement was made 

at or very near critical flow, and 100 measurements were made 

under subcritical flow. Comparisons of cross-section data 

obtained during high-flow measurements with pre- or postrunoff cross-section data generally indicated a lack of substantial scour-hole infilling, which also supports the conclusion 

that clear-water conditions prevailed. Of the measurements 

made, 50 percent had an approach velocity (Vo) that equaled or 

exceeded 70 percent of the critical velocity (Vc50) for incipient 

motion of bed material upstream from the bridge, which might 

indicate that scour was measured very near the threshold 

between clear-water and live-bed scour (Vo /Vc50 = 1.00) where 

maximum pier scour was shown in laboratory studies (Chiew, 

1984; Melville, 1984).

The pier and stream characteristics associated with the 

103 measurements made for this study were variable but some 

general observations can be made about the measurements 

based on the individual measurements and summary statistics 

(tables 2 and 5). Despite reasonably high approach velocities, measured pier scour was less than a couple of feet in 

depth for 70 percent of the measurements made at the coarsebed sites. Particle size of sampled surface bed material was 

predominantly very coarse gravel based on the D50 particle 

size. Surface bed material at one-half of the sites lacked 

substantial gradation, as indicated by gradation coefficients 

less than about 2.0. Drainage areas for most sites ranged from 

a few hundred square miles to a few thousand square miles 

in drainage area. About 60 percent of the sites were located 

on high-gradient streams based on water-surface slopes that 

exceeded 0.002 (Jarrett, 1984). The approach depth of flow 

typically was less than twice the effective pier width. This 

condition is probably common in high-gradient streams in 

the northern Rocky Mountains where flow depths tend to be 

shallower compared to flatter-gradient streams having similar 

streamflow magnitudes and bridge pier widths. There were no 

instances where pier-scour measurements involving relatively 
Pier Scour in Coarse-Bed Streams in Montana 29

shallow approach flow depths in relation to effective pier 

width required later adjustment of computed pier-scour depth 

for wide-pier effect (Richardson and Davis, 2001, p. 6.7). All 

sites were subjected to hydraulic conditions where real-time 

scour was in progress at the time of measurement, as indicated 

by relative velocities (Vo /V′

c50 )in excess of 1.00 (table 5). 

Maximum relative pier scour from this study is less than 

values associated with previously published envelope curves, 

and therefore, data from this study do not redefine the upper 

magnitude of those envelope-curve relations. 

The effects of high-flow duration on pier scour were not 

investigated because most scour data for a site were collected 

in a single day. Continuous scour measurements made at three 

sites using fixed acoustic transducers, however, indicated that 

scour typically increased as streamflow increased. Pier scour 

at site 33, with a 10-ft wide pier (fig. 5, table 2), included 

remnant scour of about 5 ft, with scour depth fluctuating from 

a minimum of about 4 ft (some infilling) to a maximum of 

about 6.6 ft. Maximum pier scour at this site was attained for 

hydraulic conditions near the threshold between clear-water 

and live-bed scour (Vo /Vc50 = 1.05). Transducer soundings indicate that a similar magnitude of maximum pier scour (about 

6.6 ft) was attained several times during May–June 2006 

(fig. 5) when flows ranged from less than to slightly greater 

than bankfull discharge. These continuous data from the 

transducers provide some insight on the site-specific effects of 

high-flow duration on pier scour.

The effective pier-width adjustment for flow angle of 

attack (equation 21) was made for 11 of the 103 scour measurements. The adjustment was applied when α ranged from 

5 degrees to a maximum of 10 degrees (table 2) and when the 

flow angle of attack probably contributed to a small to moderate increase in scour depth at the upstream nose of the pier. 

When reporting relations between pier and streambed characteristics and scour depths, effective pier width was always 

used.

Comparisons with Other Pier-Scour Data

Pier-scour data collected for this study were compared 

to pier-scour data from other USGS studies that were used to 

develop pier-scour equations for coarse-bed streams. Historical data were retrieved from the BSDMS described by Landers 

and others (1996). The 508 measurements in the BSDMS database collected between 1965 and 1998 by USGS were initially 

screened using the following criteria:

1. Bed material was noncohesive and composed of 

sand and coarser particles.

2. Data for particle-size distribution of the bed material 

were reported in the BSDMS database.

3. Flow angle of attack on the pier (α) was reported 

as zero degrees. Sites with flow angles of attack 

exceeding zero degrees were not considered because 

it was not known what adjustments might have been 

made to account for α effects on scour depth.

4. Large, woody debris accumulation on piers was not 

indicated to be “substantial.”

5. Measured approach velocity (Vo) exceeded the critical velocity to initiate scour of bed material at the 

pier (V′

c50), where Vo /V′c50 > 1.00.

6. All variables needed for computing scour depth were 

reported in the BSDMS database for the measurement or could be derived from other variables 

reported in BSDMS.

The initial screening resulted in 272 pier-scour measurements that met the above criteria for highway bridges in 

15 States, referred to hereinafter as the BSDMS All dataset. 

The measurements used for this study and measurements 

from the BSDMS All dataset were classified based on the 

size of bed material relative to pier width using the system 

described by Ettema (1980). In that classification, sediment 

size is considered: fine for b/D50 ≥ 130; intermediate for 

130 > b/D50 ≥ 30; and coarse for 30 > b/D50 ≥ 8. For b/D50 < 8, 

sediment size is considered large and sediment typically is not 

eroded. 

A substantial portion (>75 percent) of the measurements in the BSDMS All dataset was made at sites with fine 

bed material based on Ettema’s classification system (fig. 6, 

table 6). In contrast, about two-thirds of pier-scour measurements made for this study were at sites with coarse bed 

material, and a small percentage (about 7 percent) of those 

measurements were at sites on the threshold between coarse 

and large bed material. 

All the data collected for this study and the BSDMS 

All dataset plot below the line reported by Raudkivi and 

Sutherland (1981) that showed the relation between relative 

pier-scour depth and relative bed-material size (fig. 6) near the 

clear-water and live-bed threshold (Vo /Vc50 = 0.9). Data plotting below the line indicate that scour is not underpredicted.

After the initial screening, the 272 measurements that 

make up the BSDMS All dataset were screened again for measurements where the D50 and D95 particle sizes were equal to 

or greater than 2 mm and 20 mm (respectively) so K4 could be 

less than 1.0. This screening reduced the BSDMS All dataset 

from 272 to 90 measurements. All subsequent comparisons of 

BSDMS data either use the BSDMS All dataset for noncohesive bed material (n=272) or the final screened dataset (n=90), 

referred to hereinafter as the BSDMS Coarse dataset. The 

BSDMS Coarse dataset is comparable with data collected for 

this study based on FHWA criteria for D50 and D95 particle size 

that defines coarse bed material.

In addition to using Ettema’s classification to compare 

relative bed material size (fig. 6), surface bed-material data 

collected for this study were compared with the BSDMS 

All and Coarse datasets (table 7) using the D50 particle size 

and the bed-material size classification (table 3) to generally characterize bed material size. Using very coarse gravel 
Table 5. Summary statistics for pier-scour measurement data collected for this study.

[Number of measurements (n) for each variable was 103. Abbreviations: ft, feet; ft/s, feet per second; mm, millimeters]

Pier-scour variable

Summary statistics

Minimum Maximum Mean Standard deviation Coefficient of variation Skewness 25th percentile Median 75th percentile

Pier-scour depth (y′

s), in ft 0.32 7.00 1.72 1.31 0.76 2.31 0.88 1.38 2.09

Approach velocity (

Vo ), in ft/s 3.03 11.1 6.24 1.55 .25 .38 5.20 6.11 7.19

Approach depth of flow (yo ), in ft .80 19.4 6.54 4.12 .63 1.07 3.20 5.67 8.54Bed-material particle size (D50 ), in mma 9.63 108 49.0 24.4 .50 .50 29.8 49.3 62.7Bed-material particle size (D95 ), in mma 38.2 265 125 57.3 .46 .71 81.6 122 159Gradation coefficient (σg), dimensionlessa 1.48 4.14 2.23 .69 .31 1.71 1.82 2.01 2.32Bed-material particle size (D50), in mmb 3.64 119 40.1 27.5 .69 .97 18.1 32.4 57.7Bed-material particle size (D95), in mmb 38.2 174 101 44.3 .44 .36 62.0 86.0 122.0Gradation coefficient (σg), dimensionlessb 1.79 7.63 4.14 1.32 .32 .42 3.44 4.14 4.80Nominal pier width (bn), in ftc 1.00 10.0 3.19 1.38 .43 2.51 2.59 3.00 3.50Effective pier width (b), in ftd 1.00 10.9 3.81 2.49 .65 1.94 2.59 3.00 3.64Pier length (L), in ft 1.00 68.0 29.5 13.6 .46 -.57 25.0 32.0 39.0Drainage area, in square miles 3.13 19,930 4,085 5,520 1.35 1.74 482 1,881 3,597Water-surface slope (

S ), dimensionless .00033 .01568 .00359 .00365 1.02 1.90 .00093 .00245 .00486

Relative flow depth (yo/b), dimensionless .50 5.41 1.91 1.10 .57 1.20 1.14 1.58 2.45Relative bed-material size (b/D50), 

dimensionless

5.72 132 32.6 29.3 .90 1.72 12.5 21.0 40.9

Froude number (Fr), dimensionless .19 1.18 .49 .18 .37 1.16 .37 .46 .59Relative velocity (

Vo

/Vc50), dimensionlesse .55 1.23 .74 .15 .20 1.11 .61 .70 .81

Relative velocity (

Vo

/V′

c50), dimensionlessf 1.01 2.33 1.35 .29 .21 1.19 1.12 1.25 1.51

Relative pier scour (y′

s

/b), dimensionless .12 1.00 .46 .18 .39 .46 .32 .43 .60

aBased on surface bed material.bBased on shallow-subsurface bed material.cPier width measured perpendicular to pier length without any adjustment for flow angle of attack.dNominal pier width adjusted for flow angle of attack, which affected 11 of 103 measurements.eRelative velocity associated with incipient motion of bed material upstream from the bridge.fRelative velocity associated with initiation of scour of bed material at the pier.

30 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Figure 6. Comparison of pier-scour measurements collected for this study and from the Bridge Scour Data Management System 

(BSDMS) All dataset. Relative bed-material size criteria from Ettema (1980).

0.01

0. 1

1

10

Data collected for this study (n=103)

BSDMS All dataset (n=272)

is the number of measurements

Large Coarse Intermediate Fine

RELATIVE BED-MATERIAL SIZE ( b/D ), DIMENSIONLESS 50

Relation based 

on Raudkivi and 

Sutherland (1981) 

RELATIVE PIER-SCOUR DEPTH (y's /b), DIMENSIONLESS

n

is pier-scour depth from field measurements, in feet

is effective width of the pier measured perpendicular to 

the approach flow, in feet

is the bed-material particle size where 50 percent of 

particles are finer by weight, in millimeters or feet

y's

b

D50

EXPLANATION

Variables

1 10 100 1,000 10,000 100,000

Table 6. Comparison of the relative bed-material size associated with pier-scour measurements made in this 

study with the Bridge Scour Data Management System (BSDMS) All dataset.

[Symbols: >, greater than; ≥, greater than or equal to; <, less than; (b/D50), relative bed-material size; n=, number of data values in 

dataset]

Interval for relative bed-material 

size (b/D50), dimensionless

Classification for 

bed-material size1

Percent of data in size classification

This study (n=103) BSDMS All dataset (n=272)

b/D50 ≥ 130 Fine 0.00 76.5

130 > b/D50 ≥ 30 Intermediate 33.7 16.9

30 > b/D50 ≥ 8 Coarse 59.2 6.6

b/D50 < 8 Large 7.1 .00

1

Bed-material size critera defined by Ettema (1980).

Pier Scour in Coarse-Bed Streams in Montana 31

as the size reference, about 67 percent of the measurements 

from this study are classified very coarse gravel or larger, only 

about 8 percent of the BSDMS All dataset is classified very 

coarse gravel or larger, and about 26 percent of the BSDMS 

Coarse dataset is classified very coarse gravel or larger. The 

BSDMS datasets do not include data for shallow-subsurface 

bed material, however, 54 percent of the shallow-subsurface 

measurements from this study are classified very coarse gravel 

or larger. None of the data from this study or the comparable 

BSDMS Coarse dataset (table 7) are ripple-forming sediments 

(D50 < 0.6 mm), whereas a substantial portion (about 32 percent) of the BSDMS All dataset are ripple-forming sediments. 

Ripple-forming sediments are more applicable to sand-bed 

streams, where fine noncohesive particle sizes and bed forms 

such as dunes are involved in the pier-scour process. 

Approach velocity (Vo) is among the most important 

hydraulic variables that can affect pier-scour depth. The 

median approach velocity for measurements collected for this 

study was slightly lower than the median approach velocity for 

measurements in the BSDMS Coarse dataset but was higher 

than the measurements in the BSDMS All dataset (fig. 7). The 

approach velocity for the measurements in the BSDMS All 

dataset is more variable. 
(n=103) (n=90) (n=272)

0

2

4

6

8

10

12

14

THIS STUDY BSDMS COARSE BSDMS ALL

APPROACH VELOCITY (

Vo), IN FEET PER SECOND

Outlier

Maximum value less

than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater

than Q25-1.5xIR

Outlier

EXPLANATION

Interquartile Range (IR)

n is the number of measurements

is the approach velocity directly

 upstream from the pier, in feet

 per second

Vo

Variable

Figure 7. Comparison of approach velocities for pier-scour measurements collected for this study with the Bridge Scour Data 

Management System (BSDMS) Coarse and All datasets.

Table 7. Comparison of bed-material size classification associated with pier-scour measurements made for this study and 

measurements from the Bridge Scour Data Measurement System (BSDMS) Coarse and All datasets.

[Abbreviations: mm, millimeter. Symbols: ≥, greater than or equal to; <, less than; D50, median particle size of the sample; --, not applicable; n=, number of data 

values]

Bed-material size classification

Number of pier-scour measurements in each bed-material size classification1

Surface bed material Shallow-subsurface bed material2

This study

(n=103)

BSDMS All dataset

(n=272)

BSDMS Coarse dataset

(n=90)

This study

(n=103)

Small cobble 26 16 16 21

Very coarse gravel 43 7 7 35

Coarse gravel 27 35 35 32

Medium gravel 7 13 13 12

Fine gravel 0 19 19 0

Very fine gravel 0 0 0 3

Sand (very fine to very coarse) 0 182 0 0

Ripple-forming sediments3 0 88 0 --

Nonripple-forming sediments4 103 184 90 --

1

Based on D50 particle size of sample.

2

No data reported in BSDMS for shallow-subsurface bed material.

3

D50 < 0.6 mm.

4

D50 ≥ 0.6 mm.

32 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
THIS STUDY BSDMS COARSE

RELATIVE SCOUR VELOCITY, (

Vo /V'c50), DIMENSIONLESS

0

1.0

2.0

3.0

4.0

5.0

6.0

(n=103)

(n=90)

Outlier

Maximum value less

than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater

than Q25-1.5xIR

Outlier

EXPLANATION

Interquartile Range (IR)

is the approach velocity directly upstream 

 from the pier, in feet per second

Vo

V'

c50

Scour

No scour

Threshold for initiating

scour of bed material

at the pier

n is the number of measurements

Variables

is the critical velocity to initiate scour of 

 bed material at the pier for the 

 D50 particle size, in feet per second

Figure 8. Comparison of relative scour velocities for pier-scour measurements collected for this study and from the Bridge Scour Data 

Management System (BSDMS) Coarse dataset.

Pier Scour in Coarse-Bed Streams in Montana 33

Relative scour velocities (Vo /V′

c50 ) determined from 

measurements collected for this study also were compared 

with the BSDMS Coarse dataset (fig. 8). The threshold for 

initiating scour of bed material at the pier (Vo /V′

c50 > 1.00) was 

exceeded to a greater degree by measurements in the BSDMS 

Coarse dataset. These higher relative scour values are likely 

due to the smaller bed-material size of measurements in the 

BSDMS Coarse dataset, compared to measurements collected 

for this study. Lower approach velocities can initiate scour at 

a pier when bed material is smaller. The median D50 and D95

bed-material size for data collected for this study are about 1.8 

and 1.6 times (respectively) larger than the D50 and D95 of the 

BSDMS Coarse dataset (fig. 9).

