In Cooperation with the American Water Works Association Research Foundation

Augmenting Two-Dimensional Hydrodynamic 

Simulations with Measured Velocity Data to Identify 

Flow Paths as a Function of Depth on Upper St. Clair 

River in the Great Lakes Basin

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2005-5081
Augmenting Two-Dimensional Hydrodynamic 

Simulations with Measured Velocity Data to Identify 

Flow Paths as a Function of Depth on Upper St. Clair 

River in the Great Lakes Basin

By David J. Holtschlag and John A. Koschik

Prepared in cooperation with the 

American Water Works Association Research Foundation

Scientific Investigations Report 2005-5081

U.S. Department of the Interior

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

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia 2005

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Suggested citation:

Holtschlag, D.J., and Koschik, J.A., 2005, Augmenting two-dimensional hydrodynamic simulations with 

measured velocity data to identify flow paths as a function of depth on Upper St. Clair River in the 

Great Lakes Basin: U.S. Geological Survey, Scientific-Investigations Report 2005-5081, 42 p.
Contents

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

Introduction.......................................................................................................................................................1

Purpose and Scope ........................................................................................................................................2

Location.........................................................................................................................................................2

Previous Studies............................................................................................................................................5

Approach...........................................................................................................................................................5

Velocity Measurements in Upper St. Clair River..........................................................................................5

Hydrodynamic Simulation of Upper St. Clair River...................................................................................11

Integration of Velocity Measurements With Flow Simulations..................................................................13

Results.............................................................................................................................................................17

Velocity Survey ...........................................................................................................................................17

Model Enhancements..................................................................................................................................18

Mesh Refinement.....................................................................................................................................18

Bendway Correction................................................................................................................................22

Peclet Number .........................................................................................................................................24

Integration of Velocity Information with Hydrodynamic Simulations .......................................................26

Identification of Flow Path as a Function of Flow Depth ...........................................................................30

Summary and Conclusions .............................................................................................................................32

Acknowledgments.........................................................................................................................33

References Cited .............................................................................................................................................33

Appendix 1 Summary of acoustic Doppler current profiler velocity data for transects on the 

Upper St. Clair River ......................................................................................................................................35

Figures

1-6. Maps showing-- 

1. Upper St. Clair River study area in the Great Lakes Basin ...................................................................3

2. Water depths in the northern part of Upper St. Clair River...................................................................4

3. Acoustic Doppler current profiler (ADCP) transects in Upper St. Clair River (panel 1)......................6

4. ADCP transects in Upper St. Clair River (panel 2)...............................................................................7

5. ADCP transects in Upper St. Clair River (panel 3)...............................................................................8

6. ADCP transects in Upper St. Clair River (panel 4)...............................................................................9

7. Histogram of velocity measurement distribution by layer during the July 2003 ADCP survey 

 of Upper St. Clair River ................................................................................................................. 10

8. Graph showing flows and water levels during the July 2003 ADCP survey of Upper 

St. Clair River....................................................................................................................................11

9. Map showing the finite-element mesh for the public water intake version and the refined 

version of the Upper St. Clair River model.......................................................................................12

10. Variogram showing spatial structure of horizontal velocity innovations in layer 1 ..........................15

11. Graph showing the cumulative distribution function of flow velocities measured during the

July 2003 ADCP survey of Upper St. Clair River.............................................................................18

12-14. Graphs showing--

12. Relation between easting velocity components simulated by the public water intake 

version and the refined version of the Upper St. Clair River hydrodynamic model .........................19

13. Relation between northing velocity components simulated by the public water intake version 

and the refined versions of the Upper St. Clair River hydrodynamic model. ...................................20

14. Relation between water levels simulated by the public water intake version and the refined 

iii
iv

version of the Upper St. Clair River hydrodynamic model. ..............................................................20

15-17. Maps showing--

15. Simulated velocities downstream from the Blue Water Bridge on Upper St. Clair River 

using the refined mesh without the bendway correction and with a Peclet number of 21.9. ............21

16. Simulated vorticity in the northern part of Upper St. Clair River .....................................................22

17. Change in velocities simulated by use of the refined mesh with implementation of the Bendway

Correction and a Peclet number of 21.9. ...........................................................................................23

18. Graph showing relation between simulated and measured velocities averaged within 

elements for the initial Peclet number of 21.9 and the revised Peclet number of 200. .....................24

19. Map showing simulated velocity field downstream from the Blue Water Bridge on Upper 

St. Clair River on refined mesh with Bendway Correction and Peclet number of 200 .....................25

20. Graph showing the relation between average measured and integrated velocities in layer 1 

of Upper St. Clair River ....................................................................................................................27

21-22. Maps showing--

21. Integrated near-surface velocities in layer 1 of Upper St. Clair River downstream from 

the Blue Water Bridge .......................................................................................................................29

22. Flow paths as a function of depth (layer) in Upper St. Clair River from reverse particletracking for average flow conditions during July 1–3, 2003. ..........................................................31

Tables

1. Distribution of acoustic Doppler current profiler velocity innovations among layers. .......................14

2. Parameters used in model variograms of velocity innovations ............................................................17

3. Summary of relations between element-averaged measured and integrated velocities .......................28
v

Conversion Factors and Datums

Multiply By To obtain

Length

International foot (Int. ft) 0.3048 (exactly) meter (m)

U.S. Survey foot (ft) 0.3048006 (approximately) meter (m)

mile (mi) 1.609 kilometer (km)

meter (m) 3.2808 foot (ft)

centimeter (cm) 0.3937 inch (in.)

Area

square foot (ft2) 0.09290 square meter (m2)

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)

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

°F=(1.8×°C)+32

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

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the International Great Lakes Datum of 1985 (IGLD 85). 

Horizontal coordinate information is referenced to the Michigan State Plane Coordinate System (MSPCS), Southern Zone 

(2113), based on the North American Datum of 1983 (NAD 83). MSPCS units are in International feet. 

Altitude, as used in this report, refers to distance above the vertical datum. 

Sampling and signal frequencies of acoustic Doppler current profilers are given in hertz (Hz) and kilohertz (kHz).
 1

Augmenting Two-Dimensional Hydrodynamic Simulations 

with Measured Velocity Data to Identify Flow Paths as 

a Function of Depth on Upper St. Clair River in the Great 

Lakes Basin

By David J. HoltschlagF

1

F and John A. KoschikF

2

F

Abstract

Upper St. Clair River, which receives outflow from Lake Huron, is characterized by flow velocities that exceed 7 feet per 

second and significant channel curvature that creates complex flow patterns downstream from the Blue Water Bridge in the 

Port Huron, Michigan, and Sarnia, Ontario, area. Discrepancies were detected between depth-averaged velocities previously 

simulated by a two-dimensional (2D) hydrodynamic model and surface velocities determined from drifting buoy deployments. 

A detailed ADCP (acoustic Doppler current profiler) survey was done on Upper St. Clair River during July 1–3, 2003, to help 

resolve these discrepancies. 

As part of this study, a refined finite-element mesh of the hydrodynamic model used to identify source areas to public water 

intakes was developed for Upper St. Clair River. In addition, a numerical procedure was used to account for radial accelerations, 

which cause secondary flow patterns near channel bends. The refined model was recalibrated to better reproduce local velocities 

measured in the ADCP survey. ADCP data also were used to help resolve the remaining discrepancies between simulated and 

measured velocities and to describe variations in velocity with depth. 

Velocity data from ADCP surveys have significant local variability, and statistical processing is needed to compute reliable 

point estimates. In this study, velocity innovations were computed for seven depth layers posited within the river as the differences between measured and simulated velocities. For each layer, the spatial correlation of velocity innovations was characterized by use of variogram analysis. Results were used with kriging to compute expected innovations within each layer at 

applicable model nodes. Expected innovations were added to simulated velocities to form integrated velocities, which were used 

with reverse particle tracking to identify the expected flow path near a sewage outfall as a function of flow depth. 

Expected particle paths generated by use of the integrated velocities showed that surface velocities in the upper layers 

tended to originate nearer the Canadian shoreline than velocities near the channel bottom in the lower layers. Therefore, flow 

paths to U.S. public water intakes located on the river bottom are more likely to be in the United States than withdrawals near 

the water surface. Integrated velocities in the upper layers are generally consistent with the surface velocities indicated by drifting-buoy deployments. Information in the 2D hydrodynamic model and the ADCP measurements was insufficient to describe 

the vertical flow component. This limitation resulted in the inability to account for vertical movements on expected flow paths 

through Upper St. Clair River. A three dimensional hydrodynamic model would be needed to account for these effects. 

