## Abstract

The one-dimensional HEC-RAS multi-purpose open channel flow modeling software was successfully used, with ArcMap and HEC-GeoRAS, to simulate flow over the Wappapello Dam limited-use Ogee spillway (Wappapello, Missouri). Initial computational hydraulic modeling results predicted a lake elevation of 132.9 m (405.0 ft) [NAVD 1988] would be required for the resulting floodwaters overtopping the spillway to reach the nearby Wappapello Lake Management Office. An intense rainfall event during 2011 led to the spillway being overtopped for the first time since 1945. Spillway performance during the 2011 event was analyzed afterwards. Results indicated that the spillway crest was not submerged by backwater. A technique was employed which successfully estimated the design energy head of 7.160 m (23.49 ft) for the spillway. Hydraulic modeling developed after the 2011 event incorporated this estimated design energy head, allowing the spillway discharge coefficient to vary with discharge in the course of an unsteady modeling run. Results indicated that, while the spillway did perform as designed, the performance is limited by the shallow approach depth.

## Introduction

Headwaters of the St. Francis River are in St. Francois County, Missouri, about 129 km (80 mi) south of St. Louis. The flow of this river is regulated by Wappapello Dam which controls runoff originating from a 3354 km^{2} (1310 mi^{2}) watershed. The St. Francis forms the western boundary of the Missouri Bootheel before joining the Mississippi River near Helena, Arkansas. The dam consists of a rolled earth embankment, a concrete outlet structure, and a limited-use concrete spillway. Construction on Wappapello Dam began in 1938 and was completed in 1941.

Before 2011 the only year during which overtopping of the spillway occurred was 1945. During 2011 the authors were engaged in hydraulic modeling and analysis of the St. Francis River when an unusually intense rainfall event occurred. The limited-use concrete spillway was overtopped for two weeks. This event provided a rare opportunity to observe the spillway in operation, and various data were collected during and after it. The 2011 overtopping event also motivated additional study. Consequently, the new data were incorporated into subsequent hydraulic modeling and analyses.

The original goal of the authors, prior to the 2011 event, was to develop a realistic hydraulic model which included the spillway - and was capable of inundation mapping. One purpose was to identify lake elevations associated with hypothetical storm events capable of allowing floodwaters to reach United States Army Corps of Engineers (USACE) facilities located a short distance downstream of the dam. After the 2011 event this goal was expanded to include evaluation of spillway performance, specifically, the rating curve expressing spillway flow as a function of water surface elevation in the lake.

Wappapello Dam and vicinity are shown in Figure 1. The 823.0 m (2700 ft) long rolled earth dam has a crown elevation of 127.29 m (417.60 ft) [NAVD 1988] which is 33.2 m (109 ft) above the old channel bed. (The authors found that the geographical extent of the hydraulic model was small enough to allow use of a constant elevation datum conversion. NAVD 1988 (m) = NGVD 1929 (m) – 0.0290 (m) or NAVD 1988 (ft) = NGVD 1929 (ft) – 0.0951 (ft).) The dam has a concrete outlet structure with three 3.0 m by 6.1 m (10 ft by 20 ft) vertical gates, and a 226 m (740 ft) long limited-use concrete Ogee spillway with crest at 120.29 m (394.64 ft) [NAVD 1988]. In the entrance channel leading to these vertical gates, some flow is diverted into a 1.52 m (5.00 ft) diameter penstock. This diverted flow passes through a 75.00 kVa (100.5 hp) hydroelectric unit. *Tunnel* discharge consists of flow from the hydroelectric unit and the flow passing under the vertical gates. Discharge over the limited-use concrete spillway is referred to as *spillway* discharge. Discharge over the spillway flows down an open channel for approximately 660.5 m (2167 ft) to a confluence with the main stem of the St. Francis River 520.3 m (1707 ft) downstream of the dam. This is the *spillway channel* feature shown in Figure 1. Various USACE facilities, including the Wappapello Lake Management Office, are located alongside this channel.