In addition to comparing data used for this study to the 

BSDMS All and BSDMS Coarse datasets, data collected for 

this study also were compared to historical pier-scour measurements made in Montana by USGS during large floods. 

The measurements were not part of the BSDMS database. 

Pier-scour measurements made at sites 60, 61, and 62 (fig. 1, 

table 8) in 1996–97 were selected for this comparison. Nine 

measurements made at these three sites were for peak streamflows approximating the 100-, 10-, and 5-year recurrence 

intervals, which are much greater than any recurrence intervals 

associated with measurements made for this study and provide 

an independent check on the envelope curves bounding the 

upper end of both the data collected for this study and BSDMS 

data combined.

While conducting research leading to the development of 

K4, Mueller (1996) introduced the concept of an idealized K4:

, (22)

where

y′

s is pier-scour depth from field measurements, 

in feet;

ys is pier scour computed by using the basic 

form of the HEC-18 equation without K4

applied (equation 4), in feet; and 

K4 is dimensionless. 

The idealized K4 is the relative error between measured scour 

and computed scour. Idealized K4 values were computed for 

pier-scour measurements collected for this study along with 

Mueller’s velocity-intensity ratios (equation 10). The resulting 

idealized K4 and Mueller’s velocity-intensity ratios were then 

plotted in relation to each other for the pier-scour measurements collected for this study, the BSDMS Coarse dataset, and 

historical pier-scour measurements from sites 60–62 (fig. 10). 

The dashed line (fig. 10) defines an idealized K4 according to 

HEC-18 criteria currently (2011) used (Richardson and Davis, 

2001), and represents the envelope curve under which all idealized K4 values derived from field data should plot. 

Data collected for this study included lower Mueller 

velocity-intensity ratios than the BSDMS Coarse dataset and 
Figure 9. Comparison of bed-material particle size (D50 and D95) for pier-scour measurements collected for this study with the Bridge 

Scour Data Management System (BSDMS) Coarse dataset. 

than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater

than Q25-1.5xIR

is the bed-material particle size where 

 50 percent of particles are finer by weight, 

 in millimeters or feet 

is the bed-material particle size where 

 95 percent of particles are finer by weight, 

 in millimeters or feet 

Interquartile Range (IR)

THIS STUDY BSDMS

COARSE

THIS STUDY BSDMS

COARSE

(n=103)

(n=103)

(n=90)

(n=90)

D50 PARTICLE SIZE D95 PARTICLE SIZE

BED-MATERIAL PARTICLE SIZE, IN MILLIMETERS

0

100

200

300

400

D50

D95

n is the number of measurements

Variables

Maximum value less

EXPLANATION

34 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

historical data from sites 60–62, which supports Mueller’s 

(1996) envelope curve for K4 for lower velocity-intensity ratios 

(fig. 10). Nearly all data collected for this study plot below the 

idealized K4-envelope curve; one data point plots on the curve 

and six data points for five different sites (sites 20, 37, 51, 55, 

and 57) plot slightly above the curve. Idealized K4 values plotting above the envelope curve indicate that current methods 

(2011) underpredict pier scour and that factors not described 

by current (2011) equations led to scour underprediction. This 

underpredicted scour depth had a median residual, or difference between measured and computed scour depth, of about 

0.2 ft. The most likely factor causing scour underprediction 

for the five sites where pier scour was measured for this study 

was the presence of woody debris on piers, noted as a moderate effect at four of the five sites (table 2). While this and other 

factors were not considered substantial enough to remove 

measurements from the dataset, the factors very likely contributed to the pier-scour depth measured in the field but not 

predicted by the HEC-18 equations (fig. 10). Additionally, five 

sites from the BSDMS Coarse dataset show underpredicted 

scour with a median residual of 0.41 ft (fig. 10). The five 

BSDMS sites were discussed by Mueller and Wagner (1995), 

where four of the five idealized K4 values were found to plot 

below the envelope line when the D50 and D95 variables used in 

the prediction equations were derived from composite samples 

of bed-material obtained at each site.

Mueller and Wagner (2005) proposed a different equation 

for computing K4 (equation 20), which related K4 and relative bed-material size. When plotted, the equation defines an 

envelope curve that represents the maximum idealized K4 for a 

given bed-material size. Idealized K4 and relative bed-material 

sizes for the pier-scour measurements collected for this study, 

measurements from the BSDMS Coarse dataset, and historical pier-scour measurements from sites 60–62 were plotted 

along with the equation Mueller and Wagner (2005) proposed 

(fig. 11). All data plot below the K4-envelope curve proposed 

by Mueller and Wagner (2005), indicating that scour was not 

underpredicted. The lower end of the relative bed-material size 

scale is mostly populated with idealized K4 values computed 

from data collected for this study because of the larger bedmaterial sizes involved (fig. 11). The relation between idealized K4

 and relative bed-material size are similar for data collected for this study, the BSDMS Coarse dataset, and historical 

data for sites 60–62 and therefore, all data support the Mueller 

and Wagner (2005) envelope curve because pier scour was 

always overpredicted. 

The maximum value of relative pier scour (y′

s /b) 

obtained in the study was 1.00 (table 5), and the maximum 

values observed in the two BSDMS datasets (All, Coarse) 

were 2.09 and 1.25, respectively. Research indicates that 

maximum scour generally does not exceed about 2.4 times 

the effective pier width (y′

s = 2.4b) for measurements with 

Froude numbers less than about 0.8 (Richardson and Davis, 

2001), which is supported by the data collected for this study 

and BSDMS datasets. The largest maximum of 2.09 (BSDMS 

All dataset) includes data for sites with sand bed material, 

whereas the lower maxima (1.00 and 1.25) include data collected for this study and from the BSDMS Coarse dataset for 
Table 8. Selected historical pier-scour measurements (1996–97) used in this study1.

[Abbreviations: DMS, degrees, minutes, seconds; USGS, U.S. Geological Survey. Symbols: ≈ , approximately; n, number of measurements]

Site

number

(fig. 1)

Site name Description Major river basin

Drainage 

area above 

bridge, 

in square 

miles

Bridgestructure

number2

Total 

length of 

bridge, 

in feet

Latitude, 

in DMS

Longitude, 

in DMS

Year of 

pier-scour 

measurement

Number 

of piers 

measured

Number of 

pier-scour 

measurements 

(n)

Recurrence 

interval, 

in years3

60 Yellowstone 

River southwest of Emigrant

U.S. Highway 89 

about 11 miles southwest of Emigrant

Yellowstone 2,810 P00011 

020+04171

455 45°22′04′′ 110°43′36′′ 1996, 1997 3 6 ≈ 100

61 Clarks Fork 

Yellowstone near Bridger 

U.S. Highway 310 

about 2 miles south of Bridger

Yellowstone 1,809 P00004 

023+07461

289 45°15′43′′ 108°54′38′′ 1997 2 2 ≈ 10

62 Bitterroot 

River near Darby

U.S. Highway 93 

about 4 miles southeast of Darby at USGS streamflowgaging station 12344000

Columbia 1,049 P00007 

026+04491

250 45°58′20′′ 114°08′26′′ 1996 1 1 ≈ 5

1Pier-scour measurements (n=9) associated with peak streamflows of greater recurrence interval than data collected for this study.2From Montana Department of Transportation (1994).3Recurrence interval is for peak streamflow associated with the scour measurement.

Pier Scour in Coarse-Bed Streams in Montana 35
Figure 10. Comparison of idealized K4

 to Mueller velocity-intensity ratios for pier-scour measurements collected for this study, 

historical pier-scour measurements from Montana, and pier-scour measurements in the Bridge Scour Data Management System

(BSDMS).

Figure 11. Comparison of idealized K4 to relative bed-material size for pier-scour measurements collected for this study, historical pierscour measurements from Montana, and pier-scour measurements in the Bridge Scour Data Management System (BSDMS) Coarse 

dataset.

Data from sites 60–62 for 1996–97 (n=9) associated with 

 peak streamflows of greater recurrence interval than 

 data collected for this study

1

5

0.1

0.05

Data collected for this study (n=103)

MUELLER VELOCITY-INTENSITY RATIO , DIMENSIONLESS

V V' o c50

Vc50 V'C95

IDEALIZED K4 , DIMENSIONLESS y's ys

EXPLANATION

y' is pier-scour depth from field measurements, in feet s

is pier-scour depth computed by an equation, in feet

is the approach velocity directly upstream from 

 the pier, in feet per second 

Vo

is the critical velocity for incipient motion of bed material 

 upstream of the bridge for the D50 particle size, in feet 

 per second

Vc50

is the critical velocity to initiate scour of bed material at 

 the pier for the D50 particle size, in feet per second 

V'c50

is the critical velocity to initiate scour of bed material at 

 the pier for the D95 particle size, in feet per second 

V'c95

y

s

is the correction factor for the coarse bed-material 

 armoring effect on pier-scour depth, dimensionless 

K4

0.01 0.1 1 10 100 1,000

Data from BSDMS Coarse dataset (n=90)

n is the number of measurements

Variables

Envelope curve for K4 as proposed by 

Mueller (1996) and applied in current 

HEC-18 criteria (Richardson and 

Davis, 2001)

1 10 100 1,000

0.05

0.1

1

 5

RELATIVE BED-MATERIAL SIZE (b/D ), DIMENSIONLESS 50

Envelope curve proposed by 

Mueller and Wagner (2005)

K4 = 0.35 (b/D50)

0.19

EXPLANATION

is the correction factor for the coarse bed-material armoring 

 effect on pier-scour depth, dimensionless 

K4

is pier-scour depth from field measurements, in feet 

is the pier-scour depth computed by an equation, in feet 

is effective width of the pier measured perpendicular to the 

 approach flow, in feet 

b

is the bed-material particle size where 50 percent of 

 particles are finer by weight, in millimeters or feet 

D50

y's

y

s

Data collected for this study (n=103)

Data from BSDMS Coarse dataset (n=90)

Data from sites 60–62 for 1996–97 (n=9) associated with peak 

 streamflows of greater recurrence interval than data 

 collected for this study

IDEALIZED K4 , DIMENSIONLESS y's ys

n is the number of measurements

Variables

36 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Pier Scour in Coarse-Bed Streams in Montana 37

sites characterized by much coarser bed material. These results 

reaffirm the general finding that pier scour is inversely related 

to bed-material size.

Comparisons of Measured and Computed Scour 

The most reliable pier-scour equation minimizes underpredicted scour depth (positive residual) as the principal 

criterion but also does not unreasonably overpredict scour 

depth (negative residual). Scour depth was computed for the 

measurements collected in this study using the basic HEC-18 

equation and five pier-scour equations described previously 

that use the K4 correction factor for the armoring effect of 

coarse-bed material: 

1. The basic HEC-18 equation with no K4 factor (equation 3 or 4). This is the original equation from which 

other equations have evolved and is the equation 

(equation 4) used in comparisons involving the idealized K4 (equation 22). 

2. The HEC-18 equation (equation 5), with K4 determined by equation 11 as proposed by Mueller (1996, 

where 0.4 ≤ K4 ≤ 1.0) and used by Richardson and 

Davis (2001). This equation is currently (2011) recommended by the FHWA and is referred to hereinafter as the HEC-18-K4

Mu equation. 

3. The HEC-18-K4Mu equation (equation 5) with the 

current (2011) FHWA constraint (0.4 ≤ K4 ≤ 1.0) 

removed and 0.0 ≤ K4 ≤ 1.0 is applied instead. Values 

of K4 determined by equation 11 to be less than 0.4 

are, therefore, used in equation 5. This version of 

equation 5 is referred to hereinafter as the HEC-18-

K4Mu′ equation.

4. The equation developed using nonuniform sediment 

mixtures (equation 13) by Molinas (2003), referred 

to hereinafter as the Molinas (2003) equation. 

5. A variation of the HEC-18 equation (equation 19) 

proposed by Molinas (2003), with K4 determined by 

equation 17 and referred to hereinafter as the HEC18-K4

Mo equation.

6. The HEC-18 equation (equation 5), with K4 determined by equation 20 as proposed by Mueller and 

Wagner (2005) and referred to hereinafter as the 

HEC-18-K4MW equation.

Computed scour depths were compared to measured 

scour depths using scatter plots (fig. 12) and boxplots of 

residuals (fig. 13) to evaluate the reliability of the equations to 

predict scour. Additionally, positive and negative residuals of 

scour were computed and reported here because of the implication each type of residual separately has to bridge foundation design. The evaluation showed:

• The basic HEC-18 equation (fig. 12A and fig. 13) did 

not underpredict scour depth at any site, but scour 

depth was substantially overpredicted (median and 

maximum overpredicted residuals of about -4.0 ft and 

-12.4 ft, respectively). 

• The HEC-18-K4Mu equation (fig. 12B and fig. 13) 

underpredicted scour depth six times at five different 

sites (median and maximum underpredicted residuals 

of about 0.2 ft and 0.5 ft, respectively) and moderately 

overpredicted scour depth (median and maximum 

overpredicted residuals of about -0.9 ft and -4.4 ft, 

respectively). 

• The HEC-18-K4Mu′ equation (fig. 12C and fig. 13) 

underpredicted scour depth 12 times at 9 different sites 

(median and maximum underpredicted residuals of 

about 0.3 ft and 0.8 ft, respectively) and moderately 

overpredicted scour depth (median and maximum 

overpredicted residuals of about -0.7 ft and -4.4 ft, 

respectively) similar to the HEC-18-K4Mu equation. 

• The Molinas (2003) equation (fig. 12D and fig. 13) 

underpredicted scour depth 35 times at 23 different 

sites (median and maximum underpredicted residuals 

of about 0.4 ft and 2.0 ft, respectively) and moderately 

overpredicted scour depth (median and maximum 

overpredicted residuals of about -0.6 ft and -2.8 ft, 

respectively) similar to the HEC-18-K4Mu and HEC18-K4

Mu′ equations. 

• The HEC-18-K4Mo equation (fig. 12E and fig. 13) 

slightly underpredicted scour depth at three sites 

(median and maximum underpredicted residuals of 

about 0.1 ft and 0.2 ft, respectively), but overpredicted scour depth more substantially (median and 

maximum overpredicted residuals of about -2.0 ft and 

-8.6 ft, respectively) compared to the HEC-18-K4Mu, 

HEC-18-K4Mu′, and Molinas (2003) equations. 

• The HEC-18-K4MW equation (fig. 12F and fig. 13) 

did not underpredict scour depth at any site, but scour 

depth was overpredicted (median and maximum overpredicted residuals of about -1.9 ft and -9.9 ft, respectively) similar to the HEC-18-K4

Mo equation. 

Finally, boxplots summarizing residuals for the six equations (fig. 13) are consistent with the conclusion of Chase and 

Holnbeck (2004), which indicated that the HEC-18-K4Mu 

equation generally predicted scour depth in closer agreement 

to measured scour than the other equations used for computing 

pier-scour depth in coarse-bed streams. Furthermore, the magnitude of the residuals of underpredicted and overpredicted 

scour depth was the lowest using the HEC-18-K4Mu equation.
Figure 12. Comparison of measured with computed pier-scour depths for data collected for this study using selected pier-scour 

equations. A, Using the basic HEC-18 equation without K4. B, Using the HEC-18-K4Mu equation, where 0.4 ≤ K4 ≤ 1.0. C, Using the 

HEC-18-K4Mu′ equation, where 0.0 ≤ K4 ≤ 1.0. D, Using the Molinas (2003) equation. E, Using the HEC-18-K4Mo equation. 

F, Using the HEC-18-K4MW equation.