Introduction

Simulated velocities from a two-dimensional (2D) hydrodynamic model were used to describe the advective movement of 

water through the St. Clair-Detroit River Waterway (Holtschlag and Koschik, 2004). This advective component was used with 

particle tracking to identify source areas to U.S. public water intakes. Depth-integrated velocities simulated by the model generally paralleled the shoreline in Upper St. Clair River. Source areas for the city of Port Huron’s public water intakes were consequently mapped to the U.S. side of the waterway upstream from the intakes. Simulated depth-integrated velocities, however, 

were inconsistent with surface velocities indicated by drifting-buoy deployments, which showed that near-surface water flowed 

from the Canadian to the U.S. side of the waterway within Upper St. Clair River (Holtschlag and Aichele, 2001). Therefore, the 

F

1

F By David J. Holtschlag

F

2

F John A. Koschik
simulated velocities may not be sufficient to describe the complex flow pattern in Upper St. Clair River, and source areas to the 

city of Port Huron’s public water intakes may be inadequately identified. 

In response to the discrepancies between near-surface flow velocities indicated by drifting-buoy deployments and depthaveraged velocities indicated by model simulations in Upper St. Clair River, the U.S. Geological Survey (USGS) in cooperation 

with the American Water Works Association Research Foundation (AwwaRF), began a study to resolve these discrepancies. 

Purpose and Scope

The purpose of this report is to refine the identification of flow paths to the city of Port Huron’s public water intake. 

Improvements are based on enhancing the calibration and resolution of depth-integrated velocities simulated by an existing 2D 

hydrodynamic model and on adjusting depth-integrated velocities to reflect systematic variations in flow velocities with depth, 

as measured by means of detailed ADCP (acoustic Doppler current profiler) surveys of flow velocity in Upper St. Clair River. 

Only that part of the existing hydrodynamic model from the head of St. Clair River to the mouth of Black River, a distance of 

about 2.5 mi, was recalibrated and refined. 

Location

Upper St. Clair River is part of the international boundary between the United States and Canada (XFigure 1X). The river, 

which connects Lake Huron with Lake St. Clair, is a major navigational and recreational resource of the Great Lakes Basin. St. 

Clair River extends about 39 mi from its head at the outlet of Lake Huron near Port Huron, Mich., to an extensive delta area 

that discharges into Lake St. Clair. Throughout the river’s length, water levels (water-surface elevations) decrease about 5 ft as 

an average of 182,000 ft3/s is discharged from a drainage area of about 222,400 mi2. Local tributaries to Upper St. Clair River 

include Black River, which has a drainage area of about 746 mi2 and an average flow of about 489 ft3/s. 

In this report, Upper St. Clair River refers to the reach of St. Clair River from the National Oceanic and Atmospheric 

Administration (NOAA) water-level gaging station Fort Gratiot, Mich., (station 9014098), which is near the head of St. Clair 

River, to the NOAA station Mouth of the Black River, Mich. (station 9014090), a distance of about 2.5 mi. Within this reach, 

water levels commonly fall about 0.6 ft. Depths of flow can be as great as 78 ft, with maximum depths in the northern half of 

the river (fig. 2). Upper St. Clair River bends toward the left (east) as water flows generally towards the south. In some areas, 

flow velocities exceed 7 ft/s. Complex secondary flow circulation patterns, which are associated with high flow velocities and 

channel curvature, have been measured downstream from the Blue Water Bridge. 

2 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 1. Upper St. Clair River study area in the Great Lakes Basin.

Introduction 3
Figure 2. Water depths in the northern part of Upper St. Clair River.

4 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Previous Studies

Numerous investigators have contributed to the understanding of flow on the St. Clair-Detroit River Waterway. Schwab and 

others (1989) compared currents measured on Lake St. Clair with particle-tracking results based on 2D hydrodynamic simulations. Tsanis and others (1996) implemented the RMA2 hydrodynamic model on St. Clair and Detroit Rivers; results indicated 

that simulated currents closely matched field measurements of drifting buoys. Williamson and others (1997) developed a 2D 

finite-element model of the St. Clair-Detroit River system for the Canadian Coast Guard for water-level prediction and assessment of structures in the river systems. The U.S. Army Corps of Engineers (USACE) Waterway Experiment Station (WES) 

in Vicksburg, Miss., developed a prototype 2D model of the St. Clair-Detroit River Waterway for the Detroit District USACE 

(Ron Heath, USACE-WES, written commun., 1999). The resulting model was modified and adapted for use in a joint study 

by Environment Canada and the Detroit District USACE, to assess the effects of encroachments on water levels in St. Clair and 

Detroit Rivers (Aaron Thompson, Environment Canada, written commun., July 2000). An RMA2 hydrodynamic model was 

initially applied to the St. Clair-Detroit River Waterway by matching water-surface elevations and flows in numerous branches to 

measured values (Holtschlag and Koschik, 2002). Subsequently, this model was refined by including depth-averaged velocities 

from ADCP surveys in the calibration criteria. Techniques used to map source areas to public water intakes are documented by 

Holtschlag and Koschik (2004). 

Approach

The study described in this report integrates velocity measurements and hydrodynamic model simulations in Upper St. 

Clair River to identify flow paths near sewage outfalls used by the city of Port Huron, Mich. The velocity data were a basis for 

evaluating and enhancing part of an existing hydrodynamic model of the St. Clair-Detroit River Waterway that was developed 

for the Michigan Source Water Assessment Program (Holtschlag and Koschik, 2004). Finally, the velocity data were integrated 

with model simulations to describe the variation of horizontal flow velocities with depth. 

Velocity Measurements in Upper St. Clair River

A detailed velocity survey of the upper 2.5-mile reach of St. Clair River was done jointly by the USGS and the USACE 

during July 1-3, 2003. Workhorse Rio Grande ADCP units by RD Instruments were used for the survey. ADCP units used in 

this study emit acoustic signals through four ceramic transducers. The transducers are on the underside of the unit and are oriented downward with a 20-degree offset from the vertical and outward with a 90-degree horizontal angle of separation. Following a signal emission, the four transducers detect reflected signals from backscattering material in the water column. Returning 

signals are subdivided into a set of time intervals that correspond to small (25-100 cm) intervals of flow depth. The phase shift 

in returning signals is used to compute the northing, easting, and vertical velocity components for each depth interval by use 

of the Doppler principle (Simpson, 2001). A collection of velocity data with depth for each acoustic signal is referred to as a 

“velocity ensemble.” 

During the 2003 survey, ADCP units were attached to boats that made single-transect measurements across the river. For 

each transect, an attempt was made to maintain the boat speed at a rate lower than the local water speed. With acoustic signals 

typically emitted at twice per second (sampling at 2 Hz), this resulted in velocity ensembles at about 1-ft intervals along each 

transect. Transects were separated by about 100-ft intervals, thus providing 127 single-transect ADCP flow measurements 

across Upper St. Clair River (figs. 3-6). 

Approach 5
Figure 3. Acoustic Doppler current profiler (ADCP) transects in Upper St. Clair River (panel 1).

6 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 4. ADCP transects in Upper St. Clair River (panel 2).

Approach 7
Figure 5. ADCP transects in Upper St. Clair River (panel 3).

The 2.1 million velocity measurements obtained during the survey were subdivided into seven discrete layers corresponding to depth intervals having a thickness of 6.56 ft (2 m) or more (fig. 7). Among these intervals, the top (first) layer was thicker 

(8.4 ft) than most other layers because no velocity measurements were available at depths less than 3.4 ft below the water surface. Near-water-surface velocities are lacking because deployed ADCP units were submerged below the water surface by about 

1 ft. In addition, ADCP measurements require time for vibration of the ceramic transducer plates to dampen sufficiently before 

detection of incoming signals is possible. This time delay results in a loss of data just below the ADCP unit that is referred to 

as the blanking distance. The bottom (seventh and deepest) layer also has a thickness greater than 6.56 ft because it contains all 

velocity measurements greater than 41.2 ft below the water surface. The subdivision of measurements served as a mechanism to 

detect possible differences in velocity with depth. 

8 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 6. ADCP transects in Upper St. Clair River (panel 4).

Approach 9
Figure 7. Histogram of velocity measurement distribution by layer during the July 2003 ADCP survey of Upper St. Clair River.