## Background

The work described in this article is a part of a larger study involving hydraulic and hydrologic analyses and modeling of the watershed controlled by Wappapello Dam, and a distance downstream from the structure. Therefore, the area of interest includes Wappapello Lake, and the St. Francis River extending from the headwaters downstream to the vicinity of St. Francis, Arkansas. The authors collected a number of corroborative documents and technical data in support of this work.

Wappapello Dam was designed on the basis of results from a series of physical model studies (U.S. Waterways Experiment Station 1938a and b) conducted at the U.S. Waterways Experiment Station in 1938. (In 1999, the U.S. Waterways Experiment Station was renamed the Vicksburg Site of the U.S. Engineer Research and Development Center.) The results of this study were documented in two reports. The first of these (U.S. Waterways Experiment Station 1938a) dealt with the geometry of the vertical gates, and related structures. A 1:25 scale model was constructed and used for testing. The second (U.S. Waterways Experiment Station 1938b) concerned the design of the limited-use spillway and downstream channel. A 1:100 scale model was used.

The latter report is the vital starting point for any study concerning the hydraulics of the limited-use spillway at Wappapello Dam. To summarize, the report documents development of a basic design which was then optimized by the testing of five alternative designs. The physical model was operated using a fixed bed for some simulations, and a moveable bed for others. Differences in the alternative designs included variable positioning of baffle blocks on the lip of the spillway bucket, alignment and depth of the downstream channel, placement of a dike between the spillway and tunnel discharge, and placement of bridge piers along the spillway. The last design mentioned is the only one with bridge piers absent, and is the one upon which construction was based. A spillway rating curve for this particular geometry was developed and is presented in the report. It is this rating curve that was the single most important piece of technical data relevant to the current study, and is hereafter referred to as the *1938 Rating Curve*. It is, to the authors' knowledge, the only such rating curve in existence for the limited-use spillway of Wappapello Dam.

A set of design drawings (U.S. Army Engineer Office, Memphis, Tennessee 1944) were supplied by the St. Louis District, USACE. These drawings contained important technical information including dimensions and measurements concerning the shape of the Ogee crest of the limited-use spillway. The physical model studies and design drawings comprise a valuable set of documents specifically relevant to Wappapello Dam and its outlet structures.

Widely applicable technical data concerning spillway performance was obtained from two U. S. Bureau of Reclamation (BR) reports (Bureau of Reclamation 1948 and 1987). These two reports are referenced from a USACE Engineering Manual (EM) containing widely applicable technical information regarding spillways (U. S. Army Corps of Engineers 1992). These three documents were used separately, and in combination, to assess spillway performance via historically accepted empirical coefficients, and associated mathematical relationships.

## Modeling Software and Analysis

### Hydraulic Modeling Software

Computational hydraulic modeling was performed using the widely accepted and successful one-dimensional open channel flow software available from the USACE through their Hydrologic Engineering Center (HEC). Hydrologic Engineering Center-River Analysis System (HEC-RAS 4.1.0) is a software package capable of performing one dimensional open channel hydraulic simulations (Hydrologic Engineering Center 2010b). HEC-RAS uses an extensively documented finite difference scheme (Hydrologic Engineering Center 2010a) to obtain numerical solutions to the Saint-Venant Equations (Sturm 2009). HEC-RAS can accommodate irregularly shaped channels, steady or unsteady flow, and subcritical or supercritical flow regimes. Computational routines are equipped to handle mixed (sub- and supercritical) regimes as well. Designed for practical applications, the software has the ability to incorporate the effects of many commonly encountered artificial structures or naturally occurring geographic features. Computational routines and dialog boxes enable implementation of bridges, culverts, inline structures, pump stations, basins, levees, flood plain encroachments, and many other features into a simulation for analysis. Although designed for gradually varying flow, HEC-RAS can handle rapidly varying flow over structures and other features with appropriate empirical coefficients. It is the direct descendent of an earlier, similar software named Hydrologic Engineering Center – 2 (HEC-2). HEC-RAS was released in 1999, is windows-based, and continues to be as widely accepted and used as was its predecessor. Numerous examples of successful applications using HEC-RAS can be found in the literature (Ahmed and Freeman 2004; Brunner 2003; Brych *et al*. 2002; Hicks and Peacock 2005; Horritt and Bates 2002).