MEASURED PIER-SCOUR DEPTH (y's), IN FEET

COMPUTED PIER-SCOUR DEPTH (ys), IN FEET

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

ys equals y's

HEC-18 (no K ) 4

A Overpredicted

Underpredicted

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Overpredicted

Underpredicted

B

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Overpredicted

Underpredicted

D

Molinas (2003)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Overpredicted

Underpredicted

F

HEC-18- K MW 4

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Overpredicted

Underpredicted

E

HEC-18-K Mo 4

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Overpredicted

Underpredicted

C

12 measurements with

10 feet < ys < 20 feet 

were not plotted

ys equals y's ys equals y's

ys equals y's ys equals y's

ys equals y's

COMPUTED PIER-SCOUR DEPTH (ys), IN FEET

COMPUTED PIER-SCOUR DEPTH (ys), IN FEETCOMPUTED PIER-SCOUR DEPTH (ys), IN FEET

COMPUTED PIER-SCOUR DEPTH (ys), IN FEETCOMPUTED PIER-SCOUR DEPTH (ys), IN FEET

EXPLANATION

MEASURED PIER-SCOUR DEPTH (y's), IN FEET

MEASURED PIER-SCOUR DEPTH (y's), IN FEET

MEASURED PIER-SCOUR DEPTH (y's), IN FEET

MEASURED PIER-SCOUR DEPTH (y's), IN FEET

MEASURED PIER-SCOUR DEPTH (y's), IN FEET

< is less than

> is greater than

≤ is less than or equal to K4

5 measurements with

10 feet < ys < 20 feet 

were not plotted

6 measurements with

10 feet < ys < 20 feet 

were not plotted

HEC-18 -K4Mu

(0.4 ≤ K4 ≤ 1.0)

HEC-18-K4 Mu'

(0.0 ≤ K4 ≤ 1.0)

is the correction factor for the coarse bed-material armoring 

 effect on pier-scour depth, dimensionless

y’s is pier-scour depth from field measurements, in feet

ys is the pier-scour depth computed by an equation, in feet

38 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
K4 is the correction factor for the 

coarse bed-material armoring effect on pier-scour 

depth, dimensionless 

RESIDUAL PIER-SCOUR DEPTH, IN FEET

–15

–10

–5

0

5

Underpredicted

Overpredicted

(n=103 for all boxplots)

Outlier

Maximum value less than 

 Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater than 

 Q25-1.5xIR

Outlier

is the number of measurements

Residual pier-scour depth is measured 

pier-scour depth minus computed 

pier-scour depth. 

EXPLANATION

Interquartile Range (IR)

Molinas

(2003)

Basic HEC-18 HEC-18-K4Mo HEC-18-K4MW

(no K4

)

HEC-18-K4

Mu

(0.4 ≤ K4 ≤ 1.0)

HEC-18-K4

Mu′

(0.0 ≤ K4 ≤ 1.0)

Variables

n

Figure 13. Comparison of residual pier-scour depth for six pier-scour equations evaluated for this study.

Pier Scour in Coarse-Bed Streams in Montana 39

Effects of Bed-Material Characteristics on Pier 

Scour 

Bed-material data were collected for the surface or armor 

layer as part of each measurement. In addition, similar data 

also were collected for bed material in the shallow-subsurface 

bed material immediately below the surface layer. The 

shallow-subsurface layer included only that part of the streambed where scour and depositional processes are active (active 

layer). These detailed bed-material data for each layer (table 9) 

were used to assess the effects of bed-material characteristics 

on pier scour in coarse-bed streams in Montana.

Assessment of the effects of bed-material characteristics 

on pier scour began with a detailed examination of the relation of bed-material size and gradation of the surface layer to 

relative pier scour. This effort largely involved applying the 

work of previous researchers to the pier-scour measurements 

made on coarse-bed streams for this study. Next, the effects of 

shallow-subsurface bed-material characteristics on pier scour 

were assessed and compared to the effects of the surface bedmaterial characteristics. In particular, the differences in pierscour prediction caused by using bed-material characteristics 

of one layer or the other were examined. Finally, the effects 

of differences in bed-material size and gradation between the 

layers on pier scour were each assessed separately and then 

the combined effects of differences in bed-material size and 

gradation between the two layers were assessed. 

Effects of Surface Bed-Material Characteristics

Early bridge scour studies did not consider surface 

bed-material characteristics, but in the mid–1960s researchers began to note that surface bed-material size had an effect 

on pier-scour depth (Shen and others, 1966). Later studies 

showed that the gradation of the surface bed material also 

affected pier-scour depth (Ettema 1976, 1980; Chiew, 1984; 

Baker, 1986; and Abdou, 1993). The effect of bed-material 

size and gradation on scour depth was investigated using 

data collected for this study, the BSDMS All dataset, and the 

BSDMS Coarse dataset to confirm and extend the findings of 

previous studies. Data from these three datasets were compared to previously published relations of relative pier scour 

(y′

s /b) and relative velocity (Vo /Vc50). Relative velocity was 

computed based on the critical velocity for incipient motion 

of bed material upstream from the bridge (Vc50). Comparisons 

also were made using relative velocity computed based on 

the critical velocity to initiate scour of bed material at the pier 

(V′

c50). This was considered appropriate because most pierscour data collected for this study reflect clear-water scour 

conditions, where incipient motion of the bed material and 

substantial sediment supply to the scour hole was not a factor. 

Recognizing the relation between bed-material characteristics and pier-scour depth, researchers have continued to 

improve scour-prediction equations by incorporating mathematical expressions that attempt to account for bed-material 
Table 9. Particle-size distribution and gradation coefficient for surface and shallow-subsurface bed-material data collected for this 

study.

[All pier-scour measurement data were obtained for bed material that was noncohesive. Symbols: Di, bed-material particle size where i percent of particles are 

finer by weight, in millimeters; σg,s, gradation coefficient of the surface bed material, dimensionless; σg,ss, gradation coefficient of the shallow-subsurface bed 

material, dimensionless]

Site 

number

(fig.1)

Site name

Surface bed material 

D16 D35 D50 D84 D85 D90 D95 D99 σg,s 

Missouri River Basin 

1 Big Hole River east 

of Wise River

40.5 73.3 102 176 179 209 263 309 2.09 

2 Big Hole River west 

of Divide

23.7 51.9 71.3 155 159 177 190 200 2.56 

3 Beaverhead River 

north of Dillon

20.3 28.8 35.8 59.2 60.1 66.2 80.3 103 1.71 

4 Big Hole River near 

Melrose

28.2 42.7 59.5 104 107 123 157 192 1.92 

5 Boulder River near 

Basin

29.1 44.4 58.0 111 114 127 163 284 1.96 

6 Wisconsin Creek near 

Sheridan

23.2 36.0 47.8 78.8 80.2 87.2 104 123 1.84 

7 Gallatin River 

northwest of West 

Yellowstone

32.7 51.2 64.7 117 118 126 167 272 1.89 

8 Big Hole River 

southwest of Twin 

Bridges

20.8 30.6 38.5 60.9 61.6 68.2 80.1 89.6 1.71 

9 Jefferson River north 

of Twin Bridges

28.8 44.9 60.7 114 116 125 148 174 1.99 

10 Madison River south 

of Cameron

29.9 56.7 79.8 149 155 206 253 291 2.23 

11 Boulder River at I-90, 

near Cardwell

2.58 10.3 17.1 44.1 45.8 61.8 82.9 188 4.14 

12 Boulder River near 

Cardwell

2.58 10.3 17.1 44.1 45.8 61.8 82.9 188 4.14 

13 South Boulder River 

near Cardwell

29.3 43.2 55.2 111 114 129 155 175 1.95 

14 South Willow Creek 

near Harrison

8.76 29.1 38.3 77.3 79.0 87.0 109 151 2.97 

15 Gallatin River west 

of Bozeman

45.5 58.5 70.4 109 112 127 143 179 1.55 

16 Jefferson River west 

of Three Forks

5.91 12.4 22.3 57.0 59.3 73.6 89.6 126 3.11 

17 Gallatin River near 

Logan

17.8 26.7 33.3 57.4 58.3 63.9 76.0 102 1.80 

18 Jefferson River north 

of Three Forks

8.85 16.1 19.1 31.1 31.5 38.2 47.5 63.7 1.87 

19 Madison River near 

Three Forks

19.7 25.3 29.8 43.2 43.7 48.2 56.8 63.7 1.48 

20 Missouri River near 

Townsend

11.9 19.9 29.7 64.1 66.2 76.8 87.4 118 2.32 

21 Sun River north of 

Augusta

26.8 45.7 57.4 111 114 128 156 179 2.03 

40 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Shallow-subsurface bed material

D16 D35 D50 D84 D85 D90 D95 D99 σg,ss

Missouri River Basin

11.8 39.1 66.4 139 142 154 167 177 3.44

7.29 31.2 57.3 109 110 116 122 127 3.86

3.26 12.9 21.8 51.8 53.2 60.4 72.6 86.5 3.99

4.95 18.7 47.5 104 106 113 121 127 4.59

4.35 22.4 40.4 74.4 75.4 80.3 85.1 89.0 4.14

1.97 17.3 31.1 71.8 73.0 78.6 84.3 88.9 6.03

6.06 24.5 41.3 73.8 74.9 79.9 85.0 89.0 3.49

6.43 19.6 25.5 48.5 49.4 54.3 59.1 63.0 2.75

14.3 38.9 57.7 86.6 87.4 95.7 112 125 2.46

22.7 93.8 119 161 163 168 174 179 2.66

1.03 1.66 3.64 27.6 28.5 33.5 41.4 57.0 5.18

1.03 1.66 3.64 27.6 28.5 33.5 41.4 57.0 5.18

6.89 29.5 53.8 108 109 116 122 127 3.96

1.87 12.1 32.8 105 106 114 121 127 7.49

6.05 28.1 58.4 147 149 159 170 178 4.93

2.77 10.4 16.0 36.6 38.0 45.0 54.5 62.1 3.64

9.24 25.2 35.4 67.4 68.9 75.9 83.0 88.6 2.70

1.86 5.44 9.92 24.9 25.9 30.9 40.1 56.9 3.66

1.91 14.5 19.0 35.0 36.3 42.8 52.6 61.7 4.28

1.85 9.39 18.2 40.9 41.7 45.6 54.8 62.2 4.71

2.62 12.4 32.4 153 154 163 171 178 7.63

Pier Scour in Coarse-Bed Streams in Montana 41
Site 

number

(fig.1)

Site name

Surface bed material 

D16 D35 D50 D84 D85 D90 D95 D99 σg,s 

Missouri River Basin—Continued 

22 Marias River near 

Shelby

7.36 17.2 22.7 44.0 45.0 52.1 59.3 74.4 2.44 

23 Smith River southeast 

of Ulm 

6.67 10.4 13.6 26.7 27.5 31.2 41.5 58.7 2.00 

24 Smith River at old 

truss bridge, southeast of Ulm

19.8 30.9 39.2 86.0 89.7 106 123 165 2.08 

25 Belt Creek 7 miles 

south of Belt

15.1 29.6 44.6 88.8 90.2 108 125 168 2.42 

26 Belt Creek 6 miles 

south of Belt

14.5 28.1 38.0 75.1 77.0 86.5 105 123 2.28 

27 Belt Creek 2 miles 

southeast of Belt

23.1 31.3 39.5 76.3 78.2 87.9 109 127 1.82 

Yellowstone River Basin 

28 Yellowstone River at 

Emigrant

14.6 25.0 33.2 67.6 69.3 78.9 94.4 110 2.15 

29 Shields River near 

Wilsall

24.8 40.2 50.6 88.0 89.6 102 115 125 1.88 

30 Mill Creek near Pray1 32.2 51.2 67.4 130 133 149 164 177 2.01 

31 Yellowstone River 

near Pray1

41.5 71.5 92.1 133 136 150 165 177 1.79 

32 Yellowstone River 

near Pine Creek

10.9 32.2 54.5 114 116 126 174 244 3.23 

33 Yellowstone River 

south of Livingston

24.1 44.5 55.5 88.8 89.9 103 117 128 1.92 

34 Boulder River near 

McLeod 

31.9 42.9 55.6 108 110 119 128 178 1.84 

35 Boulder River south 

of Big Timber

43.2 61.3 76.6 125 127 148 171 269 1.71 

36 Boulder River at I-90 

near Big Timber 

46.9 67.7 84.6 136 140 164 196 266 1.70 

Columbia River Basin 

37 Little Blackfoot River 

near Avon

13.2 23.2 29.2 55.2 56.4 62.6 77.9 119 2.06 

38 Blackfoot River (old 

bridge) west of 

Lincoln1

1.68 4.31 9.63 26.8 27.6 31.6 38.2 43.6 4.00 

39 Blackfoot River (new 

bridge) west of 

Lincoln1

1.68 4.31 9.63 26.8 27.6 31.6 38.2 43.6 4.00 

40 Little Blackfoot River 

near Garrison

14.6 42.6 57.4 98.1 101 114 127 169 2.59 

Table 9. Particle-size distribution and gradation coefficient for surface and shallow-subsurface bed-material data collected for this 

study.—Continued

[All pier-scour measurement data were obtained for bed material that was noncohesive. Symbols: Di, bed-material particle size where i percent of particles are 

finer by weight, in millimeters; σg,s, gradation coefficient of the surface bed material, dimensionless; σg,ss, gradation coefficient of the shallow-subsurface bed 

material, dimensionless]

42 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Shallow-subsurface bed material

D16 D35 D50 D84 D85 D90 D95 D99 σg,ss

Missouri River Basin—Continued

4.54 14.9 25.3 65.2 66.8 74.5 82.3 88.5 3.79

1.59 3.99 9.10 30.6 31.6 37.9 44.3 59.8 4.39

10.4 20.1 29.0 55.0 56.0 60.8 70.4 86.1 2.30

1.98 7.77 18.0 63.8 65.3 73.6 81.8 88.4 5.68

1.70 14.6 27.2 78.5 79.2 82.8 86.4 89.3 6.79

2.00 9.41 20.0 67.3 69.0 75.8 82.9 88.6 5.80

Yellowstone River Basin

1.94 15.3 27.9 60.4 61.2 67.7 78.9 87.8 5.57

2.30 10.3 19.3 68.8 70.3 78.7 87.3 94.3 5.47

32.2 51.0 67.4 130 133 149 164 177 2.01

41.5 72.0 92.1 133 136 150 165 177 1.79

5.74 22.5 59.1 151 152 162 171 178 5.13

7.06 22.5 42.2 147 149 159 170 178 4.56

8.86 48.5 67.8 110 111 117 122 127 3.52

7.11 38.0 66.4 104 106 113 121 127 3.83

13.8 56.8 101 158 159 166 173 179 3.38

Columbia River Basin

1.99 6.80 14.8 46.0 47.5 54.9 62.3 83.3 4.81

1.68 4.31 9.63 26.8 27.6 31.6 38.2 43.6 4.00

1.68 4.31 9.63 26.8 27.6 31.6 38.2 43.6 4.00

9.39 20.1 31.3 58.1 58.8 62.4 73.3 86.7 2.49

Pier Scour in Coarse-Bed Streams in Montana 43
Site 

number

(fig.1)

Site name

Surface bed material 

D16 D35 D50 D84 D85 D90 D95 D99 σg,s 

Columbia River Basin—Continued 

41 Clark Fork near Gold 

Creek

8.62 19.8 30.9 79.8 82.1 97.2 159 165 3.04 

42 North Fork Blackfoot River west of 

Lincoln

22.4 29.8 36.6 60.6 61.5 69.7 82.2 108 1.65 

43 North Fork Blackfoot 

River near Ovando

25.9 39.6 50.1 93.7 96.8 112 128 169 1.90 

44 Clark Fork south of 

Drummond

5.46 14.4 34.4 86.4 87.9 102 118 153 3.98 

45 Lost Horse Creek 

north of Darby

24.1 52.0 73.0 122 123 139 168 264 2.26 

46 Clark Fork near 

Drummond

15.8 21.8 27.8 43.6 44.0 51.4 62.0 83.6 1.66 

47 Clark Fork at Turah 

Bridge

20.9 37.1 49.3 87.1 88.3 103 122 165 2.04 

48 Swan River east of 

Ferndale 

34.8 57.6 72.9 115 117 124 146 173 1.82 

49 Clark Fork at Missoula28.5 47.2 67.3 123 125 144 168 197 2.08 

50 Clark Fork near 

Alberton

18.1 29.7 46.1 95.4 98.9 116 140 172 2.30 

51 Clark Fork near 

Superior

20.0 36.9 53.4 105 108 121 144 174 2.29 

52 St. Regis River at 

Ward Creek

22.7 35.5 50.7 88.6 89.8 105 121 161 1.97 

53 St. Regis River west 

of St. Regis

27.2 35.8 41.0 69.6 71.1 78.8 86.4 114 1.60 

54 St. Regis River at St. 

Regis

12.7 19.5 24.5 50.0 52.2 63.2 84.8 127 1.99 

55 Flathead River near 

Perma

16.4 21.4 27.6 54.7 56.3 64.3 81.6 125 1.82 

56 Clark Fork above 

Flathead River, 

near Paradise

16.2 38.4 54.9 115 117 128 159 240 2.67 

57 Clark Fork below 

Flathead River, 

near Paradise

18.7 25.5 31.3 51.1 52.4 58.8 69.4 102 1.65 

58 Kootenai River near 

Libby

27.4 45.8 58.0 93.0 96.0 111 126 168 1.84 

59 Yaak River near Troy 46.1 80.6 108 189 196 231 265 293 2.02 

1

The particle size of the shallow-subsurface bed material was estimated in the field to be essentially the same as the surface bed material.