USGS measurements were made using a 600-kHz ADCP unit and a bin size of 1.64 ft (50 cm). In contrast, USACE measurements were made using a 1200-kHz unit and bin sizes of 0.82 ft (25 cm) and 3.28 ft (100 cm). The 600-kHz unit obtained 

velocity information throughout the deepest water transected during the survey. The 1200-kHz unit was unable to measure some 

isolated areas where depths of water exceeded 60 ft, but could yield higher resolution of velocities (smaller bin sizes) than the 

600-kHz unit. Despite minor differences in instrumentation, mounting characteristics, and boats, flow measurements by USGS 

and USACE crews were consistent (fig. 8). Some measurements that indicate less than expected flows at transects 0480, 0481, 

0500, and 0520 may be related to shoreline obstructions, such as docked freighters, which resulted in shortened transects and 

underestimated flows on the U.S. side of the river. With the exception of these four measurements, the measured flow had a 

mean of 176,000 ft3/s and a standard deviation of 3.1 percent. 

10 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 8. Flows and water levels during the July 2003 ADCP survey of Upper St. Clair River.

Hydrodynamic Simulation of Upper St. Clair River

The generalized 2D hydrodynamic code RMA2 (Donnell and others, 2003) was applied to the St. Clair-Detroit River 

Waterway as part of the Michigan Source Water Assessment Program (Holtschlag and Koschik, 2004). The finite-element 

method is used within RMA2 to solve the continuous equations governing dynamically changing horizontal velocity components and water levels in surface-water bodies. The finite-element method facilitates this solution by discretizing the waterway 

geometry into a mesh composed of quadrilateral and triangular elements. Elements are defined by corner nodes at the vertices 

of the elements and by midside nodes midway between vertices. 

Hydrodynamic simulations result in predictions of easting UNode( x y, ) and northing V Node( x y, ) velocity components and 

water levels at model nodes based on conditions specified at the boundaries of the waterway, the description of the waterway 

geometry, and model parameters. Horizontal flow components and water levels elsewhere within the waterway are estimated by 

quadratic interpolation across elements from the defining nodes. The density of nodes and elements determines the maximum 

resolution of flow simulations and affects the stability of the numerical solution. 

The mesh for the public water intake version of the hydrodynamic model developed for source water assessment of the St. 

Clair-Detroit River Waterway consisted of 30,306 elements defined by 90,306 nodes (Holtschlag and Koschik, 2004). Within 

Upper St. Clair River, this mesh contained 1,513 quadratic elements and 42 triangular elements defined by 4,868 nodes. To better simulate the complex flow patterns observed in Upper St. Clair River and to enhance the numerical stability of the solution, 

the mesh for the public water intake version was refined: each element was subdivided into four subelements, with a corresponding increase in the number of nodes. The resulting mesh, referred to as the “refined mesh” in this report, consists of 6,052 

quadrilateral elements and 168 triangular elements defined by 18,981 nodes (fig. 9). Typical elements were 35 ft wide and 120 

ft long, thus having an aspect ratio in the direction of flow of 0.29. 

Approach 11
Figure 9. The finite-element mesh for the public water intake version and the refined version of the Upper St. Clair River model.

12 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
In addition to refining the mesh, a numerical procedure termed the “Bendway Correction” was implemented to improve 

predictions of flow around channel bends. Conventional RMA2 depth-averaged calculations of flow around a bend frequently 

overestimate streamwise velocities on the inside bank of a river bend (Donnell and others, 2003). When water flows around a 

bend, a radial acceleration is developed that forces the near-surface waters to the outside bank and the water near the bed to the 

inside bank. This results in a secondary or helical flow pattern that causes sediment aggradation on the inside of curves and 

soil erosion on the outside of curves. RMA2 normally cannot simulate this effect accurately. To improve predictions of depthaveraged velocities around curves, a secondary flow corrector, termed the “Bendway Correction,” was made available with the 

release of RMA2 version 4.35 (Donnell and others, 2003). 

The Bendway Correction works by solving an additional equation for the transport of streamwise vorticity, where vorticity is a measure of flow rotation. Streamwise vorticity occurs in the vertical plane, perpendicular to the direction of flow, and is 

related to the radial accelerations that cause helical flow patterns around channel bends (Donnell and others, 2003). To lessen 

memory requirements for RMA2, the calculation of velocities and depths (phase 1) is separated from the calculation of vorticities (phase 2). 

To implement the Bendway Correction in the Upper St. Clair River model, a conventional hydrodynamic solution was 

obtained before activating the Bendway Correction. Although not activated initially, a BVL card (Donnell and others, 2003) 

was used to specify a vorticity of zero at the upstream flow boundary. Then, using the REV card (Donnell and others, 2003), 

the simulation was revised to activate the Bendway Correction by specifications on the VO and TV cards (Donnell and others, 

2003). Parameters on the VO card were used to switch the Bendway Correction on, to specify a 0.001-ft convergence criterion, 

and to specify vorticity coefficients. Default values for coefficients listed by Donnell and others (2003) were used. The TV card 

controls the number of iterations for vorticity calculations. In this simulation, the maximum number of iterations between phase 

1 and phase 2 computations was set to 100. The maximum number of phase 2 iterations was set to 100. Finally, the minimum 

number of successfully converging passes between phase 1 and phase 2 iterations was set to 10 (Donnell and others, 2003). The 

hydrodynamic model of Upper St. Clair River was recalibrated to improve the match between measured and simulated velocities. Recalibration was accomplished by adjusting the Peclet number, which automatically assigns eddy viscosities to individual 

elements. The Peclet formula can be written as 

where E is the eddy viscosity, ρ is the fluid density,

u is the average elemental velocity, 

dx is the length of the element in the streamwise direction, and 

P is the Peclet number. 

Higher Peclet numbers (lower eddy viscosities) allow simulated velocities to readily change direction. Minimum eddy 

viscosity values are required, however, to maintain numerical stability. Although it is difficult to theoretically establish a value 

for Peclet numbers or eddy viscosities, analogy with physical conditions suggests that turbulence exchanges depend on the 

momentum of the fluid, spatial gradients of the velocity, and the scale of the flow phenomenon, characterized here by length of 

the element in the streamwise direction (Donnell and others, 2003). Therefore, as the element size or flow velocities increase, 

eddy viscosity also should increase. The Peclet method automatically adjusts the eddy viscosity after each iteration, on the basis 

of element sizes and simulated velocities. 

Integration of Velocity Measurements With Flow Simulations

This section describes the procedures used to integrate velocity measurements with simulated flows. Essentially, simulated 

velocities were estimated at ensemble locations by quadratic interpolation from simulation results at model nodes. Then, the differences between simulation estimates and measured velocities were computed for each model layer at all ensembles. These differences are referred to as “innovations.” The spatial coherency of these velocity innovations was described by use of variogram 

analysis. A measure of this coherency provided a basis for computing the expected innovation across a grid of 20-ft-square cells 

spanning the model area by use of the kriging equations. The expected velocity innovations at model nodes were estimated by 

use of bilinear interpolation from this grid. The expected innovations at model nodes were added to simulated flows to form the 

integrated velocity information for each layer. 

The simulated (depth-averaged) easting and northing velocity components were estimated for each ADCP ensemble by 

quadratic interpolation, Q

f , using the easting and northing coordinates of the ensemble, Ens(x,y), and simulated velocities and 

specified locations at the set of nodes forming the element containing the ensemble, Node(x.,y.). The resulting depth-averaged 

estimates for the easting and northing velocity components are designated as UEns(x,y ) Q ,y f ~ ( . .)

, ( , ) UNode x Ens x y ~ = ( )

 Approach 13 

E = ρ.u.dx /P
and , V f Ens x( ,y Q ) ,y

~ ( . .), ( , ) V E Node x ns x y ~ = ( ) respectively. 

Easting and northing velocity components measured in bins of 0.25- to 1.00-m heights throughout most of the water column were averaged within their nominal 2-m layers (XTable 1X) and designated as uEns x( ,y ),l

_ and v Ens x( ,y ),l

_ . Finally, velocity innovations were defined as the differences between average measured velocities in each layer and simulated velocities as 

Ens x( ,y ),l Ens(x,y ),l E U ns(x,y )

~ υ = u

_ _ . and 

Variogram analyses were used to characterize a measure of the spatial correlation structure of the velocity innovations as a 

function of separation distance, h; this correlation measure is referred to as the variogram and is denoted g(h). Easting innovations were analyzed independently from northing innovations. Preliminary analysis of vertical velocity components indicated 

that they were highly variable and contained no appreciable spatial correlation structure. Therefore, only horizontal velocity 

components are evaluated in this report. 

Table 1. Distribution of acoustic Doppler current profiler velocity innovations among layers.