Table 9. Particle-size distribution and gradation coefficient for surface and shallow-subsurface bed-material data collected for this 

study.—Continued

[All pier-scour measurement data were obtained for bed material that was noncohesive. Symbols: Di, bed-material particle size where i percent of particles are 

finer by weight, in millimeters; σg,s, gradation coefficient of the surface bed material, dimensionless; σg,ss, gradation coefficient of the shallow-subsurface bed 

material, dimensionless]

44 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Shallow-subsurface bed material

D16 D35 D50 D84 D85 D90 D95 D99 σg,ss

Columbia River Basin—Continued

1.92 17.5 31.5 72.2 73.3 78.8 84.4 88.9 6.13

1.49 5.08 13.2 44.0 45.1 51.4 57.7 62.7 5.44

3.30 10.7 18.5 61.9 63.9 72.6 81.3 88.3 4.33

4.25 23.4 40.5 76.4 77.3 81.5 85.8 89.2 4.24

11.4 46.4 69.1 100 102 111 119 126 2.96

4.69 13.2 19.3 45.9 47.0 52.7 58.3 62.9 3.13

3.20 20.8 33.7 62.2 62.9 70.8 80.4 88.1 4.41

8.60 30.6 52.7 107 108 115 121 127 3.52

4.09 18.9 36.0 136 138 152 166 177 5.76

2.40 10.7 21.8 76.8 78.8 88.8 108 124 5.66

5.31 21.0 52.3 97.1 99.1 109 118 126 4.27

25.0 62.4 91.1 116 117 121 124 127 2.16

1.90 7.20 15.2 36.9 37.4 40.0 42.5 44.5 4.41

2.49 16.3 32.9 88.0 89.7 102 115 125 5.93

5.17 16.4 26.9 72.1 73.2 78.8 84.4 88.9 3.73

5.95 20.8 39.1 77.6 78.3 82.2 86.1 89.2 3.61

1.78 13.0 17.4 38.5 39.5 44.3 76.0 87.2 4.65

12.1 36.0 66.4 86.8 87.4 92.4 110 124 2.68

17.9 47.9 73.1 111 112 117 123 127 2.49

Pier Scour in Coarse-Bed Streams in Montana 45
46 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

effects, as cited in the section “Development of Pier-Scour 

Equations for Coarse-Bed Streams.” Some of these improvements were based on laboratory studies and some were based 

on field studies. Many of the equations developed or improved 

based on laboratory studies are not able to successfully predict 

pier-scour depth in natural streams with highly variable bedmaterial characteristics. Even equations developed based on 

field data are not able to successfully predict pier scour under 

all field conditions. For example, the most recent (2011) 

HEC-18 equation (equation 5) incorporates a correction factor 

for coarse bed streams (K4 ). K4 is calculated using a velocityintensity ratio (equation 10) that is dependent on the D50 and 

D95 particle sizes. Under most natural conditions, the D95

moderately exceeds the D50, the velocity-intensity ratio and 

K4 can be calculated, and the resulting scour predictions are 

acceptable. Mueller and Jones (1999, p. 306) and Richardson 

and Davis (2001, p. 6.6) reported that the configuration of the 

denominator in equation 10 can produce anomalous results 

(negative velocity ratio) for rare combinations of D50 and D95. 

None of the surface bed-material particle-size data associated 

with pier-scour measurements collected for this study produced anomalous results when used in equation 10. To better 

understand conditions where equation 10 produced anomalous 

results, selected pier-scour data collected for this study were 

used in equation 10 and manipulated (increased or decreased) 

until anomalous results were achieved. Anomalous results 

were produced when D95 substantially exceeded D50 and when 

D95 was similar in magnitude to D50 for certain D50 and D95 size 

combinations. Because the solution of equation 10 depends on 

D95 being larger in some modest proportion to D50, the surface 

bed-material data from this study were analyzed to statistically 

determine the relation between the D50 and D95 particle sizes 

based on regression analysis (Helsel and Hirsch, 2002) and 

evaluation of r 2 and p. The coefficient of determination (r 2) 

has a value that ranges from 0 to 1 and the strength of the relation increases as the value of r 2 approaches 1. The p-value is 

the probability of obtaining a result that is more extreme than 

a given value, where a p–value less than 0.05 indicates statistical significance (significance level) of the regression relation. 

Based on the regression analysis, the D50 and D95 particle sizes 

have a positive relation with each other that is statistically significant (r 

2

 = 0.73 and p < 0.001). In other words, for the data 

collected for this study, sites that have a larger D50 particle size 

also tend to have a larger D95 particle size. This observation is 

similar to the observation of Mueller and Jones (1999) about 

the bed-material data originally used to develop the equation 

for the K4 correction factor.

Bed-material size is an important characteristic that determines the magnitude of the velocity required to cause pier 

scour. The effect of bed-material size on pier scour typically 

is investigated in conjunction with hydraulic characteristics 

like velocity and flow depth because these characteristics 

also influence the mobility of bed material. Relative velocity 

(Vo /Vc50) incorporates all three of these characteristics. Previous researchers have examined the relation between relative 

velocity and relative pier scour and formulated envelope 

curves. Melville (1984) and Chiew (1984) found the first peak 

in the envelope curve to be on the threshold between clearwater and live-bed scour (Vo /Vc50 = 1.00) and a second live-bed 

scour peak became asymptotic (approaching a constant value) 

with increasing relative velocity and live-bed scour conditions. They also showed that the magnitude of the second peak 

was less than the first peak for nonripple-forming sediments 

(D50 ≥ 0.6 mm), and the second peak was greater than the first 

peak for ripple-forming sediments (D50 < 0.6 mm). 

The relation between relative pier scour and relative 

velocity defined by an envelope curve for the data collected 

for this study and the BSDMS Coarse dataset (fig. 14A) were 

compared to relations observed in laboratory data by Melville (1984) and Chiew (1984). The envelope curve for data 

used for this study displays an earlier peak with a decrease 

in relative pier scour followed by an asymptotic rise. Interestingly, the first peak (fig. 14A) in the curve occurs at a 

lower relative velocity (Vo /Vc50 ≈ 0.75) than the clear-water 

and live-bed threshold (Vo /Vc50 = 1.00). Also, the magnitude 

of relative pier scour (maximum of 1.25) is much less than 

the published value of 2.4 obtained from laboratory data for 

cylindrical piers (Melville and Sutherland, 1988). Furthermore, equilibrium live-bed scour is about 18 percent shallower 

than the maximum pier scour under clear-water conditions, 

a larger reduction than the 10 percent indicated by Richardson and Davis (1995, p. 16). Montana historical pier-scour 

data from 1996–97 (fig. 14A) plot below or on the envelope 

curve, indicating that the curve developed from scour data for 

flows approximating bankfull discharge (≤ 2-year recurrence 

interval) also envelopes data for more severe hydrologic and 

hydraulic conditions with recurrence intervals up to about 

100 years (table 8).

As noted, the envelope curves that bound the pier-scour 

data collected for this study (fig. 14A), and in particular the 

magnitude of the maximum relative pier scour and the relative 

velocity associated with that maximum, were different than 

expected based on envelope-curve relations from previous 

studies. Some of these differences between data and relations 

from past studies and data and relations from this study could 

be mostly reflections of the paucity of data defining the y′

s /b 

maxima (fig. 14A), with the true maxima being slightly greater 

and corresponding relative velocity (Vo /Vc50) closer to the 

theoretical clear-water and live-bed scour threshold. However, 

differences in the bed-material characteristics and flow conditions between data collected for this study and earlier studies 

are a more likely explanation. First, the sizes of bed-material 

associated with pier-scour data collected for this study and 

included in the BSDMS Coarse dataset are much larger 

than bed-material sizes associated with pier-scour data from 

research (Melville and Sutherland, 1988) that documented 

higher maximum relative pier scour values approaching 2.4. 

Less pier scour would be expected at sites with coarser bed 

material and associated armoring. In addition, hydraulic 

conditions encountered in the field that cause scour of bed 

material at the pier typically involve variable streamflows over 

some duration (unsteady flow), compared to constant flow 
Clear-water scour Live-bed scour

RELATIVE VELOCITY ( Vo /Vc50 ), DIMENSIONLESS

RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS

Data collected for this study (n=103) 

Data from the Bridge Scour Data Management 

 System (BSDMS) Coarse dataset (n=90)

Data for sites 60-62 for 1996-97 (n=9) associated 

 with peak streamflows of greater recurrence 

 interval than data collected for this study

Envelope curve by Melville (1984) 

and Chiew (1984) for non-ripple

forming sand-size particles

0.25 1.00 1.50 2.00 2.50 3.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Envelope curve for coarse bed-material data used for this study

0.50 0.75 1.25 1.75 2.25 2.75 3.25

RELATIVE VELOCITY (Vo /V′

c50), DIMENSIONLESS

RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS

Threshold for incipient motion of bed 

material upstream from the bridge

Threshold for initiation of scour 

of bed material at the pier

EXPLANATION

n

Threshold for incipient motion of bed 

material upstream from the bridge

Clear-water scour Live-bed scour

is the number of measurements

y's

b

 Vo

Vc50

V'c50

is the critical velocity for incipient motion of bed 

 material upstream of the bridge for the 

 D50 particle size, in feet per second

is pier-scour depth from field measurements, 

 in feet

is the effective width of the pier measured 

 perpendicular to the approach flow, in feet

is the approach velocity directly upstream from 

 the pier, in feet per second

is the critical velocity to initiate scour of bed 

 material at the pier for the D50 particle size, 

 in feet per second

Variables

Envelope curve for coarse bed-material 

data used for this study

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

0.00

0.25

0.50

0.75

1.00

1.25

1.50

A

B

Figure 14. Relation between relative pier scour and: A, relative velocity based on critical velocity for incipient motion of bed 

material upstream from the bridge; and B, relative velocity based on critical velocity to initiate scour of bed material at the pier.

Pier Scour in Coarse-Bed Streams in Montana 47

conditions over a duration (steady flow) typically used in the 

laboratory. Raudkivi and Sutherland (1981) investigated the 

effects that steady and unsteady flow conditions and flow duration had on the relation between pier scour and relative velocity in the laboratory. They found that pier scour attained under 

unsteady-flow conditions was only about 50 to 70 percent 

of pier scour attained under steady-flow conditions. Mueller 

(1996) made the observation based on the work of Raudkivi 

and Sutherland (1981) that the nearer the relative velocity of 

the flood peak is to the threshold for live-bed conditions and 

the longer the duration of the flood wave, the nearer the scour 

depth approaches the steady-state scour depth corresponding 

to the peak discharge. The magnitude of relative pier scour 

values for data collected for this study and from the BSDMS 

Coarse dataset (fig. 14A) reflects unsteady-flow conditions 

with peak flows of limited duration and is, therefore, consistent with findings of past research (Raudkivi and Sutherland, 

1981; Mueller, 1996, p. 70–72). Armoring associated with 

nonuniform coarse-bed material and unsteady-flow conditions 

in the field probably account for differences between envelope 

curves developed for this study and those from previous studies conducted in the laboratory using fine-grained material and 

steady-flow conditions.

The very coarse nature of the bed material associated 

with pier-scour data collected for this study and in the BSDMS 

Coarse dataset most likely also contributed to the narrow 
Figure 15. Comparison of gradation coefficient for surface bed material for pier-scour measurements collected for this 

study with the Bridge Scour Data Management System (BSDMS) Coarse dataset.

(n=90)

(n=103)

THIS STUDY BSDMS COARSE

GRADATION COEFFICIENT (σg ), DIMENSIONLESS 

Outlier

Maximum value less

than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater

than Q25-1.5xIR

Outlier

EXPLANATION

Interquartile Range (IR)

1

2

3

4

5

6

7

8

9

10

n is the number of measurements

σg is the bed-material gradation 

 coefficient, dimensionless

Variables

48 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

range of relative velocity (Vo /Vc50 ) ratios (about 0.55 to 1.55, 

with one value near 2.4, fig. 14A). Finer sediment has been 

shown (Chiew, 1984) to produce a wider range of Vo /Vc50

ratios (about 0.5 to 5.0). The prevalence of clear-water conditions associated with pier-scour measurements collected for 

this study indicates that the critical velocity to initiate scour 

of bed material at the pier (V′

c50, equation 2) might be used in 

place of the critical velocity for incipient motion of bed material upstream from the bridge (Vc50) for demonstrating relations between relative pier scour and relative velocity. Replacing Vo /Vc50 with Vo

/V′

c50 as a measure of velocity intensity for 

pier scour (fig. 14B) produces similar results (fig. 14A), but in 

addition, shows that maximum relative pier scour (as measured by y′

s /b) occurs when the approach velocity (Vo ) is about 

1.3 to 1.4 times the critical velocity to initiate scour of bed 

material at the pier (V′

c50). If the approach velocity increases 

beyond that point, relative pier scour decreases until approach 

velocity reaches a minimum of about 1.8 to 1.9 times V′

c50, 

then increases asymptotically to reach equilibrium scour under 

live-bed conditions. 

The threshold for incipient motion of the streambed that 

defines the boundary between clear-water and live-bed scour 

conditions (Vo /Vc50 = 1.00) in figure 14B was determined by 

regression analysis of corresponding pairs of Vo /Vc50 and 

Vo /V′

c50 ratios computed from the pier-scour data collected 

for this study and the BSDMS Coarse dataset. The resulting 

regression equation was used to compute the ratio of Vo /V′

c50

that statistically corresponded to Vo /Vc50 = 1.00, the boundary 

between clear-water and live-bed scour conditions. Based on 

data collected for this study and the BSDMS Coarse dataset, 

the relative velocity associated with the threshold between 

clear-water and live-bed scour conditions is 1.94 times the 

relative velocity associated with initiation of scour at the pier 

(Vo/V′

c50 ).

In addition to surface bed-material size, previous studies showed that surface bed-material gradation also affected 

pier-scour depth. According to Ettema (1976), when gradation 

coefficients (σg ) are more than about 1.4 to 1.5, scour holes 

can become armored limiting further pier scour. Gradation 

coefficients associated with surface bed-material data collected 

for this study ranged from 1.48 to 4.14, with a median of 2.01 

(table 5). Bed-material gradation associated with pier-scour 

measurements collected for this study was less variable than 

bed-material gradation associated with pier-scour data in the 

BSDMS Coarse dataset (fig. 15), and bed-material gradation 

for all the pier-scour measurements collected for this study 

was sufficient for streambed armoring.

The effect of surface bed-material gradation on pier 

scour was assessed as part of this study by relating the gradation coefficient (σg ) to relative pier scour (fig. 16), similar to 

Mueller and Wagner (2005). Ripple-forming (fine) sediments 

(D50 < 0.6 mm) and nonripple-forming (coarser) sediments 

(D50 ≥ 0.6 mm) each have different relations, but the maximum 

relative pier scour for both kinds of sediments is concentrated 

near a value of σg between about 2.0 and 2.5. After reaching a 

maximum value, relative pier scour decreases as σg increases. 

Relative pier scour is greater for ripple-forming sediments 

when the gradation coefficient is less than about 2.5, except 
Figure 16. Relation between relative pier scour and gradation coefficient for pier-scour measurements collected for this study, 

historical pier-scour data from Montana, and pier-scour measurements in the Bridge Scour Data Management System (BSDMS) All 

dataset in ripple-forming (D50 < 0.6 millimeter) and nonripple-forming (D50 ≥ 0.6 millimeter) sediments.

10.0

0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0

GRADATION COEFFICIENT (σg

) FOR SURFACE BED MATERIAL, DIMENSIONLESS

RELATIVE PIER SCOUR (y's /b), DIMENSIONLESS

Variables

1.0 3.0 5.0 7.0 9.0

D50 = 0.34 millimeters

D50 = 15 millimeters

D50 = 49 millimeters

Data collected for non-ripple forming sediments for 

 this study (n=103)

Envelope curves bound the datasets represented

 by the D50 labeled for each curve

Data from the Bridge Scour Data Management 

 System (BSDMS) for non-ripple forming sediments 

 (n=184)

Data from BSDMS for ripple-forming sediments 

 (n=88)

Data from sites 60-62 for 1996-97 (n=9) associated 

 with peak streamflows of greater recurrence 

 interval than data collected for this study

n is the number of measurements

y' is pier-scour depth from field measurements, in feet s

is effective width of the pier measured perpendicular 

 to the approach flow, in feet

b

D50 is the mean of all the D50 particle sizes for data 

 associated with each envelope curve

EXPLANATION

σg is the bed-material gradation coefficient, 

 dimensionless

Pier Scour in Coarse-Bed Streams in Montana 49

perhaps for small gradation coefficients (less than about 

1.3) where the envelope curves are poorly defined. Relative 

pier-scour for ripple-forming sediments reaches a maximum 

when the gradation coefficient is slightly less than or about the 

same magnitude as for nonripple-forming sediments, but the 

envelope curve for ripple-forming sediments (n=88) declines 

more sharply with increasing gradation coefficient compared 

to the envelope curve for the nonripple-forming sediments 

(n=184). Maximum relative pier-scour is substantially less 

for nonripple-forming sediments from this study because 

the bed material was much coarser compared to the BSDMS 

nonripple-forming sediment data. The envelope-curve decline 

rate associated with data from this study is not compared to 

the BSDMS envelope-curve decline rates because data from 

this study having gradation coefficients greater than about 4.0 

are lacking. Mueller and Wagner (2005) noted that the larger 

reduction in relative pier scour for ripple-forming sediments 

having more gradation may be due to armoring caused by the 

coarser sediment fractions. The average median particle size 

( ) for data points defining each of the three curves (fig. 16) 

also indicates that maximum relative pier scour decreases with 

an increase in bed-material size.

Gradation effects also were investigated by using σg as a 

grouping variable when relating relative pier scour to relative 

velocity. Scour data were grouped into selected σg intervals, 

and the average σg value within an interval was used to define 

each group. No obvious relation was observed between relative pier scour and relative velocity for any of the σg

 intervals 

when the data were plotted, which is consistent with the 

conclusion of Mueller and Wagner (2005). However, Ettema 

(1980) showed that relative pier scour for σg > 2.5 is reduced 

for both ripple- and nonripple-forming sediments. Based on 

that observation, data collected for this study and the BSDMS 

Coarse dataset were grouped by σg, and two envelope curves 

were hand drawn (fig. 17A) that distinguish between σg ≤ 2.5 

and σg > 2.5. As shown by Ettema (1980) for data collected for 

this study and the BSDMS Coarse dataset, maximum relative 

pier scour is greater when σg ≤ 2.5.

Relations between relative pier scour, relative velocity, 

and gradation coefficient also were investigated using the 

critical velocity to initiate scour of bed material at the pier 

(V′

c50) in place of the critical velocity for incipient motion of 

bed material upstream from the bridge (Vc50) to define relative velocity (fig. 17B). With relative velocity expressed as 

Vo / V′

c50 , differences in maximum relative pier scour between 

the two envelope curves (fig. 17B) that distinguish between 

σg ≤ 2.5 and σg > 2.5 are the same as before (fig. 17A), but the 

maxima demonstrate better symmetry with each other and are 

located at about 1.2 to 1.4 times the relative velocity defined 

by Vo / V′

c50. The threshold for incipient motion of bed material 

upstream from the bridge that defines the boundary between 

clear-water and live-bed scour conditions for each envelope 

curve in figure 17B was determined by regression using corresponding ratios of Vo /Vc50 and Vo /V′

c50 computed from pierscour measurements collected for this study and the BSDMS 

Coarse dataset. The resulting regression equation was used to 

compute the ratio of Vo /V′

c50 that statistically corresponded to 

Vo /Vc50 = 1.00 for each envelope curve. Based on data from 
RELATIVE VELOCITY, (Vo /V'c50), DIMENSIONLESS

RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

1. 2

1. 4

Envelope curve and data from this study and the Bridge 

 Scour Data Management System (BSDMS) Coarse 

 dataset for σg ≤ 2.5 (n=137) 

Data from sites 60-62 for 1996-97 (n=9) associated with 

 peak streamflows of greater recurrence interval than 

 data collected for this study

Envelope curve and data from this study and the BSDMS 

 Coarse dataset for σg > 2.5 (n=56)

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Clear-water scour

Clear-water scour

Live-bed scour

Live-bed scour

0.25 0.50 0.75 1.00 1.25 1.50 1.75

0.

0.

0.

0.

0.

1.

1.

1.

A

B

Threshold for 

initiation of 

scour of bed 

material at 

the pier

0

2

4

6

8

0

2

4

is pier-scour depth from field measurements, in feet

is the effective width of the pier measured perpendicular 

 to the approach flow, in feet

is the approach velocity directly upstream from the pier, 

 in feet per second

is the critical velocity for incipient motion of bed material 

 upstream from the bridge for the D50 particle size, in 

 feet per second

is the critical velocity to initiate scour of bed material at 

 the pier for the D50 particle size, in feet per second

is the number of measurements

Less than or equal to

Greater than

Variables

y's

b

Vo

Vc50

V'c50

σ is the bed-material gradation coefficient, dimensionless g

n

≤

>

Threshold for incipient 

motion of bed material

upstream of the bridge

EXPLANATION

RELATIVE VELOCITY, (Vo /V c50), DIMENSIONLESS

Threshold between clear-water 

and live-bed scour for σg > 2.5

Threshold between clear-water 

and live-bed scour for σg ≤ 2.5

Envelope

curves

Envelope curves

Figure 17. Relation between relative pier scour and: A, relative velocity based on critical velocity for incipient motion of bed material 

upstream from the bridge; and B, relative velocity based on critical velocity to initiate scour of bed material at the pier.

50 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Table 10. Percentage of pier-scour 

measurements1 with a gradation coefficient 

greater than 2.5.

[Abbreviations: BSDMS, Bridge Scour Data Management System. Symbols: σg

, gradation coefficient, 

dimensionless; >, greater than; K4, correction factor 

for bed-material armoring effect; n, number of data 

values in dataset]

Description class for 

surface bed material2

Percent of data3 

with σg > 2.5

Small cobble 1

Very coarse gravel 2

Coarse gravel 3

Medium gravel 2

Fine to very fine gravel 0

Sand 15

1

Includes all pier-scour measurements collected 

for this study (n=103) and BSDMS All dataset 

(n=272) used in comparisons that meet HEC-18 criteria (Richardson and Davis, 2001) for applying K4

.

2

Surface bed material classified based on the D50

particle size.

3

Number of observations within a description class 

(with σg > 2.5) divided by the total number of data 

(n=375), expressed as a percent.

Pier Scour in Coarse-Bed Streams in Montana 51

this study and the BSDMS Coarse dataset, the relative velocity 

associated with the threshold between clear-water and live-bed 

scour conditions is 1.88 times the relative velocity associated 

with initiation of scour at the pier when σg ≤ 2.5 (r 2 = 0.98 and 

p < 0.001), and is 2.00 times the relative velocity associated 

with initiation of scour at the pier when σg > 2.5 (r 2 = 0.99 

and p < 0.001). Coarse-bed material with a larger gradation 

coefficient (σg >2.5) attains live-bed conditions at a higher 

relative velocity, compared to similar coarse-bed material with 

a smaller gradation coefficient (σg ≤ 2.5). This observation 

is consistent with previous studies (Chiew, 1984 and Baker, 

1986) that noted the tendency for well-graded bed material 

to armor at lower relative velocities and resist further scour. 

Similarities in the magnitude of maximum relative scour for 

the envelope curve (fig. 17B) associated with larger gradation coefficients (σg

 > 2.5) also may indicate that distinctions 

between clear-water and live-bed scour may not be important 

when surface bed material is coarse and well graded (σg > 2.5). 

Comparison of the two envelope curves indicates that a 

relation between relative pier scour and relative velocity that 

depends on σg may exist under field conditions but less conclusively than was indicated in laboratory research (Baker, 1986), 

where D50 was held constant while the size of the coarser 

fraction of the sediments was increased. A lack of field data 

for ripple-forming sediments with large gradation coefficients 

indicates that surface bed material having this characteristic 

may not be commonly found in natural streams. 

Pier-scour measurements collected for this study and the 

BSDMS All dataset confirm a general lack of data for coarsebed sites with σg

 greater than about 2.5 (figs. 15 and 16). 

Grouping these data by size classification based on the D50 of 

the bed material also illustrates the lack of data for coarse-bed 

sites with σg greater than 2.5 (table 10). The percentage of 

scour data from sites (n=375) with coarse-bed material (larger 

than sand, table 10) having σg greater than 2.5 is only 8 percent of the data. Furthermore, only 15 percent of scour data 

from sites with bed material classified as sand (table 10) have 

gradation coefficients exceeding 2.5. Mueller and Wagner 

(2005) found that the small number of sites with ripple-forming sediments with larger gradation coefficients made it difficult to draw conclusions on the effect of gradation on scour. 

Similarly, the relatively small amount of pier-scour data from 

sites with coarse bed material and large gradation coefficients 

makes it difficult to understand the effects of gradation under 

these conditions. The narrow range in gradation coefficients 

displayed by all available data and the nonuniform conditions 

present in the field continue to complicate efforts to isolate the 

effects of gradation on pier scour.

Effects of Shallow-Subsurface Bed-Material 

Characteristics

Since the mid–1960s, pier-scour researchers have considered the effects of bed-material characteristics on pier scour 

as cited in the section “Development of Pier-Scour Equations 

for Coarse-Bed Streams.” That research focused primarily 

on surface bed material because vertical variations in bedmaterial characteristics were not a concern in flume studies 

where the experimental bed material was composed of either 

uniform particles or a nonuniform mixture of sediment with 

the same size and gradation characteristics throughout the 

thickness of the bed. While collecting coarse-bed pier-scour 

data in the field for this study, differences were observed in 

surface and shallow-subsurface bed-material size and gradation (fig. 4). As a result, bed-material samples for both the 

surface and shallow-subsurface layers were collected and 

analyzed (table 9). The particle-size distribution and gradation 
BED-MATERIAL PARTICLE SIZE, (Di

), IN MILLIMETERS

0

50

100

150

200

250

300

D50 PARTICLE SIZE D95 PARTICLE SIZE

SURFACE

BED 

MATERIAL

SHALLOWSUBSURFACE

BED 

MATERIAL

SURFACE

BED 

MATERIAL

SHALLOWSUBSURFACE

BED 

MATERIAL

Outlier

Maximum value less

than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater

than Q25-1.5xIR

Outlier

EXPLANATION

Interquartile Range (IR)

(n=103 for all boxplots)

n is the number of measurements

is the bed-material particle size, 

 D, where i percent of particles 

 are finer by weight, 

 in millimeters or feet

(Di

)

Variable

Figure 18. Comparison of surface and shallow-subsurface bed-material particle size for pier-scour measurements collected 

for this study.

52 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

for the bed-material samples for each layer were compared 

and the differences in particle size that might relate to armoring were described by using ratios of the particle sizes for 

each layer. Differences in particle size and gradation between 

the two layers were apparent (table 9). Therefore, the effect 

these differences might have on pier-scour computations was 

investigated. 

Particle-size distribution curves for paired samples of 

surface and shallow-subsurface bed material collected for this 

study were analyzed and the D50 and D95 particle sizes were 

compared. The difference between the surface and shallowsubsurface bed-material D50 and D95 particle sizes (fig. 18) confirmed what was observed in the field—due to armoring, larger 

particles were more commonly found on the streambed surface 

than in the shallow subsurface (fig. 4). However, coarse 

particles still made up a substantial portion of the shallowsubsurface bed material at most sites. Median D50 particle sizes 

for 103 surface and shallow-subsurface bed-material samples 

were about 49 mm and 32 mm, respectively, and median D95 

particle sizes for 103 surface and shallow-subsurface bed 

material samples were 122 mm and 86 mm, respectively 

(fig. 18, table 5). These differences may result in reduced pier 

scour because bed-material size has been shown in this (figs. 6 

and 11) and other studies cited in the section “Development of 

Pier-Scour Equations for Coarse-Bed Streams” to be inversely 

related to pier-scour depth. 

Differences in gradation between surface and shallowsubsurface bed-material samples also were examined for this 

study. The median gradation coefficients for surface and shallow-subsurface bed material associated with measurements 

made for this study were 2.01 and 4.14, respectively (fig. 19 

and table 5). The median gradation coefficient of the shallowsubsurface bed material was about twice that of the surface 

bed material (fig. 19) owing to the presence of finer-grained 

sediment in the shallow-subsurface bed material. These differences in gradation coefficient may result in reduced pier 

scour because gradation coefficients have been shown in this 

(figs. 16 and 17) and other studies cited in the section “Development of Pier-Scour Equations for Coarse-Bed Streams” to 

be inversely related to pier-scour depth in coarse-bed streams 

like those where pier-scour measurements were collected for 

this study. The combined effect of the larger surface bedmaterial size, relative to shallow-subsurface bed-material 

size, and the wider range of gradation coefficients in shallowsubsurface bed material is typical for the pier-scour measurements made for this study and may contribute to reduced pier 

scour in these coarse-bed streams. Differences in particle size 

and gradation coefficient between the two bed-material layers 

are likely similar to the variability in particle size and gradation coefficient that might be observed between multiple sets 

of pebble counts conducted to quantify average surface bedmaterial characteristics. This variability has been recognized 
GRADATION COEFFICIENT (σg), DIMENSIONLESS

0

1

2

3

4

5

6

7

8

SURFACE BED MATERIAL SHALLOW-SUBSURFACE

BED MATERIAL

(n=103 for each boxplot)

Outlier

Maximum value less

than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater

than Q25-1.5xIR

Outlier

EXPLANATION

Interquartile Range (IR)

n is the number of measurements

Variable

is the bed-material gradation 

 coefficient, dimensionless

σg

Figure 19. Comparison of surface with shallow-subsurface bed-material gradation coefficients for pier-scour 

measurements collected for this study.

D50 particle size

D95 particle size

10 100 1,000

1,000

100

10

1

1

D95,ss = 2.11D95,s

0.79

r 2 = 0.647

p-value < 0.001

n = 59

r 2

 = 0.690

p-value < 0.001

SHALLOW SUBSURFACE-BED MATERIAL SIZE (

Di,ss ), IN MILLIMETERS 

SURFACE BED-MATERIAL SIZE (Di,s), IN MILLIMETERS

EXPLANATION

 is the coefficient of determination

 is the significance level

r2

p-value

 D(i, s) is the surface bed-material particle size, D, where 

 i percent of particles are finer by weight, in millimeters 

 or feet

D(i, ss) is the shallow-subsurface bed-material particle size, D, 

 where i percent of particles are finer by weight, in 

 millimeters or feet

n = 59

D50,ss = 0.40D50,s

1.16 n is the number of measurements

Variable

Figure 20. Relations between corresponding surface and shallow-subsurface bed-material particle size (D50 and D95) for pier-scour 

measurements collected for this study.