Layer 

number

Depth 

range of 

layer (feet)

Number 

of velocity 

measurements in 

layer

Number of 

depth-averaged 

velocity 

measurements

Easting velocity innovation 

(feet per second)

Northing velocity innovation 

(feet per second)

Mean Standard 

deviation

Mean Standard 

deviation

1 < 8.38 315,993 134,357 0.0719 0.6824 -0.2838 0.9109

2 8.38 – 14.9 449,134 131,720 .0594 .6413 -.2144 .8704

3 14.9 – 21.5 422,182 124,485 .0236 .6492 -.1588 .8506

4 21.5 – 28.1 371,164 112,051 -.0130 .6651 -.0772 .8078

5 28.1 – 34.6 269,060 85,352 -.0203 .7119 .0652 .8271

6 34.6 – 41.2 155,688 50,505 -.0130 .7429 .2422 .8718

7 > 41.2 98,233 19,622 .0783 .8263 .5579 .9681

In this report, Euclidian separation distances were computed for all possible pairs of innovations on the basis of ensemble 

locations. Separation distances were subdivided into bins having a width of 2 ft, up to a maximum separation distance of 300 

ft. Thus, all ensembles that were separated by less than 2 ft were placed in the first bin; ensembles with separation distances of 

2 to 4 ft were placed in the second bin, and so on, until the 150th bin containing the maximum separation distances of 298 to 

300 ft was formed. For each separation bin shown on the x-axis, the mean value of one-half the squared differences between all 

innovation pairs is plotted on the y-axis as the empirical (semi) variogram, (fig. 10). 

14 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin

U f Ens x( ,y Q ) ,y

~ ( . .), ( , ) U E Node x ns x y ~ = ( )

Ens x( ,y ),l Ens(x,y ),l E V ns(x,y )

~ ν = v

_ _ .

γ( ) h
Figure 10. Variogram showing spatial structure of horizontal velocity innovations in layer 1.

The empirical variograms show a generally increasing magnitude from 0 to 300 ft of separation distance. The rate of 

increase, however, generally decreases with increasing separation distance until the variograms appear to approach an asymptotic limit, or sill, that is less than the variance of the variogram. This indicates that the velocity innovations are consistent with 

the hypothesis of intrinsic stationarity (de Marsily, 1986), which is assumed in the subsequent application of the kriging equations for estimating expected innovations. 

The empirical variogram is approximated by model variograms to provide a mathematically useful form for estimating 

expected innovations. In this report, a set of three nested variogram models was fit to the empirical variograms for use in estimation, such that the model variogram was equal to ( ) ( ) ( ) N h h + + γS γE h , where, γN ( ) h =CN is a nugget model 

component, γ ( ) S h is a spherical model component (equation 2), and γ ( ) E h is an exponential model component (equation 3). 

All variogram models were assumed to be isotropic. 

(2)

(3)

3 1 5 0 5 . for 

 otherwise

. / . ( / )

( )={ S S S S

S

S

C h L h L h L

h

C

_ [ ] <

γ }

γS (h)= CE [1-e-h / LE ]

The estimated parameters of all variogram models, CN, CS, LS, CE, and LE are shown in table 2. 

Approach 15

γ( ) h ~
Results of variogram analyses were used to solve the point kriging equations, which provide the best linear unbiased 

estimates (de Marsily, 1986) of velocity innovations. In an area containing n ensembles where velocity innovations have been 

computed near a target point indexed by 0, the point kriging equations can be written as

n n n 

n

n

.

.

. .

.

. .

.

. .

.

. .

.

. .

.

. .

.

.

... .

.. .

.. .

..

.

 . .

n

n

 

 ,

where, , for example, is the magnitude of the model variogram between velocity measurements indexed by 1 and 2. 

Similarly, , represents the model variogram between sample point 1 and target point 0. The λ's are weights applied to nearby 

velocity innovations to compute a local average, and μis a Lagrange multiplier that constrains the sum of the λ's to 1. An 

estimate of the easting innovation magnitude at target point zero based on the nearest n easting innovations is 

.

Estimates of northing innovations are computed similarly. 

The shape of the variograms (fig. 10), which is inversely related to the weights, indicates that the weights would generally 

decrease with separation distance from about 0 to 250 ft. Beyond 250 ft, which is considered the range of the variogram, all 

innovations would be weighted the same. This range was anticipated from analysis of previous ADCP data on St. Clair River 

and was used to design the 2003 ADCP survey so that the transect interval would be about 100 ft. When estimating the expected 

innovation from a network of sample points, the geometric configuration of ensemble locations relative to the point of estimation 

also will affect the weighting factors. 

In this report, block kriging (de Marsily, 1986) was used to map the expected value of the easting velocity innovation, 

Ε υ x y, ,l , and the northing velocity innovation, Ε ν x y, ,l , at the center of each 20-ft-square cell within a 282-row 

by 633-column grid spanning the model area. In block kriging, a 3-by-3 array of equally spaced points is targeted for each cell. 

The average of the nine point-kriging estimates is used to represent the block (cell) average. The maximum search radius used 

in kriging was 300 ft. The minimum easting of the grid was 13,642,100 ft (MSPC 83, southern zone, Int. ft.), and the minimum 

northing of the grid was 542,410 (MSPC 83, southern zone Int. ft). 

16 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin

0 0 0 1 , , , , ˆ n i

x y l x y l i = υ = λ ⋅ υ

12 γ

~

10 γ

~
Table 2. Parameters used in model variograms of velocity innovations.

Layer Depth 

range

(feet)

Velocity

innovation

component

Variogram model parameters

Nugget

(CN)

Spherical 

model

Exponential

model

Variance (feet

squared

per second 

squared) Scale

(CS)

Length

(LS)

Scale

(CE)

Length

(LE)

1 <8.38 Easting 0.175 0.045 10 0.185 95 0.466

Northing .160 .020 10 .470 110 .830

2 8.38-

14.9

Easting .130 .070 10 .170 110 .411

Northing .120 .040 10 .420 100 .758

3 14.9-

21.5

Easting .140 .090 12 .140 90 .421

Northing .125 .040 10 .400 85 .724

4 21.5-

28.1

Easting .160 .110 10 .130 100 .442

Northing .154 .050 14 .350 90 .653

5 28.1-

34.6

Easting .210 .120 10 .130 120 .507

Northing .190 .075 10 .310 70 .684

6 34.6-

41.2

Easting .235 .120 12 .140 100 .552

Northing .205 .080 10 .330 60 .760

7 >41.2 Easting .290 .150 20 .200 100 .683

Northing .270 .120 10 .490 90 .937

Results

Velocity Survey

During the July 2003 ADCP survey of Upper St. Clair River, 2,100,954 measurements of velocity were obtained. Easting 

velocities generally (98 percent of the time) ranged from –4.1 to 3.3 ft/s, where negative velocities indicate a westerly direction 

(fig. 11). Similarly, northing velocities generally ranged from –6.9 to 1.3 ft/s. Resultant speeds generally ranged from 0.3 to 

7.3 ft/s. In contrast to the predominantly southerly flow direction, northerly velocities occurred along the Canadian shoreline 

near a large eddy that was about 1,800 ft downstream from the Blue Water Bridge near the waterfront building (fig. 2). The 

eddy, which circulated counterclockwise, was approximately 450 ft wide and 1,000 ft long. 

Results 17
Figure 11. Cumulative distribution function of flow velocities measured during the July 2003 ADCP survey of Upper St. Clair River.

Ninety-nine percent (2,081,454) of the 2003 ADCP velocity survey measurements were located within the finite-element 

mesh of the hydrodynamic model. Among the elements forming the mesh, about 35 percent (2,175 of 6,220) had no ADCP 

measurements, 50 percent had 240 or more measurements, 25 percent had 497 or more measurements, and 10 percent had 826 

or more measurements. Element-averaged velocities are used subsequently to compare measured and simulated results. 

Model Enhancements

As mentioned previously, a subreach of the public water intake version of hydrodynamic model of the St. Clair-Detroit 

River Waterway (Holtschlag and Koschik, 2004) was enhanced to improve identification of flow paths near public water intakes 

operated by the city of Port Huron, Michigan. The subreach, referred to as Upper St. Clair River, was enhanced by (1) refining 

the finite-element mesh to provide additional detail on the local flow conditions and greater numerical stability, (2) implementing the Bendway Correction for improved flow simulations around channel bends, and (3) recalibrating the model on the basis of 

velocity data obtained during the July 2003 ADCP survey. The following sections describe the results of these enhancements. 