Pier Scour in Coarse-Bed Streams in Montana 53
D16 D35 D50 D84 D85 D90 D95 D99

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

BED-MATERIAL SIZE RATIO , DIMENSIONLESS D

D

i s

i ss

,

,

(n=59 for all boxplots)

Outlier

Maximum value less than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater than Q25-1.5xIR

Outlier

EXPLANATION

Interquartile Range (IR)

BED MATERIAL SIZE, (Di), IN MILLIMETERS

Di,s is the surface bed-material particle size, 

 Di,s, where i percent of surface particles 

 are finer by weight, in millimeters or feet

Di, ss is the shallow subsurface bed-material 

 particle size, Di,ss, where i percent of 

 surface particles are finer by weight, 

 in millimeters or feet 

n is the number of measurements

Variables

Figure 21. Comparison of ratios of surface to shallow-subsurface bed-material particle size based on selected percentiles (Di) from 

the particle-size distribution curves for each site where pier-scour measurements were collected for this study.

54 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

as a potential source of uncertainty in accurate prediction of 

pier-scour depth (Mueller and Wagner, 2005, p. 61).

The relation between selected particle sizes of the two 

layers, independent of site location or specific particle size, 

was investigated using regression analysis of paired values 

of particle sizes for the surface and shallow-subsurface bedmaterial data collected for this study (fig. 20). A significant 

relation was indicated between surface and shallow-subsurface 

bed-material size for the D50 and D95 particle sizes (for D50, 

r 2 = 0.690 and p < 0.001; for D95, r 2 = 0.647 and p < 0.001). 

In other words, as the particles in the surface bed material 

increase in size, the particles in the shallow-subsurface bed 

material also tend to increase in size.

Recognizing that armoring is a process that commonly 

leads to reduced scour in coarse-bed streams, a previously 

published concept relating to armoring was used to further 

evaluate variations in particle size between surface and 

shallow-subsurface bed material. Lisle and Madej (1992, 

p. 277) proposed the ratio of surface-to-subsurface bedmaterial particle sizes as a measure of armoring. The variation 

in bed-material size over a selected range of the particle-size 

distribution curve (D16 to D99 ) was, therefore, computed for the 

surface and shallow-subsurface bed material associated with 

the pier-scour data collected for this study using the equation: 

 , (23)

where

Ri is the bed-material size ratio of the 

corresponding surface and shallowsubsurface bed-material particle size;

Di, s is the surface bed-material particle size, D, 

where i percent of particles are finer by 

weight, in millimeters or feet; and

Di, ss is the shallow-subsurface bed-material 

particle size, D, where i percent of particles 

are finer by weight, in millimeters or feet.

Bed-material size ratios (median values for Ri) for the 

coarser portion of the particle-size distribution curve (D50 to 

D99 ) did not vary substantially, and ranged from about 1.1 to 

1.4 (fig. 21). The interquartile ranges (IR) also did not vary 

substantially and were all less than 1. Bed-material size ratios 

(median values for Ri) for the finer portion of the particle-size 

distribution curve (D16 to D35) were more variable than for 

bed-material size ratios for the coarser portion of the curve, 

and ranged from about 1.8 to about 4.0. The interquartile 

ranges also were more variable and ranged from about 1.2 to 

about 4.6. Thus, the greatest variability between surface and 

shallow-subsurface particle sizes, as defined by Ri, is at the 

fine-grained end of the particle-size distribution curve. 

The streambed upstream from the bridge likely was 

armored at most measurement sites because the surface bed 

material was more coarse when compared to the shallowsubsurface bed material, and because of predominant 
Pier Scour in Coarse-Bed Streams in Montana 55

clear-water conditions (Vo < Vc50 ) and gradation coefficients 

greater than 1.4 (table 9). At the piers, the relative scour 

velocities (fig. 8) were great enough (Vo /V′

c50 > 1.00) that the 

armor layers in the scour holes were mobilized and all sites 

were undergoing pier scour. Based on observed field conditions, and the relative coarseness of the bed-material particle 

sizes for the two layers (figs. 20 and 21), D50 and D95 particlesize data for the shallow-subsurface bed material underlying 

the armor layer (table 9) also were used to compute pier scour 

by the HEC-18-K4Mu equation (equation 5) with K4 determined by equation 11. 

Scour depths were computed using the D50 and D95 shallow-subsurface particle-size data for 94 of the 103 pier-scour 

measurements made for this study. Nine of the measurements 

either did not meet the criteria to be considered as a valid 

real-time scour measurement, or anomalous results were produced when the D50 and D95 shallow-subsurface bed-material 

particle sizes were used in equation 11 to compute K4. Scour 

depths computed based on shallow-subsurface particle sizes 

were then compared to both measured scour depths (table 2) 

and scour depths computed based on the surface bed-material 

particle-size data (D50 and D95, table 9). Differences in D50 and 

D95 particle size between the surface and shallow-subsurface 

bed material did not cause substantial differences in computed 

scour (table 11). The frequency and magnitude of the residual 

pier-scour depth (fig. 22) associated with underpredicted and 

overpredicted scour was about the same regardless of which 

particle-size data were used with the HEC-18-K4Mu equation. Computed scour was underpredicted in only 5 out of 

94 instances when surface bed-material particle-size data were 

used with the HEC-18-K4Mu equation and in only 3 out of 

94 instances when shallow-subsurface bed-material particlesize data were used. Positive (underpredicted) and negative 

(overpredicted) residuals of scour are noted as separate statistics in table 11 because of the implication each type of residual 

separately has to bridge foundation design. The median underpredicted (positive) residual scour using surface and shallowsubsurface data with the HEC-18-K4

Mu equation (table 11) 

was 0.34 and 0.29 ft, respectively, and the maximum underpredicted residual scour was 0.52 and 0.34 ft, respectively. The 

median overpredicted (negative) residual scour using surface 

and shallow-subsurface data with the HEC-18-K4Mu equation 

(table 11) was -0.91 and -1.02 ft, respectively, and the greatest 

overpredicted residual scour was -4.41 ft using either bedmaterial layer. 

Computed scour generally was the same or slightly 

greater when shallow-subsurface bed-material particle-size 

data were used with the HEC-18-K4Mu equation because 

shallow-subsurface bed-material particle sizes (D50 and D95) 

typically were slightly smaller than corresponding surface bedmaterial particle sizes. In addition, other factors contributed 

to the close agreement between pier-scour computations using 

the surface and shallow-subsurface bed-material size data. For 

example, of the 94 measurements made for this study satisfying criteria for real-time scour at the pier (Vo

 /V′

c50 > 1.00), 37 

had computed values of K4 less than 0.4, regardless of whether 

surface or shallow-subsurface bed-material data were used 

with equation 11. The minimum allowed value of K4 (0.4) was 

substituted for the computed K4 in both instances according to 

HEC-18 criteria (Richardson and Davis, 2001). This substitution resulted in identical computed scour depths for these 

measurements, regardless of which bed-material data were 

used. Furthermore, 34 of 94 measurements had both computed 

values of K4 exceeding 0.4 for surface and shallow-subsurface 

bed-material data—the median difference in computed K4

was 0.03. In addition, 21 of 94 measurements had at least one 

computed value of K4 exceeding 0.4 when either surface or 

shallow-subsurface bed-material data were used with equation 11—the median difference in computed K4

 was 0.05 after 

substituting 0.4 for the calculated K4 that was less than 0.4. 

Finally, substantial differences for the computed Mueller 

velocity-intensity ratio (VRM , equation 10) for some sites typically resulted in much smaller differences between the resulting values of K4

 computed by equation 11 using corresponding 

surface and shallow-subsurface data. The modest response 

of K4 (dependent variable) to more widely varying values of 

VRM (independent variable) is partly due to the small exponent 

(0.15) in equation 11, which indicates a relatively flat slope 

(weak relation) for the regression line relating K4 to VRM . In 

summary, despite vertical variations in coarse bed-material 

characteristics between the surface and shallow-subsurface 

bed-material layers, pier-scour depths computed using either 

set of characteristics were quite similar. The HEC-18-K4Mu 

equation was not very sensitive to vertical variations in bedmaterial characteristics within the active layer at coarse-bed 

sites where pier-scour data were collected for this study. 

Combined Effects of Surface and ShallowSubsurface Bed-Material Characteristics

In addition to examining the effects that differences 

in bed-material size and gradation between the surface and 

shallow-subsurface layers might have on predicted pier-scour 

depth, the combined effect that the bed-material characteristics 

of both layers might have on measured pier scour also was 

examined. The combined effect is of interest because if the 

critical velocity to initiate scour of bed material at the pier is 

exceeded, the surface and shallow-subsurface layers could 

interact and exchange bed material within the active layer until 

an immobile armor layer re-forms. Lagasse and others (1991) 

describe the armoring process as the interaction between the 

surface and shallow-subsurface layers of bed material. For 

gravel-bed streams where the bed is not composed of immobile armor but is paved so that coarse particles are frequently 

mobilized, Parker and others (1982, p. 546) indicate that the 

particle-size distribution of the bedload typically is more 

similar to the size distribution of the shallow-subsurface layer 

(subpavement) than to that of the surface layer (pavement). It 

is therefore conceivable that the bed-material size and gradation characteristics of the two layers in combination can affect 

pier scour. For example, substantial pier-scour depth might 
Table 11. Statistical summary of measured pier scour and computed pier scour based on surface and shallow-subsurface bed-material data.

[Computed pier scour was obtained using the HEC-18-

K Mu equation 4

a, and the 

D50 and 

D95 particle-size variables used in the equation were determined from surface and shallow-subsurface bed-material data 

collected for this study. Number of measurements (n) was equal to 94, unless indicated otherwise. Abbreviation: ft, feet; Symbols: ≈, approximately equal to; 

K4, correction factor for bed-material armoring 

effect, dimensionless; ≤, less than or equal to]

Pier-scour or related data

Summary statistics

Minimum Maximum Mean Standard deviation Skewness 25th percentile Median 75th percentile

Measured pier scour, in ft 0.32 7.00 1.78 1.35 2.19 0.90 1.42 2.10

Computed pier scour based on surface bed-material particle 

size, in fta

.87 7.46 2.70 1.48 1.80 1.75 2.34 2.93

Computed pier scour based on shallow-surface bed-material 

particle size, in fta

.99 8.88 2.92 1.80 2.08 1.88 2.44 3.05

Residual pier scour based on surface bed-material particle size, 

in ftb

-4.41 .52 -.93 .77 -1.69 -1.23 -.86 -.44

Residual pier scour based on shallow-subsurface bed-material 

particle size, in ftb

-4.41 .34 -1.14 .89 -1.59 -1.40 -.98 -.58

Positive residual (n=5) for underpredicted pier scour based on 

surface bed-material particle size, in ftc

.08 .52 .29 .20 -.17 .09 .34 .43

Positive residual (n=3) for underpredicted pier scour based on 

shallow-subsurface bed-material particle size, in ftd

.08 .34 .24 .14 -1.43 .18 .29 .31

Negative residual (n=87) for overpredicted pier scour based on 

surface bed-material particle size, in ftc

e-4.41 -.14 -1.02 .73 -2.16 -1.23 -.91 -.54

Negative residual (n=90) for overpredicted pier scour based on 

shallow-subsurface bed-material particle size, in ftd

e-4.41 -.13 -1.20 .87 -1.78 -1.42 -1.02 -.63

Mueller velocity-intensity ratio based on surface bed-material 

particle size, dimensionless

.02 4.40 1.02 .87 1.26 .38 .74 1.56

Mueller velocity-intensity ratio based on shallow-subsurface 

bed-material particle size, dimensionless

.04 f21.0 2.61 3.52 2.58 .55 1.17 2.71

K4 based on surface bed-material particle size 

(0.0 ≤ 

K4 ≤ 1.0), dimensionless

.23 .50 .38 .06 -.43 .35 .38 .43

K4 based on shallow-subsurface bed-material particle size 

(0.0 ≤ 

K4 ≤ 1.0), dimensionless

.24 .63 .42 .08 .41 .37 .41 .46

K4 based on surface bed-material particle size 

(0.4 ≤ 

K4 ≤ 1.0), dimensionless

.40 .50 .41 .02 1.58 .40 .40 .43

K4 based on shallow-subsurface bed-material particle size 

(0.4 ≤ 

K4 ≤ 1.0), dimensionless

.40 .63 .44 .06 1.34 .40 .41 .46

aEquation based on Richardson and Davis (2001).bEqual to measured pier scour minus computed pier scour.cTotal number of values (n=94) equals the number of positive residuals (n=5) and negative residuals 

(n=87) combined, plus two computed pier-scour values essentially equal to measured pier scour (residuals ≈ 0).

dTotal number of values (n=94) equals the number of positive residuals (n=3) and negative residuals (n=90) combined, plus one computed pier-scour value essentially equal to measured pier scour 

(residual ≈ 0).

eValue shown as the minimum negative residual corresponds to the maximum residual of overpredicted pier scour.fLarge value (21.0) computed based on small number in denominator in equation.

56 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007
Figure 22. Comparison of residual pier-scour depth based on pier-scour measurements and bed-material data collected for this study 

with pier-scour depth computed by the HEC-18-K4Mu equation (Richardson and Davis, 2001).

RESIDUAL PIER-SCOUR DEPTH, IN FEET

SURFACE 

BED MATERIAL

SHALLOW-SUBSURFACE

BED MATERIAL

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

Overpredicted

Underpredicted

(n=94 for each boxplot)

Outlier

Maximum value less than Q75+1.5xIR

75th percentile (Q75)

Median

25th percentile (Q25)

Minimum value greater than Q25-1.5xIR

Outlier

Residual pier-scour depth is measured pier-scour 

depth minus computed pier-scour depth.

EXPLANATION

Interquartile Range (IR)

n is the number of measurements

Variable

Table 12. Groups of pier-scour measurements based on classification of particle sizes of the 

surface and shallow-subsurface bed material.

Combination

Bed-material size classification modified 

from Lane and others (1947)

Number 

of 

measurements

Group 

number

Surface Shallow-subsurface (fig. 23)

1 Small cobble Small cobble 17 1

2 Small cobble Very coarse gravel 9 2

3 Very coarse gravel Small cobble 3 2

4 Very coarse gravel Very coarse gravel 23 3

5 Very coarse gravel Coarse gravel 14 4

6 Coarse gravel Very coarse gravel 4 4

7 Very coarse gravel Medium gravel 3 4

8 Coarse gravel Coarse gravel 19 5

9 Coarse gravel Medium gravel 1 5

10 Medium gravel Medium gravel 7 5

11 Coarse gravel Very fine gravel 3 5

Pier Scour in Coarse-Bed Streams in Montana 57
Figure 23. Relation between relative pier 

scour and relative velocity for different 

groups of combinations of surface 

and shallow-subsurface bed-material 

size classifications for the pier-scour 

measurements collected for this study.

RELATIVE VELOCITY, (Vo /Vc50,s ), DIMENSIONLESS

RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS

Surface layer Shallow-subsurface layer

Small cobble Small cobble

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.4 0.6 0.8 1.0 1.1 1.2 1.4

Clear-water scour Live-bed scour

Very coarse gravel Very coarse gravel

Coarse gravel to 

 medium gravel

Coarse gravel to very 

 fine gravel

Small cobble to very 

 coarse gravel 

Small cobble to very 

 coarse gravel 

Very coarse gravel 

to coarse gravel

Envelope curves Size classification for corresponding layers

and group data

EXPLANATION

Very coarse gravel to 

medium gravel

0.5 0.7 0.9 1.3

Group number 

 (table 12)

1

2

3

4

5

n is the number of measurements

Variables

Theshold for incipient motion of 

surface bed material upstream 

from the bridge 

is pier-scour depth from field measurements, in feet 

is effective width of the pier measured perpendicular to 

 the approach flow, in feet 

b

y's

is the measured approach velocity directly upstream 

 from the pier, in feet per second

Vo

Vc50,s is the critical velocity for incipient motion of surface bed 

 material upstream from the bridge for the D50 particle size, 

 in feet per second

(n=103)

58 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

be expected where a thin layer or pavement of moderately 

coarse surface bed material like gravel is underlain by much 

finer subsurface bed material like sand (American Society of 

Civil Engineers, 1975, reprinted 1977, p. 182). In contrast, 

less scour might be expected when the bed material of the two 

layers includes coarse gravel and cobbles and the bed material 

of one layer has a greater degree of gradation than the other 

(fig. 4).