Mesh Refinement

In Upper St. Clair River, mesh refinement quadrupled the number of elements to 6,220 and increased the number of nodes 

to 18,981 with respect to the public water intake version of the hydrodynamic model developed for the Source Water Assessment Program. The additional nodes provide more points for velocity and water-level simulation, thus increasing the resolution 

of flow information. Simulation results at individual nodes common to both model versions, however, were expected to produce 

generally consistent results. 

18 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
To verify the consistency of simulation results between the two model versions, both versions were simulated by use of 

steady-state boundary specifications that approximated conditions during the July 2003 ADCP survey. The average measured 

flow of 175,128 ft3/s was specified as the upstream boundary at the head of St. Clair River near the Fort Gratiot, Mich., gaging 

station. An average measured water level of 576.739 ft was specified as the downstream boundary on Upper St. Clair River, 

based on records at the Mouth of the Black River, Mich. gaging station. No wind boundary conditions were specified. 

Simulated flow velocities and water levels by the two model versions were in close agreement (figs. 12–14). Minor 

discrepancies may be related to greater flow resolution in the refined mesh and variability due to the precision specified for 

numerical convergence. Velocity patterns on Upper St. Clair River downstream from the Blue Water Bridge simulated by use 

of the refined mesh (XFigure 15X) are consistent with those simulated by use of the mesh developed for the public water intake 

version, given that they are simulated with the same numerical options (no Bendway Correction) and parameterization (same 

Peclet number). The large eddy structure identified during the July 2003 ADCP survey was not reproduced by either version. 

Figure 12. Relation between easting velocity components simulated by the public water intake version and the refined version of the 

Upper St. Clair River hydrodynamic model.

Results 19
Figure 13. Relation between northing velocity components simulated by the public water 

intake version and the refined version of the Upper St. Clair River hydrodynamic model.

Figure 14. Relation between water levels simulated by the public water intake version and 

the refined version of the Upper St. Clair River hydrodynamic model.

20 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 15. Simulated velocities downstream from the Blue Water Bridge on Upper St. Clair River using the refined mesh without the 

Bendway Correction and with a Peclet number of 21.9. 

Results 21
Bendway Correction

Implementation of the Bendway Correction results in the simulation of vorticity at all model nodes. For a Peclet number 

of 21.9 and boundary specifications representing flow and water-level conditions during the 2003 ADCP survey, results show 

that the vorticity simulated with the refined mesh is uniform and of relatively low magnitude near the waterfront building along 

the Canadian shoreline (fig. 16). Local variations in simulated vorticity near the flow boundary specifications may be related to 

local irregularities in the defined shoreline geometry. 

Significant changes in the pattern of simulated velocities, however, result from implementation of the Bendway Correction 

for boundary conditions prevalent during the July 2003 ADCP survey. In particular, simulated velocities generally decreased 

along the inside bank of Upper St. Clair River downstream from the Blue Water Bridge near the waterfront building (fig. 17). 

Correspondingly, simulated velocities generally increased along the outside of the bank along the U.S. shoreline. Although the 

pattern of velocity changes is significant, the magnitudes of these changes do not account for most of the discrepancies between 

simulated and measured velocities. Donnell and others (2003) suggest that the Bendway Correction makes little difference 

where helical flows have fully evolved a shallow point bar on the inside bank and a deep channel on the outer bank. This channel geomorphology is evident in figure 2.

Figure 16. Simulated vorticity in the northern part of Upper St. Clair River. 

22 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 17. Change in velocities simulated by use of the refined mesh with implementation of the Bendway Correction and a Peclet 

number of 21.9. 

Results 23
Peclet Number

A Peclet number of 21.9 was estimated for the hydrodynamic model representing the St. Clair-Detroit River Waterway 

(Holtschlag and Koschik, 2004) based on 2002 ADCP surveys of St. Clair River (Holtschlag and Koschik, 2003a) and Detroit 

River (Holtschlag and Koschik, 2003b). This value was determined on the basis of parameter estimation techniques and fell 

within the range of 15 to 40 commonly used for Peclet numbers (Donnell and others, 2003). Although perhaps the best single 

value for the entire waterway, this parameterization did not enable the model to simulate the large eddy measured on Upper St. 

Clair River near the waterfront building along the Canadian shoreline. Increases in the Peclet number were used to better reproduce the large variability in velocities measured within Upper St. Clair River. 

A series of model simulations were done with the refined mesh and Bendway Correction under boundary specifications 

consistent with conditions on the Upper St. Clair River model during the 2003 ADCP survey. The series was initiated with a 

Peclet number of 21.9, and computations were revised by step increases to 210, where simulations failed to converge. A Peclet 

number of 200, which was near the maximum value needed to ensure numerical stability, was used. 

To assess the improvement in velocity matching, ADCP measurements were located within their corresponding model elements. Average measured easting and northing velocities were computed for each element. In addition, the average simulated 

velocities were computed for each element by quadratic interpolation of simulated velocities at ADCP measurement locations. 

The match between element-averaged measured and simulated velocities increased with increasing Peclet number. The best 

match in velocities occurred at a Peclet number of 200 (fig. 18), with greater improvements in the match between measured and 

simulated northing velocity components than for easting components. Resulting simulations using a Peclet number of 200 show 

further decreases in northing velocities near the waterfront building, with some areas of reverse flow appearing (fig. 19). 

Figure 18. Relation between simulated and measured velocities averaged within elements for the initial Peclet number of 21.9 and the 

revised Peclet number of 200. 

24 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 19. Simulated velocity field downstream from the Blue Water Bridge on Upper St. Clair River on refined mesh with Bendway Correction and Peclet number of 200. 

Results 25
Mesh refinements, implementation of the Bendway Correction, and increasing the Peclet number from 21.9 to 200 

improved the match between simulated and measured velocities. In particular, improvements in velocities were noted near the 

waterfront building, where earlier simulations had failed to suggest a large eddy structure that was measured during the July 

2003 ADCP survey. A Peclet number of 200, however, is outside the reference range of 15 to 40, so its utility in other areas of 

the waterway or for describing water-quality mixing characteristics is uncertain. It has apparent utility in Upper St. Clair River, 

perhaps because it allows the 2D model to better approximate 3D flow conditions. 

Integration of Velocity Information with Hydrodynamic Simulations

Despite improvements in the match between simulated and measured velocities associated with enhancements to the 

hydrodynamic model of Upper St. Clair River, significant discrepancies remain between simulated and measured velocities 

downstream from the Blue Water Bridge. In addition, the 2D simulations do not provide information about possible differences 

in flow velocities with depth. To address these limitations, ADCP velocity data were integrated with simulation results for flow 

conditions during the July 2003 survey. The integrated results were converted into a binary solution-file format that was used to 

visualize the results. 

Integrated velocities combine 2D simulated velocities and layer-dependent expected velocity innovations. Integrated 

velocities were computed for all ensembles, where present in a layer. In particular, the integrated easting and northing velocity components were computed as the sum of 2D simulated velocities, from quadratic interpolation, and the expected value of 

velocity innovations, based on bilinear interpolation, B L f , from block-kriging estimates at ensemble locations: 

To assess the results, integrated velocities were averaged for all ADCP ensembles contained within each element, Elem(j), 

and represented as and . These results were compared with depth-averaged-within-layer, 

element-averaged values of measured easting and northing velocities represented as and . 

Results generally show close agreement between integrated and measured velocities (fig. 20). 

26 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin

( , ), ( ., .)

( , ), ( ., .)

ˆ , ( , ) + , ( , )

ˆ , ( , ) + , , , ( , )

Ens x y l = Q Node x y BL

Ens x y l = Q Node x y BL

U f U Ens x y f Ens x y

V f V Ens x y f x y l Ens x y

~

~

( ) (Ε[ υ x y, , l]

( ) (Ε[ ν ]

( , ), ( ) ˆ ( ) U j E ns x y l Elem j ( , ), ( ) ˆ ( ) V j E ns x y l Elem j

( , ), ( ) ˆ ( )E ns x y l Elem v j j ( , ), ( ) ˆ ( )E ns x y l Elem u j j
Figure 20. Relation between average measured and integrated velocities in layer 1 of Upper St. Clair River.

A statistical summary of element-averaged velocities measured in the 2003 ADCP survey and the integrated velocities is 

shown in table 3. Summary statistics include the bias, which indicates the average difference between measured and integrated 

values; the standard deviation, which indicates the standard deviation of differences between measured and integrated values; the 

root mean square error (RMSE), which is computed as the square root of the sum of the bias squared plus the standard deviation squared (variance); and the coefficient of determination (r2

), which is the square of the linear correlation coefficient between 

measured and simulated values. 

Resulting RMSEs for northing velocity components were consistently lower than those for easting velocity components, 

indicating a slightly more precise match for northing velocity components than for easting components. Furthermore, RMSEs 

generally increased and r2 generally decreased with increasing layer number, indicating greater uncertainty in the match between 

measured and integrated velocities with depth below water surface. This greater uncertainty may be related to greater sensitivity of measured velocities to erratic boat movements, such as pitch, yaw, and roll, with increasing depth below the transducer. 

There is no indication of statistically significant discrepancies between element-averaged values of measured and integrated 

flows. 

The integrated velocities at model nodes were computed as the sum of 2D simulated velocity components plus the expected 

values of velocity innovations estimated by use of bilinear interpolation from the kriging grid as 

Results 27

( , ),

( , ),

ˆ

ˆ

Ens x y l = 

Ens x y l =

U

V

( , ) +

( , ) +

Node x y

Node x y

U

V

~

~

)

, Node ( , )

, , , Node ( , )

BL

BL

f x y

f x y l x y

(Ε[ υ x y, , l]

(Ε ν[ ]

)
Integrated velocities were computed for each layer and transformed into a binary solution file, by use of the format developed for RMA2 simulation results. Solution files with the layer-specific integrated velocities were used by the postprocessing 

software SMS (Environmental Modeling and Research Laboratory, 2003) to display layer-specific velocity patterns. Results 

indicate that the integrated velocities clearly map the recirculation zone downstream from the Blue Water Bridge near the shoreline building (fig. 21). 

Table 3. Summary of relations between element-averaged measured and integrated velocities.

Layer Number of 

elements

Velocity 

component

Bias 

(feet per 

second)

Standard 

deviation 

(feet per 

second)

Root mean 

square 

error (feet 

per second)

Coefficient of 

determination (r2)

1 4,045 Easting -0.0002 0.1302 0.1302 0.9939

Northing -.0002 .1221 .1221 .9952

2 3,962 Easting .0006 .1286 .1286 .9938

Northing .0026 .1229 .1229 .9949

3 3,802 Easting .0014 .1405 .1405 .9925

Northing .0030 .1316 .1316 .9936

4 3,508 Easting .0014 .1560 .1560 .9910

Northing .0019 .1386 .1386 .9911

5 2,864 Easting .0053 .2158 .2159 .9842

Northing .0063 .1836 .1837 .9756

6 1,928 Easting .0010 .2344 .2344 .9836

Northing .0097 .2279 .2281 .9519

7 922 Easting .0022 .2719 .2719 .9783

Northing .0026 .2493 .2493 .9403

28 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 21. Integrated near-surface velocities in layer 1 of Upper St. Clair River downstream from the Blue Water Bridge.

Results 29
Identification of Flow Path as a Function of Flow Depth 

Particle tracking provides a numerical procedure for following one or more hypothetical, massless particles through a 

velocity field. Forward particle tracking identifies the likely movement of one or more particles forward in time from an initial 

point of release to likely impact areas. Reverse particle tracking identifies areas that are likely a source or prior pass-through 

area for a terminal location. Reverse tracking changes the numerical sign of the velocities simulated at model nodes and provides a mechanism to track one or more particles backward in time. When many particles are specified at the starting location, 

independent random increments are commonly added to the advective movement specified by the velocity field to mimic dispersion or other submesh velocity components. When a single particle is specified at a starting location, particle tracking can be 

used without random increments to identify the expected flow path associated with the specified velocity field. 

In this report, the particle-tracking code ParTrac developed by the U.S. Army Corps of Engineers (John Koschik, written 

commun., 2003) was used to investigate expected particle paths as a function of starting layer. Expected particle paths from 

reverse particle tracking based on integrated velocities in the top four layers of the Upper St. Clair River and a starting location 

near the city of Port Huron, Mich., sewage treatment plant are shown in figure 22. Reverse particle tracking results indicate that 

particle paths in shallower layers tend be nearer the Canadian boundary than particle paths in deeper waters. These results are 

consistent with near-surface velocities tracked by drifting-buoy deployments in Upper St. Clair River (Holtschlag and Aichele, 

2001). Conversely, particles started in deeper waters, where public water intakes are commonly placed, are more likely to 

be consistent with particle tracks indicated by the public water intake version of the hydrodynamic model developed for the 

Michigan Source Water Assessment Program of the St. Clair-Detroit River Waterway (Holtschlag and Koschik, 2004). Particle 

paths for layers 5 through 7 are not continuous because of shallow-depth areas along the particle paths from the sewage treatment plant; therefore, those paths could not be shown. Particle paths do not account for possible vertical movement of particles 

between layers. A 3D hydrodynamic model would be needed to account for these effects and assess their importance on the 

identification of flow paths near public water intakes. 

30 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Figure 22. Flow paths as a function of depth (layer) in Upper St. Clair River from reverse particle-tracking for average flow conditions 

during July 1–3, 2003. 

 Results 31
Summary and Conclusions

In this report, Upper St. Clair River refers to a 2.5-mile reach of St. Clair River from Fort Gratiot, Mich., to a point near the 

mouth of the Black River in the Port Huron, Mich. area. St. Clair River, which forms part of the boundary between the U.S. and 

Canada, discharges an average of 182,000 ft3/s from a drainage area of about 222,400 mi2. In this upper reach, St. Clair River 

has velocities that exceed 7 ft/s, flow depths of up to 75 ft, and significant channel curvature that creates complex flow patterns 

downstream from the Blue Water Bridge. 

Simulated velocities from a two-dimensional (depth-averaged) hydrodynamic model were augmented with horizontal flow 

information from an ADCP (acoustic Doppler current profiler) survey to characterize depth-dependent velocities in Upper St. 

Clair River. Discrepancies were previously detected between depth-averaged velocities simulated by a hydrodynamic model and 

surface velocities indicated by drifting-buoy deployments in Upper St. Clair River. A detailed ADCP survey was done along the 

upper 2.5 mi of St. Clair River in the Port Huron, Mich. area during July 1–3, 2003, to resolve these discrepancies. 

During the ADCP survey, about 2.1 million velocity measurements were obtained at 127 transects, which were separated by 

an average distance of about 100 ft. Along each transect, ensembles of velocity data were obtained at about 1-ft intervals. Each 

ensemble described the instantaneous easting, northing, and vertical water-velocity components in a set of vertically aligned 

cells ranging from 0.82 ft (0.25 m) to 3.28 ft (1.00 m) in depth throughout most of the water column. Velocities near the water 

surface and near the channel bottom, however, could not be measured directly because of instrument limitations. 

To improve the resolution of simulated velocities, the finite-element mesh developed to identify flow paths near public 

water intakes on the St. Clair-Detroit River Waterway was refined within Upper St. Clair River by subdividing each of the 1,555 

elements previously defined into four subelements. Velocities and water levels simulated by use of the public water intake 

version and the refined version of the mesh were confirmed to be similar. The Bendway Correction, a numerical technique 

that improves the accuracy of simulated flows in areas of channel curvature by accounting for vorticity, was implemented in 

the refined version of the model. This implementation improved the match between simulated and measured flows. Finally, 

the refined model with the Bendway Correction was recalibrated by increasing the Peclet number, which is used to assign eddy 

viscosity and control the local variability in simulated velocities, from 21.9 to 200. Resulting simulations were better able to 

reproduce measured velocities. 

Despite the generally improved match between simulated and measured velocities, enhanced model simulations of depthaveraged velocities were not fully consistent with surface velocities indicated by drifting-buoy deployments in Upper St. Clair 

River. In a further attempt to resolve the remaining discrepancies, the ADCP velocity data were used to adjust simulated velocities and identify possible variations in horizontal-velocity components with depth. 

As a basis for adjustment, simulated velocities were computed for each ADCP ensemble by quadratic interpolation of 

velocities simulated with the enhanced model for flow boundary conditions during the July 1–3, 2003, survey. Differences 

between simulated and measured velocities were computed and are referred to as “velocity innovations.” Average easting and 

northing components of velocity innovations were computed within seven layers of approximately 6.56 ft (2 m) depth. The 

spatial correlation structure of the horizontal velocity innovations was analyzed within each layer by variogram analysis. Insufficient spatial correlation structure was detected in the vertical velocity components to permit estimation. 