The effects that differences in particle size between 

surface and shallow-subsurface bed-material layers might 

have on pier scour were investigated first. Examination of the 

differences in particle sizes between the two layers began by 

grouping the bed-material particle-size data for each layer 

into bed-material size classifications (table 3) based on the D50 

particle size. The classifications ranged from small cobble to 

very fine gravel. The different combinations of surface and 

shallow-subsurface particle-size classifications for individual 

scour measurements were examined next. The 103 scour 

measurements collected for this study exhibited 11 combinations of bed-material size classification involving the surface 

and shallow-subsurface layers, and the measurements were 

initially grouped based on these 11 combinations (table 12). 

Many of the 11 combinations were characteristic of only a 

small number (four or less) of the 103 pier-scour measurements. Therefore, one or more combinations with only slightly 

different size characteristics were lumped together into five 

groups to allow development of envelope curves. The relation 

between relative velocity, based on the D50 of the surface bedmaterial layer, and relative pier scour was examined for pierscour measurements in each group (fig. 23). Based on the five 

envelope curves drawn to define the upper bounds of the data 

for each group, relative pier scour generally was less for measurements with coarser bed material like very coarse gravel or 

cobbles in the two layers (fig. 23). Few other conclusions are 

apparent from a comparison of the envelope curves.

The relation between relative velocity, based on the average D50 for the two bed-material layers, and relative pier scour 

also was examined and produced similar results. This similarity of results may be because surface and shallow-subsurface 

bed-material size classifications for both layers were the same 

or similar for both layers for most measurements (for example, 

small cobbles in both layers), even though surface bed-
Figure 24. Comparison of relative pier scour to gradation coefficient for surface and shallow-subsurface bed-material data collected 

for this study.

0.0

0.3

0.4

0.6

0.8

1.0

1.2

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

(n=103)

GRADATION COEFFICIENT (σg), DIMENSIONLESS

RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS

EXPLANATION

y's

b

σg,s

σg,ss

n

σg,c

σg

Envelope curves and data points

Surface gradation coefficient, σg,s

Shallow-subsurface gradation coefficient, σg,ss

Composite gradation coefficient, σg,c based on 

geometric mean of σg,s and σg,ss

Variables

is the number of measurements

is pier-scour depth from field measurements,

in feet

is effective width of the pier measured perpendicular 

to the approach flow, in feet

is the surface bed-material gradation coefficient,

dimensionless

is the shallow-subsurface bed-material gradation

coefficient, dimensionless

is the composite gradation coefficient or the geometric 

mean of the corresponding surface and shallowsubsurface σg values, dimensionless

is the bed-material gradation coefficient, dimensionless 

Pier Scour in Coarse-Bed Streams in Montana 59

material size typically was larger than the shallow-subsurface 

bed-material size. There were no sites where the difference in 

bed-material particle size between the surface or armor layer 

and shallow subsurface was substantial (for example, small 

cobbles underlain by very fine gravel or sand); thus, more 

detailed assessment of such widely contrasting bed-material 

compositions was not possible.

Differences in gradation of bed material in the surface and shallow-subsurface layers and the relation of these 

differences to pier scour also were investigated using two 

approaches. First, the differences in relations between gradation coefficients for each layer and relative pier scour were 

compared by examining envelope curves for each set of measurements obtained for this study (fig. 24). Maximum relative 

pier scour corresponded to a gradation coefficient of about 

2.3 for the surface bed material and about 4.7 for the shallowsubsurface bed material (fig. 24). This fairly narrow range in 

gradation coefficients associated with maximum pier scour 

for the two layers is similar to other studies that showed the 

narrow range in gradation coefficients associated with maximum pier scour based only on surface bed material (Ettema, 

1980; Baker, 1986). A composite envelope curve (fig. 24) was 

generated as a means to account for the variability in σg as it 

pertains to the computation of pier-scour depth when results of 

multiple pebble counts or sieve analyses involving both layers 

are combined. The composite gradation coefficient (σg,c ) is the 

geometric mean of the corresponding surface and subsurface 

σg values. 

For the second approach, the variation in gradation over 

the range of the particle-size distribution curve was defined 

using the following relation: 

 , (24)

where

Rg is the relative gradation or the ratio of 

shallow-subsurface-to-surface bed-material 

gradation coefficient, dimensionless;

σg,ss is the gradation coefficient of the shallowsubsurface bed material, dimensionless; 

and

σg,s is the gradation coefficient of the surface bed 

material, dimensionless. 

The relative gradation (Rg) was calculated for each of the 

103 pier-scour measurements collected for this study and 

the measurements were ranked by Rg from largest to smallest. Selected percentiles (99th, 75th, 50th, 25th, and 10th) 

were then determined and the relative pier scour and relative 

velocity for each measurement in the percentile were plotted 

to examine the relation between relative pier scour and relative 

velocity for each percentile. Envelope curves bounding the 

measurements in each percentile were drawn. Each of the five 

envelope curves (fig. 25), therefore, bounds a group of pierscour measurements with Rg

 values that equal or exceed a particular value (smallest value) in each percentile. For example, 

10 percent of the measurements had Rg values that equaled or 
Figure 25. Relation between relative pier scour and relative velocity based on surface and shallow-subsurface bed-material gradation 

differences for pier-scour measurements collected for this study. 

75th percentile; Rg greater than or equal to 1.24 

10th percentile; Rg greater than or equal to 3.18 

25th percentile; Rg greater than or equal to 2.46 

50th percentile; Rg greater than or equal to 1.95 

Envelope curves and data points

EXPLANATION

Rg

σg,s

σg,ss

Vo

Vc50

is the number of measurements

is the approach velocity just upstream from the 

pier, in feet per second

is the critical velocity for incipient motion of 

surface bed material upstream from the 

bridge, in feet per second 

is the relative gradation or the ratio of shallow 

subsurface-to-surface bed-material 

gradation coefficients, dimensionless 

is the surface bed-material gradation coefficient, 

dimensionless 

is the shallow-subsurface bed-material 

gradation coefficient, dimensionless 

Variables

99th percentile; Rg greater than or equal to 0.96 

RELATIVE VELOCITY, (Vo /Vc50), DIMENSIONLESS

RELATIVE PIER SCOUR, (y'

s /b), DIMENSIONLESS 

σg,ss

σg,s

Rg = 

Clear-water scour Live-bed scour

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

(n=103)

n

Some data points associated with an envelope 

curve have coordinates identical to data points 

associated with other envelope curves. In those 

instances, the data points were slightly offset from 

each other in the horizontal, vertical, or both 

directions for clarity.

60 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

exceeded 3.18, so 10 data points (Rg ≥ 3.18, fig. 25) were used 

to define the 10th-percentile envelope curve. Maximum relative scour associated with each envelope curve is concentrated 

at about the same relative velocity (Vo /Vc50 ≈ 0.75 to 0.8). The 

five envelope curves (fig. 25) appear to converge when relative velocities are high enough for live-bed scour conditions 

to be reached (Vo /Vc50 > 1.00). Based on these curves, when 

the difference between the shallow-subsurface and surface 

bed-material gradation coefficients increases (higher values of 

Rg), pier scour decreases. Earlier studies reported by Mueller 

(1996) and Mueller and Wagner (2005) indicated that when 

the gradation coefficient of the surface bed material increased, 

pier scour was reduced. 

Higher variability in gradation coefficients would seem to 

indicate the presence of a higher fraction of fine-grained particles with expected deeper scour before armoring. Increased 

scour associated with finer bed material may, however, be 

more than offset if sufficiently large particles also are available to hide the smaller particles (Einstein, 1950; Parker and 

Klingeman, 1982), limiting overall mobility of bed material 

and allowing more rapid armoring.

Finally, the combined influence of relative differences in 

surface and shallow-subsurface bed-material size and gradation on pier scour was investigated by defining a bed-material 

variability index (Iv):

 , (25)

Substitution based on equations 23 and 24 yields: 

, (26)

where all terms are as previously defined. The three variables 

on the right hand side of equation 26 were derived from 

variables (D50, D95, and σg) shown to be related to pier scour in 

coarse-bed streams as noted in this section and other studies 

cited in the section “Development of Pier-Scour Equations for 
RELATIVE VELOCITY, (Vo /Vc50), DIMENSIONLESS

RELATIVE PIER SCOUR, (y's /b), DIMENSIONLESS

ss

s

ss

s

v D

D

D

D

I = 

95,

95,

50,

50,

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Clear-water scour Live-bed scour

(n=103)

+ + σg,ss

σg,s

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Envelope curves and data points

EXPLANATION

Iv

Vo

Vc50

D(i, s)

D(i, ss)

σg,s

σg,ss

y'

s

b

is the number of measurements

is pier-scour depth from field measurements, 

in feet

is effective width of the pier measured perpendicular 

to the approach flow, in feet

is the bed-material variability index, dimensionless

is approach velocity directly upstream from the pier, 

in feet per second

is the critical velocity for incipient motion of the 

surface bed material upstream from the bridge 

for the D50 particle size, in feet per second

is the surface bed-material particle size, D, where 

i percent of particles are finer by weight, in 

millimeters or feet

is the shallow-subsurface bed-material particle size, 

D, where i percent of particles are finer by 

weight, in millimeters or feet

is the surface bed-material gradation coefficient, 

dimensionless

is the shallow-subsurface bed-material gradation 

coefficient, dimensionless

Variables

n

Some data points associated with an envelope curve 

have coordinates identical to data points associated 

with other envelope curves. In those instances, the 

data points were slightly offset from each other in the 

horizontal, vertical, or both directions for clarity.

50th percentile; Iv greater than or equal to 4.60

25th percentile; Iv greater than or equal to 5.52

75th percentile; Iv greater than or equal to 3.61

10th percentile; Iv greater than or equal to 6.48

99th percentile; Iv greater than or equal to 2.63 

Figure 26. Relation between relative pier scour and relative velocity based on 

surface and shallow-subsurface bed-material size and gradation differences for 

pier-scour measurements collected for this study.

Pier Scour in Coarse-Bed Streams in Montana 61

Coarse-Bed Streams.” These variables have about the same 

magnitude and, therefore, have a similar influence on the values of Iv

. Based on research cited in the section “Development 

of Pier-Scour Equations in Coarse-Bed Streams,” that showed 

pier scour to be less for coarse-size bed material with higher 

gradation coefficients, perfect homogeneity (where R50, R95, 

and Rg each equal 1.00 in equation 26) between surface and 

shallow-subsurface bed-material size and gradation (I

v

 = 3.00) 

might be expected to result in greater relative pier scour than 

a condition where, for example, the relative homogeneity of 

size and gradation varied by a factor of two for each ratio 

(Iv = 6.00). 

The bed-material variability index (Iv) was calculated 

for each of the 103 measurements collected for this study and 

the measurements were ranked by Iv from largest to smallest. 

Selected percentiles (99th, 75th, 50th, 25th, and 10th) were 

determined and the relative pier scour and relative velocity for 

each measurement in the percentile were plotted to examine 

the relation between relative pier scour and relative velocity 

for each percentile. Envelope curves bounding the measurements in each percentile were drawn. Each of the five envelope curves (fig. 26), therefore, bounds a group of pier-scour 

measurements with Iv values that equal or exceed a particular 

value (smallest value) in each percentile group. For example, 

10 percent of the measurements had Iv values that equaled or 

exceeded 6.48, so 10 data points (Iv ≥ 6.48, fig. 26) were used 

to define the 10th-percentile envelope curve. Maximum relative scour associated with each envelope curve is concentrated 

at about the same relative velocity (Vo/Vc50 ≈ 0.80). The five 

envelope curves (fig. 26) appear to converge when relative 

velocities are high enough for live-bed scour conditions to be 

reached (Vo /Vc50 > 1.00). Based on these curves, as differences 

in particle size and gradation between the surface and shallowsubsurface bed-material layers increase (higher values of I

v

), 

pier scour decreases. These curves also exhibit the potential to 

refine estimates of maximum relative pier scour using the bedmaterial variability index or a similar variable. 
62 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

Limitations of the Study

Field measurement of scour at bridge piers requires the 

use of specialized equipment and methods to obtain accurate 

data. The collection of real-time scour data also requires that 

measurements be made during high-flow conditions. Although 

methods for conducting scour-depth measurements generally are described in various USGS publications, innovation, 

physical effort, and judgment are required in obtaining usable 

pier-scour data under difficult field conditions. A series of 

measurement and interpretive steps are needed to compute the 

measured pier-scour depth. Except for three sites where continuous scour measurements were made over the entire runoff 

period by using fixed instrumentation, most measurements for 

this study were conducted over a few hours on a single day. 

Continuous transducer soundings made at one site indicated 

that maximum pier scour was attained several times during a 

2-month period in 2006 for flows that ranged from less than 

to about bankfull discharge, which provide some insight on 

the site-specific effects of high-flow duration on pier scour. 

High-flow duration effects on pier scour, however, were not 

addressed beyond this one example.

Runoff conditions during the study mostly were limited to bankfull discharge approximating the 1.5- to 2-year 

recurrence-interval flood. Lack of higher flows having greater 

velocity intensities may indicate that scour depths were limited 

by hydrologic conditions. Still, scour measurements made 

mostly under clear-water conditions associated with maximum 

scour involved relatively high velocity intensities. In fact, 

measured velocities at one-half of study sites were at least 

70 percent of the critical velocity for incipient motion of the 

bed material upstream from the bridge, which might indicate 

that scour was measured near the threshold between clearwater and live-bed scour conditions. Laboratory research using 

fine-grained material and steady-flow conditions indicates that 

this threshold is where maximum pier scour likely occurs. 

Furthermore, historical pier-scour data from Montana with 

peak streamflows of greater recurrence interval (up to about 

100 years) plotted below envelope curves (figs. 10, 11, and 14) 

that bound the data collected for this study and BSDMS data. 

Pier-scour depths measured in this study also may have been 

limited by streambed armoring and unsteady flow conditions.

 It is critical for bed-material samples to be representative 

of the actual streambed materials if these data are to be related 

to maximum pier-scour depth and used to improve equations 

to predict pier scour. A single bed-material sample considered 

to be representative of overall conditions was obtained for 

most sites. At 10 percent of the sites, multiple sets of pebble 

counts (3 to 4, table 4) were obtained. Results indicated that 

the standard deviation of the particle-size was small in most 

instances when compared to the corresponding mean particle 

size at each percentile analyzed.

Corresponding data for surface and shallow-subsurface 

bed-material characteristics obtained at all sites provided 

an indication of the vertical variability of the particle size. 

Relations between relative pier scour and relative velocity 

involving surface and shallow-subsurface bed-material characteristics were limited to the data obtained in this study; no 

comparable data were found in the BSDMS database. Thus, 

surface-to-shallow-subsurface relations need to be considered 

as preliminary. Additional data from bridge sites with a wider 

range in bed-material size and gradation are needed to verify 

the relations presented and perhaps use this information to 

improve scour-prediction equations for use at sites with similar bed-material characteristics. An independent dataset also 

would be needed to verify the validity of surface-to-shallowsubsurface relations proposed in this study. 

Although differences between surface and shallowsubsurface bed-material size and gradation resulted in no 

substantial difference in computed scour depths, the spatial 

variability in bed-material composition needs to be considered 

in assessing the site-specific effect that bed-material size and 

gradation differences have on predicted pier scour. Furthermore, there may be combinations not investigated in this study 

where the bed-material differences in the two layers are more 

substantial and lead to greater differences in predicted scour 

depth. Future investigations of relations between bed-material 

characteristics and pier scour will require detailed description of sampling location (areally, vertically, or both) for the 

bed-material data collected. Some difficulty is likely to be 

encountered in collecting subsurface bed-material samples at 

sites where the streambed is always under a substantial depth 

of water or velocities are excessive, which is common in some 

regions of the United States. Conclusions presented for this 

study are generally limited to the range of hydraulic conditions 

and bed-material characteristics demonstrated in the data collected for this study and BSDMS data.