The results of the variogram analysis were used with block kriging to map the spatial variation in expected innovations onto 

a grid of 20-ft-square cells that spanned the model area. Bilinear interpolation from this grid was used to compute the expected 

innovation at model nodes. Integrated velocities, which were computed as the sum of simulated velocities and expected innovations within each layer, were used to map the spatial variation in flow velocities with depth. Integrated flow velocities reproduced a large secondary circulation pattern about 1,800 ft downstream from the Blue Water Bridge that was measured during the 

ADCP survey. Expected particle paths generated with the integrated velocities showed that surface velocities in the upper layers 

tended to originate nearer the Canadian shoreline than velocities near the channel bottom in the lower layers. Therefore, flow 

paths near U.S. public water intakes on the river bottom are more likely to be in the United States than those for withdrawals 

near the water surface, assuming any vertical movement of particles can be ignored. Finally, integrated velocities in the upper 

layers are generally consistent with the surface velocities indicated by drifting-buoy deployments. 

Neither the 2D hydrodynamic model nor the velocity measurements provided sufficient information to describe the vertical 

component of flow in this report. This limitation resulted in the inability to account for vertical movements on expected flow 

paths through Upper St. Clair River. A 3D hydrodynamic model would be needed to account for these effects and assess their 

importance on the identification of flow paths near public water intakes. 

32 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Acknowledgments

Christopher Rayburn, Research Management Director, American Water Works Association Research Foundation; Ramesh 

Kashinkunti, Senior Engineer, Department of Water Works, City of Cincinnati, Ohio; and Scott Wells, Professor and Chair of 

Environmental Engineering, Portland State University, provided technical direction and review. Don James, USGS Grayling 

Mich., Office, and Paul Taylor and Benjamin Harrison, USACE Detroit District, operated the ADCP units during the velocity 

survey. 

References Cited

de Marsily, Ghislain, 1986, Quantitative hydrogeology—Groundwater hydrology for engineers: Orlando, Fl., Academic Press, 

440 p. 

Donnell, B.P., Letter, J.V., Jr., McAnally, W.H., and Thomas, W.A., 2003, Users guide to RMA2-WES (version 4.5): U.S. Army 

Corps of Engineers Waterways Experiment Station, Vicksburg, Miss., 264 p., accessed March 17, 2005 at http://chl.wes.army.

mil/software/tabs/docs.htp.

Environmental Modeling Research Laboratory, 2003, SMS Surface-Water Modeling System reference manual (version 7): 

Provo, Utah, Environmental Modeling Systems, Inc., 342 p. 

Holtschlag, D.J., and Aichele, S.A., 2001, Visualization of drifting buoy deployments on St. Clair River near public water 

intakes—October 3–5, 2000: U.S. Geological Survey Open-File Report 01–17, accessed March 17, 2005, at Hhttp://

mi.water.usgs.gov/splan2/sp08903H/SCRIndex.html 

Holtschlag, D. J., and Koschik, J.A., 2002, A two-dimensional hydrodynamic model of the St. Clair-Detroit River Waterway in 

the Great Lakes Basin: U.S. Geological Survey Water-Resources Investigations Report 01–4236, 63 p. accessed March 17, 

2005, at http://mi.water.usgs.gov/pubs/WRIR/WRIR01-4236/

Holtschlag, D.J., and Koschik, J.A., 2003a, An acoustic Doppler current profiler survey of flow velocities in St. Clair River, 

a connecting channel of the Great Lakes: U.S. Geological Survey Open-File Report 03–119, accessed March 17, 2005, at 

Hhttp://mi.water.usgs.gov/pubs/OF/HOF03-119/

Holtschlag, D.J., and Koschik, J.A., 2003b, An acoustic Doppler current profiler survey of flow velocities in Detroit River, 

a connecting channel of the Great Lakes, U.S. Geological Survey Open-File Report 03–219, accessed March 17, 2005, at 

Hhttp://mi.water.usgs.gov/pubs/OFH/OF03-219/

Holtschlag, D.J., and Koschik, J.A., 2004, Hydrodynamic simulation and particle-tracking techniques for identification of source 

areas to public-water intakes on the St. Clair-Detroit River Waterway in the Great Lakes Basin: U.S. Geological Survey Scientific Investigations Report 2004–5072, 29 p., accessed March 17, 2005, at Hhttp://water.usgs.gov/pubs/sir/2004/5072H

National Oceanic and Atmospheric Administration, 1999, Recreational chart 14583: Detroit River, Lake St. Clair, and St. Clair 

River: U.S. Department of Commerce, National Ocean Service, Washington, D.C. 

Schwab, D.J., Clites, A.H., Murthy, C.R., Sandall, J.E., Meadows, L.A., and Meadows, G.A., 1989, The effect of wind on transport and circulation in Lake St. Clair: Journal of Geophysical Research, v. 94, no. C4, p. 4947–4958. 

Simpson, M.R., 2001, Discharge measurements using a broad-band acoustic Doppler current profiler: U.S. Geological Survey 

Open-File Report 01–1, 123 p., accessed March 17, 2005, at http://water.usgs.gov/pubs/of/ofr0101/text.pdf

Tsanis, I.K., Shen, Huihua, and Venkatesh, S., 1996, Water currents in the St. Clair and Detroit Rivers: Journal of Great Lakes 

Research, v. 22, no. 2, p. 213–223. 

Williamson, D.C., Scott R.D., and Lord, Stephen, 1997, Numerical modeling of the St. Clair/Detroit River system and seasonal 

variations in head loss, in Proceedings of the 1997 Canadian Coastal Conference, Guelph, Ontario, May 21–24, 1997: Canadian Coastal Science and Engineering Association, p. 176–184. 

References Cited 33
Appendix 1. Summary of acoustic Doppler current profiler velocity data for transects on the Upper St. Clair River. 

[USGS—U.S. Geological Survey, USACE—U.S. Army Corps of Engineers. Transect locations shown in figs. 

3–6]

Transect number Date and Eastern 

standard time of 

transect start

Total 

discharge 

[cubic feet 

per second]

Number of 

ensembles

Max 

depth 

[feet]

Bin size 

[centimeters]

Left (L) or 

right (R) bank 

start

Agency

0010 7/3/03 7:51 173,001 2,552 69.9 50 R USGS

0020 7/3/03 8:14 166,682 1,087 74.7 50 L USGS

0030 7/3/03 8:33 169,091 1,210 74.7 50 R USGS

0040 7/3/03 8:45 171,494 780 68.6 50 L USGS

0050 7/2/03 16:20 171,543 1,087 68.1 50 L USGS

0060 7/3/03 8:54 175,924 995 69.7 50 R USGS

0070 7/2/03 16:07 172,814 1,357 69.5 50 R USGS

0080 7/3/03 9:10 175,200 709 68.6 50 L USGS

0090 7/2/03 15:58 174,832 863 68.8 50 L USGS

0100 7/3/03 9:18 175,980 678 68.1 50 R USGS

0110 7/2/03 15:50 177,449 807 64.8 50 R USGS

0120 7/3/03 9:26 179,349 848 61.2 50 L USGS

0130 7/2/03 15:43 180,667 544 59.2 50 L USGS

0140 7/3/03 9:56 176,194 588 57.6 50 L USGS

0150 7/2/03 15:35 180,879 727 56.7 50 R USGS

0160 7/3/03 10:03 179,906 920 60.0 50 R USGS

0170 7/2/03 15:28 177,257 682 60.0 50 L USGS

0180 7/2/03 14:11 185,446 688 59.3 25 L USACE

0181 7/3/03 10:14 177,160 605 60.1 50 L USGS

0190 7/2/03 15:21 172,104 686 70.2 50 R USGS

0200 7/3/03 10:21 176,428 601 78.0 50 R USGS

0201 7/2/03 13:57 182,387 509 77.8 25 L USACE

0202 7/2/03 14:03 180,137 641 78.3 25 R USACE

0220 7/2/03 13:51 189,383 442 66.0 25 R USACE

0230 7/2/03 14:59 166,149 879 60.5 50 L USGS

0240 7/2/03 13:43 189,210 536 60.5 25 L USACE

0250 7/2/03 14:52 168,445 709 59.5 50 R USGS

0260 7/2/03 13:38 189,045 425 57.1 25 R USACE

0270 7/2/03 14:43 169,852 556 55.7 50 L USGS

0280 7/2/03 13:32 185,811 325 52.9 25 L USACE

0290 7/2/03 14:36 174,490 723 50.5 50 R USGS

0300 7/2/03 13:20 177,745 362 48.1 25 R USACE

0301 7/2/03 13:23 179,031 380 49.5 25 L USACE

0302 7/2/03 13:27 180,021 384 49.5 25 R USACE

0310 7/2/03 14:17 176,967 686 47.5 50 L USGS

0320 7/2/03 13:09 170,526 544 46.8 25 L USACE

0321 7/2/03 13:14 173,373 387 46.8 25 R USACE

0330 7/2/03 14:08 178,186 645 46.2 50 R USGS

0340 7/2/03 12:42 175,458 591 46.3 25 R USACE

0350 7/2/03 13:59 175,607 1,062 47.9 50 L USGS

0360 7/2/03 12:30 178,532 1,160 47.4 25 L USACE

0370 7/2/03 13:39 188,972 737 49.4 50 R USGS

0380 7/2/03 12:21 175,911 562 56.1 25 R USACE

0390 7/2/03 13:31 180,994 825 61.5 50 L USGS

0395 7/2/03 13:23 178,645 726 69.5 50 R USGS

0400 7/2/03 12:13 172,165 686 69.5 25 L USACE

0410 7/2/03 13:04 170,068 882 70.4 50 R USGS

0411 7/2/03 13:14 161,713 781 70.2 50 L USGS

0420 7/2/03 12:04 169,862 750 66.5 25 R USACE

0430 7/2/03 12:55 169,203 977 64.9 50 L USGS

0440 7/2/03 11:38 181,706 546 62.1 25 R USACE

34 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin
Transect number Date and Eastern 

standard time of 

transect start

Total 

discharge 

[cubic feet 

per second]

Number of 

ensembles

Max 

depth 

[feet]

Bin size 

[centimeters]

Left (L) or 

right (R) bank 

start

Agency

0450 7/2/03 12:35 161,909 1,121 64.6 50 L USGS

0455 7/2/03 12:47 163,449 826 63.7 50 R USGS

0460 7/2/03 11:50 178,591 777 61.8 25 L USACE

0470 7/2/03 12:27 166,247 775 65.8 50 R USGS

0480 7/2/03 11:16 152,740 431 62.6 25 R USACE

0481 7/2/03 11:24 158,386 496 65.4 25 L USACE

0485 7/2/03 12:15 174,812 1,108 65.6 50 L USGS

0490 7/2/03 12:05 183,429 929 52.1 50 R USGS

0500 7/2/03 11:06 161,462 694 51.8 25 L USACE

0510 7/2/03 11:55 169,916 1,050 51.3 50 L USGS

0520 7/2/03 10:57 152,865 602 48.8 25 R USACE

0530 7/2/03 11:46 173,745 874 48.5 50 R USGS

0540 7/2/03 10:17 180,751 2,271 50.4 100 L USACE

0550 7/2/03 11:37 183,114 997 51.0 50 L USGS

0560 7/2/03 10:02 177,802 1,094 49.7 100 R USACE

0570 7/2/03 11:28 175,023 887 49.7 50 R USGS

0580 7/2/03 9:47 172,610 1,637 49.0 100 L USACE

0590 7/2/03 11:15 182,370 1,102 50.2 50 L USGS

0600 7/2/03 9:38 176,172 1,044 50.6 100 R USACE

0610 7/2/03 11:05 180,691 977 52.2 50 R USGS

0620 7/2/03 9:28 173,538 1,586 50.9 100 L USACE

0630 7/2/03 10:55 173,949 987 52.9 50 L USGS

0640 7/2/03 8:59 173,470 1,777 51.5 100 R USACE

0650 7/2/03 10:48 171,808 758 51.8 50 R USGS

0660 7/2/03 8:46 177,409 1,756 51.1 100 L USACE

0670 7/2/03 10:36 166,585 1,315 52.0 50 L USGS

0680 7/2/03 8:39 170,850 997 52.2 100 R USACE

0690 7/2/03 10:26 164,563 1,142 53.7 50 R USGS

0700 7/2/03 8:26 177,778 1,779 52.4 100 L USACE

0710 7/2/03 10:15 170,727 1,199 50.8 50 L USGS

0720 7/1/03 17:21 178,076 1,302 50.5 100 R USACE

0730 7/2/03 10:07 168,526 854 49.7 50 R USGS

0740 7/1/03 17:07 179,847 2,335 49.4 100 L USACE

0750 7/2/03 9:52 177,398 1,704 48.5 50 L USGS

0760 7/1/03 16:50 177,807 1,668 48.3 100 R USACE

0770 7/2/03 9:42 174,004 1,130 48.1 50 R USGS

0780 7/1/03 16:37 176,362 1,972 48.2 100 L USACE

0790 7/2/03 9:32 173,716 1,083 51.3 50 L USGS

0800 7/1/03 16:27 177,992 1,168 52.0 100 R USACE

0810 7/2/03 9:21 177,707 1,280 49.5 50 R USGS

0820 7/1/03 16:17 174,593 1,681 48.8 100 L USACE

0830 7/2/03 8:56 174,808 1,723 49.4 50 L USGS

0840 7/1/03 16:04 176,021 1,782 49.2 100 R USACE

0850 7/2/03 8:44 180,621 1,313 48.7 50 R USGS

0860 7/1/03 15:48 180,317 2,607 48.8 100 L USACE

0870 7/1/03 17:32 171,461 1,410 49.1 50 L USGS

0880 7/1/03 15:21 174,214 1,510 48.9 100 R USACE

0885 7/1/03 17:20 182,763 1,381 48.3 50 R USGS

0890 7/1/03 17:05 169,704 1,838 46.5 50 L USGS

0900 7/1/03 15:06 170,348 2,422 45.4 100 L USACE

0910 7/1/03 16:56 176,914 861 45.1 50 R USGS

Appendix 1 35

 Appendix 1. Summary of acoustic Doppler current profiler velocity data for transects on the Upper St. Clair 

River.--Continued
Transect number Date and Eastern 

standard time of 

transect start

Total 

discharge 

[cubic feet 

per second]

Number of 

ensembles

Max 

depth 

[feet]

Bin size 

[centimeters]

Left (L) or 

right (R) bank 

start

Agency

0920 7/1/03 14:48 175,818 1,242 43.7 100 R USACE

0930 7/1/03 16:44 173,613 1,292 42.8 50 L USGS

0940 7/1/03 14:26 188,528 2,777 43.9 100 L USACE

0950 7/1/03 16:35 176,936 828 42.4 50 R USGS

0960 7/1/03 14:04 178,958 1,723 44.2 100 R USACE

0970 7/1/03 16:21 175,075 1,393 44.2 50 L USGS

0980 7/1/03 13:48 174,333 1,300 44.5 100 L USACE

0990 7/1/03 16:11 173,994 1,121 45.0 50 R USGS

1000 7/1/03 13:35 173,130 1,426 45.2 100 R USACE

1005 7/1/03 15:55 174,942 1,763 45.4 50 L USGS

1010 7/1/03 15:39 173,573 1,716 44.2 50 R USGS

1015 7/1/03 15:18 174,796 2,395 43.8 50 L USGS

1020 7/1/03 13:18 176,231 2,527 43.4 100 L USACE

1025 7/1/03 14:58 176,297 1,513 42.9 50 R USGS

1030 7/1/03 14:38 169,549 2,276 42.5 50 L USGS

1035 7/1/03 14:22 180,672 1,729 41.9 50 R USGS

1040 7/1/03 13:02 179,862 2,364 41.5 100 R USACE

1045 7/1/03 12:56 168,887 2,016 42.0 50 L USGS

1050 7/1/03 12:11 174,105 2,146 42.2 50 L USGS

1060 7/1/03 12:45 177,917 2,899 41.2 100 L USACE

1070 7/1/03 11:58 175,014 1,428 39.6 50 R USGS

1080 7/1/03 12:31 174,824 2,001 38.1 100 R USACE

1090 7/1/03 11:42 178,930 1,871 38.8 50 L USGS

1101 7/1/03 12:11 176,815 3,445 40.1 100 L USACE

1111 7/1/03 11:32 173,174 1,005 42.2 50 R USGS

36 Augementing Two-Dimensional Hydrodynamic Simulations with Measured Velocity Data to Identify Flow Paths as a 

Function of Depth on Upper St. Clair River in the Great Lakes Basin

 Appendix 1. Summary of acoustic Doppler current profiler velocity data for transects on the Upper St. Clair 

River.--Continued
Holtschlag, D.J. and Koschik, J.A. —Augmenting Two-Dimensional Hydrodynamic Simulations with measured Velocity Data 

to Identify Flow Paths as a Function of Depth on Upper St. Clair River in the Great Lakes Basin—Scientific Investigations Report 2005-5081 

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