Summary and Conclusions

Determination of pier-scour potential is an important 

consideration in the hydraulic analysis and design of highway 

bridges that cross streams, rivers, and other waterways in the 

United States. A primary goal of ongoing research in the field 

of bridge scour is to improve equations currently used (2011) 

so that scour depth is neither underpredicted nor excessively 

overpredicted. Research efforts, therefore, continue to evaluate 

and improve the accuracy of pier-scour equations. Since the 

early 1990s, research has expanded to include field measurements of pier scour during high-flow conditions. Based on 

these measurements, scour depth for piers in noncohesive, 

nonuniform streambeds with a mixture of sand, gravel, 

cobbles, and boulders (coarse-bed streams) generally is less 

than the scour depth in finer-grained (mostly sand) streambeds 

under similar hydraulic conditions. The difference in scour 

depth is attributed to formation of an armor layer. This finding, 

supported by field measurements collected by the U.S. Geological Survey (USGS), has led to modifications in the scour 

equations recommended by the Federal Highway Administration (FHWA). Pier-scour data collected by the USGS were 
Summary and Conclusions 63

used to develop a bed-material correction factor called K4. 

Collection of additional coarse-bed pier-scour data was recommended to further validate the method for computing K4

 and 

to provide additional data for continued evaluation of pierscour equations. The Montana Department of Transportation 

(MDT) was interested in pier-scour prediction in coarse-bed 

streams because coarse-bed streams are common in Montana. 

Furthermore, the USGS has collected bridge-scour data and 

conducted bridge-scour investigations since the early 1990s 

in cooperation with MDT. Consequently, the USGS and MDT 

began a cooperative study in 2001 to investigate pier scour in 

coarse-bed streams in Montana. 

This report presents the results of a study of pier scour in 

coarse-bed streams during 2001–07 in the mountain and foothill regions of three major river basins in Montana: the Missouri, Yellowstone, and Clark Fork of the Columbia. Drainage 

areas for the streams at bridge sites where measurements were 

collected ranged from about 3 mi2 to almost 20,000 mi2. The 

report also describes how coarse-bed pier-scour measurements 

were collected, shows the extent that the coarse portion of the 

national pier-scour database was expanded, discusses how 

these new data were used to evaluate the relative accuracy of 

various equations for predicting scour in coarse-bed streams, 

and demonstrates how differences in size and gradation 

between surface bed material and bed material underlying the 

surface layer (shallow-subsurface bed material) might relate to 

pier scour.

The equation widely adopted for conducting pier-scour 

analysis and design in the United States was developed at 

Colorado State University in the mid–1970s. A modified version of the equation was later published by FHWA in Hydraulic Engineering Circular No. 18 (HEC-18) and is called the 

HEC-18 equation. The current (2011) version of the HEC-18 

pier-scour equation recommended by the FHWA includes 

the computation of K4, and is referred to in this report as the 

HEC-18-K4Mu equation. Pier-scour research has continued 

and new equations have been proposed that include the K4

correction factor. 

Methods based on limited-detail field procedures 

described in literature and equipment ranging from simple 

scour rods to complex hydroacoustic instrumentation were 

used to obtain pier-scour measurements. Data collected in 

the field for this study included information needed to measure pier scour along with bed material and ancillary data 

needed to compute scour using equations. Bed-material sizes 

used in scour-prediction equations were determined from the 

surface bed material that generally reflected the armor layer. 

The potential interaction between the surface and shallowsubsurface bed-material layers with respect to pier scour 

and armoring also was of interest for this study. Therefore, 

shallow-subsurface bed material also was sampled and data 

were compared to surface bed-material data for each site. Surface and shallow-subsurface bed-material measurements were 

obtained by pebble counts and sieve analyses. Bed-material 

data were used to construct particle-size distribution curves, 

which were used to conduct various coarse-bed pier-scour 

analyses. Surface and shallow-subsurface bed material 

sampled for this study was used to evaluate the sensitivity of 

computed scour to size and gradation differences between the 

two layers, and these data for each layer were used to assess 

the effects of bed-material characteristics on pier scour. 

Pier-scour measurements were made during 2001 through 

2007 at 59 bridge sites on coarse-bed streams. Of those measurements, 103 satisfied criteria requiring that the measured 

approach velocity (Vo) exceed the critical velocity (V′

c50) to 

initiate scour of bed material at the piers and were analyzed 

for this study. Incipient motion computations based on the D50 

particle size indicated that 96 out of 103 pier-scour measurements were made under clear-water scour conditions, when 

the streambed upstream from the bridge is stable and there is 

no substantial incoming sediment supply to the bridge opening. Of the measurements made, 50 percent had an approach 

velocity that equaled or exceeded 70 percent of the critical 

velocity (Vc50) for incipient motion of bed material upstream 

from the bridge. This might indicate that scour was measured 

near the threshold between clear-water and live-bed scour 

(Vo /Vc50 = 1.00) where maximum pier scour was shown in 

laboratory studies. Despite reasonably high approach velocities, measured pier scour was less than a couple of feet in 

depth for 70 percent of the measurements made at the coarsebed sites. Particle size of sampled surface bed material was 

predominantly very coarse gravel based on the D50 particle 

size. Surface bed material at one-half of the sites lacked substantial gradation, as indicated by gradation coefficients (σg) 

less than about 2.0. About 60 percent of the sites were located 

on high-gradient streams based on water-surface slopes that 

exceeded 0.002. The approach depth of flow (yo) in relation 

to the effective pier width (b) was not substantial, with flow 

depth typically less than twice the effective pier width. The 

effects of high-flow duration on pier scour were not investigated because most scour data for a site were collected in a 

single day. Continuous scour measurements made at three sites 

using fixed acoustic transducers, however, indicated that scour 

typically increased as streamflow increased.

Pier-scour data collected for this study were compared 

to pier-scour data from other USGS studies that were used 

to develop pier-scour equations for coarse-bed streams. 

Historical data were retrieved from the Bridge Scour Data 

Management System (BSDMS). The 508 measurements in 

the BSDMS database collected between 1965 and 1998 were 

initially screened down to 272 measurements (BSDMS All) 

with bed material that was noncohesive and composed of sand 

and coarser particles, and then to 90 measurements (BSDMS 

Coarse). The BSDMS Coarse dataset is comparable with data 

collected for this study based on FHWA criteria for D50 and 

D95 particle size that defines coarse bed material. The median 

D50 and D95 bed-material size for data collected for this study 

are about 1.8 and 1.6 times (respectively) the size of D50

and D95 in the BSDMS Coarse dataset. The maximum value 

of relative pier scour (y′

s /b) in the study was 1.00, and the 

maximum values observed in the two BSDMS datasets (All, 

Coarse) were 2.09 and 1.25, respectively. Data from this study 
64 Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 through 2007

also were compared to historical pier-scour measurements 

made in Montana by USGS during large floods. Nine pierscour measurements made at three sites were for peak streamflows approximating the 100-, 10-, and 5-year recurrence 

intervals, which are much greater than any recurrence intervals 

associated with measurements made for this study and provide 

an independent check on the envelope curves bounding the 

upper end of both the data collected for this study and BSDMS 

data combined.

Pier-scour depth was computed for the measurements 

collected for this study using the HEC-18 equation with no K4

correction factor and five pier-scour equations that use K4 for 

the armoring effect of coarse bed material. Computed scour 

depths were compared to measured scour depths using scatter 

plots and boxplots of residuals to evaluate the reliability of the 

equations to predict pier scour. The HEC-18-K4Mu equation 

was the best equation for predicting pier-scour depth in coarsebed streams because the equation generally predicted pier 

scour in closer agreement to measured scour than the other 

equations used for computing pier-scour depth. Furthermore, 

the magnitude of the residuals of underpredicted and overpredicted scour depth was the lowest using the HEC-18-K4

Mu 

equation. 

The effect of bed-material size and gradation on scour 

depth was investigated using data collected for this study, 

the BSDMS All dataset, and the BSDMS Coarse dataset to 

confirm and extend the findings of previous studies. Previous research showed that the equation for computing K4

 can 

produce anomalous results (negative velocity ratio) for rare 

combinations of D50 and D95. None of the surface bed-material 

particle-size data collected for this study produced anomalous 

results in computing K4. Regression analysis conducted in 

this study showed that the D50 and D95 particle sizes have a 

positive relation with each other that is statistically significant 

(r 2 = 0.73 and p < 0.001). In other words, for the data collected for this study, sites that have a larger D50 particle size 

also tend to have a larger D95 particle size. This observation is 

similar to the observation of previous research about the bedmaterial data originally used to develop the equation for the K4

correction factor. 

The relation between relative pier scour (y′

s /b) and relative velocity (Vo

 /Vc50 ) defined by an envelope curve for the 

data collected for this study and the BSDMS Coarse dataset 

were compared to relations observed in laboratory data. The 

envelope curve for data used for this study displays an earlier 

peak, with a decrease in relative pier scour followed by an 

asymptotic rise; the first peak in the curve occurs at a lower 

relative velocity than the clear-water and live-bed threshold 

(Vo /Vc50 = 1.00). The magnitude of relative pier scour (maximum of 1.25) is much less than the published maximum value 

of 2.4 obtained from laboratory data. Furthermore, equilibrium live-bed scour is about 18 percent shallower than the 

maximum pier scour under clear-water conditions, a larger 

reduction than the 10 percent indicated in literature. Armoring 

associated with nonuniform coarse-bed material and unsteadyflow conditions in the field probably accounts for differences 

between envelope curves developed for this study and those 

from previous studies conducted in the laboratory using finegrained material and steady-flow conditions. 

Previous studies showed that surface bed-material gradation affected pier-scour depth, therefore, the relation between 

gradation coefficient (σg ) and relative pier scour was investigated using the pier-scour measurements collected for this 

study. Gradation coefficients associated with surface bed material collected for this study ranged from 1.48 to 4.14, with a 

median of 2.01. Bed-material data collected for this study had 

sufficient gradation so that streambed armoring was possible 

according to criteria developed in previous research. As shown 

in previous research, for data collected for this study and 

the BSDMS Coarse dataset, maximum relative pier scour is 

greater when σg is less than or equal to 2.5. Pier-scour measurements collected for this study and BSDMS confirm a general lack of data for coarse-bed sites with σg

 greater than about 

2.5. The narrow range in gradation coefficients displayed by 

all available data and the nonuniform conditions present in 

the field continue to complicate efforts to isolate the effects of 

gradation on pier scour.

Since the mid–1960s, pier-scour research has focused 

primarily on surface bed material effects on pier scour. 

Surface and shallow-subsurface bed material were collected 

and analyzed for both layers during this study. Median D50

particle sizes for 103 surface and shallow-subsurface bedmaterial samples were about 49 mm and 32 mm, respectively, 

and median D95 particle sizes for 103 surface and shallowsubsurface bed material samples were 122 mm and 86 mm, 

respectively. The median gradation coefficients for surface 

and shallow-subsurface bed material associated with measurements made for this study were 2.01 and 4.14, respectively. 

The combined effect of the larger surface bed-material size 

relative to shallow-subsurface bed-material size and the 

wider range of gradation coefficients in shallow-subsurface 

bed material is typical for the pier-scour measurements made 

for this study and may contribute to reduced pier scour in 

these coarse-bed streams. A significant relation was indicated 

between surface and shallow-subsurface bed-material size, 

for the D50 and D95 particle sizes (for D50, r2 = 0.690 and 

p < 0.001; for D95, r2 = 0.647 and p < 0.001). In other words, 

as the particles in the surface bed material increase in size, the 

particles in the shallow-subsurface bed material also tend to 

increase in size. Based on observed field conditions, and the 

relative coarseness of the bed-material particle sizes for the 

two layers, particle-size data for the shallow-subsurface bed 

material underlying the armor layer also were used to compute 

pier scour by the HEC-18-K4Mu equation. Scour depths computed based on shallow-subsurface particle sizes were then 

compared to both measured scour depths and scour depths 

computed based on the surface bed-material particle-size data. 

Differences in D50 and D95 particle size between the surface 

and shallow-subsurface bed material did not cause substantial 

differences in computed scour. The frequency and magnitude 

of the residual pier-scour depth associated with underpredicted 

and overpredicted scour was about the same regardless of 
References Cited 65

which particle-size data were used with the HEC-18-K4Mu 

equation. 

In addition to examining the effects that differences 

in bed-material size and gradation between the surface and 

shallow-subsurface layers might have on predicted pier-scour 

depth, the combined effect that the bed-material characteristics 

of both layers might have on measured pier scour also was 

examined. Examination of the differences in particle sizes 

between the two layers began by grouping the bed-material 

particle-size data for each layer into classifications that ranged 

from small cobble to very fine gravel. Different combinations 

of surface and shallow-subsurface bed-material size classifications were used to group the pier-scour measurements, and the 

relation between relative pier scour and relative velocity was 

examined for each of these groups of measurements. Based on 

the five envelope curves drawn to define the upper bounds of 

the data for each group, relative pier scour was generally less 

for pier-scour measurements with coarser bed material like 

very coarse gravel or cobbles in the two layers. There were no 

sites where the difference in bed-material particle size between 

the surface or armor layer and shallow-subsurface layer was 

substantial (for example, small cobbles underlain by very fine 

gravel or sand); thus, more detailed assessment of such widely 

contrasting bed-material compositions was not possible. 

Differences in gradation of bed material in the surface and shallow-subsurface layers and the relation of these 

differences to pier scour also were investigated using two 

approaches. First, the differences in relations between gradation coefficients for each layer and relative pier scour were 

compared by examining envelope curves for each set of 

measurements obtained for this study. Maximum relative pier 

scour corresponded to a gradation coefficient of about 2.3 

for the surface bed material and about 4.7 for the shallowsubsurface bed material. A composite curve was proposed 

that incorporates the separate relation of both layers. For the 

second approach, the variation in gradation over the range of 

the particle-size distribution curve was defined using ratios of 

shallow-subsurface-to-surface gradation coefficient (relative gradation, Rg) for each of 103 pier-scour measurements 

collected for this study. The measurements were ranked by 

Rg from largest to smallest, selected percentiles were determined, and the relative pier scour and relative velocity for 

each measurement in the percentile were plotted to examine 

the relation between relative pier scour and relative velocity 

for each percentile. Envelope curves bounding the measurements in each percentile were drawn. Maximum relative scour 

associated with each envelope curve was concentrated at about 

the same relative velocity. Based on these curves, when the 

difference between the shallow-subsurface and surface bedmaterial gradation coefficients increases (higher values of Rg ), 

pier scour decreases. 

Finally, the combined influence of relative differences 

in surface and shallow-subsurface bed-material size and 

gradation on pier scour was investigated by defining a bedmaterial variability index (Iv

) based on ratios of particle sizes 

and gradation coefficients between layers. The bed-material 

variability index was calculated for each of the 103 measurements collected for this study and the measurements were 

ranked by Iv from largest to smallest. Selected percentiles were 

determined and the relative pier scour and relative velocity for 

each measurement in the percentile were plotted to examine 

the relation between relative pier scour and relative velocity 

for each percentile. Envelope curves bounding the measurements in each percentile were drawn. Maximum relative scour 

associated with each envelope curve was concentrated at about 

the same relative velocity. Based on these curves, as differences in particle size and gradation between the surface and 

shallow-subsurface bed-material layers increase (higher values 

of Iv), pier scour decreases. 

Field measurement of scour at bridge piers requires the 

use of specialized equipment and methods to obtain accurate 

data under difficult field conditions, and interpretive steps are 

involved to compute the measured pier-scour depth. Runoff 

conditions during the study mostly were limited to bankfull 

discharge approximating the 1.5- to 2-year recurrence-interval 

flood, which have a 67- and 50-percent chance, respectively, 

of happening in any year. Lack of higher flows having greater 

velocity intensities may indicate that scour depths were limited by hydrologic conditions. Pier-scour depths measured in 

this study also may have been limited by streambed armoring 

and unsteady flow conditions. A single bed-material sample 

considered to be representative of overall conditions was 

obtained for most sites. Relations between relative pier scour 

and relative velocity involving surface and shallow-subsurface 

bed-material characteristics were limited to the data obtained 

in this study; no comparable data were found in the BSDMS 

database. Thus, surface-to-shallow-subsurface relations need 

to be considered as preliminary. Conclusions presented for this 

study are generally limited to the range of hydraulic conditions 

and bed-material characteristics demonstrated in the data collected for this study and the BSDMS data.

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