During the software development, HEC-RAS was interfaced with Geographic Information Systems (GIS) data via the Environmental Systems Research Institute (ESRI) ArcMap 9.3.1 software (Environmental Systems Research Institute 2009). This configuration was established by installing and running the HEC-GeoRAS 4.2 toolbar (Hydrologic Engineering Center 2009) within the ArcMap module of ArcGIS (Environmental Systems Research Institute 2009). HEC-GeoRAS capabilities include processing of digital elevations into stationing and elevation data formatted into cross sections for import into HEC-RAS. It can also process hydraulic modeling output from HEC-RAS into inundation mapping and other graphical representations for import back into ArcMap. All work with HEC-GeoRAS involves an interface with a projected set of coordinates within ArcMap. This configuration allows all information exchanged with HEC-RAS to be correctly geo-referenced and appropriately displayed by ArcMap.

### Analysis

The limited-use spillway of Wappapello Dam is of the Ogee shape. The Ogee shape is essentially the profile formed by the underside of the flow nappe over a sharp-crested weir (Chow 1959). Discharge over an Ogee shaped spillway can be computed using an equation of the form shown in Equation 1.

L is the spillway length (perpendicular to the flow) and H_{e} is the total energy head on the crest, including the velocity head just upstream in the approach channel, as shown in Figure 2. Equation 1 is derived by defining an empirically determined discharge coefficient, C. This discharge coefficient accounts for the effect of friction and vertical accelerations. Equation 2 shows how the total energy head H_{e} is defined in terms of the elevation head and velocity head, as shown in Figure 2.

The discharge coefficient C can be determined empirically and is readily available in the literature (U. S. Army Corps of Engineers 1992 and Bureau of Reclamation 1987). However, this coefficient is itself a function of the total energy head H_{e}. Consequently, for a given Ogee spillway, there exists a unique C value associated with each value of H_{e}. Therefore, in principle, an infinite number of unique values of discharge coefficient exist for a given Ogee spillway. Moreover, each pair of H_{e} and C is associated with a particular discharge. Finally, each discharge produces a unique overflow nappe, the underside of which will form its own Ogee shape. Therefore, the shape (surface) of the Ogee spillway is associated with a unique discharge.

This unique value of discharge, the *design discharge* (Q_{D}), is an important design parameter and is associated with a unique value of H_{e}, or *design energy head* (H_{D}). For clarification Equation 3 shows how these quantities are related. Note that this equation is exactly the same relationship as depicted in Equation 1. The authors were unable to find documentation of the values of design energy head, and corresponding design discharge, for the Ogee spillway of Wappapello Dam.

The quantities C_{D,} H_{D}, and approach depth P (as shown in Figure 2), are related. This relationship is displayed graphically on page 370 of a report issued by the BR (Bureau of Reclamation 1987). To facilitate computation the authors developed a curve fit of this relationship, given in Equation 4.

Flow over an Ogee spillway is characterized by variations in pressure and velocity. For a given structure, at discharges greater than the design discharge Q_{D}, it is possible for pressures along the downstream face to become low enough to cause cavitation. At even greater discharges the flow nappe can separate from the structure altogether. To avoid these and other related problems the precise shape of the Ogee spillway is of interest, and has been the subject of considerable research. Much of the early work focused on identifying a mathematical function to define the shape of the crest. Grzywienski has compiled and presented a summary of these early attempts by various investigators (Grzywienski 1951).

Subsequently, the USACE developed functions that fit a range of available measurements of the surface of Ogee crests (Chow 1959). Chow refers to these functions as the *WES Profile Shapes*. Their origin and development is described by Harrold (Harrold 1955). Equation 5 shows the mathematical relationship which describes the surface of the Ogee spillway, utilizing the rectangular x-y coordinate system shown in Figure 3. This coordinate system assigns values of x greater than zero for locations downstream of the crest. Note that the elevation head H_{d}, as it appears in Equation 5 and Figure 3, excludes velocity head. The values for the exponent n and coefficient K are recommended by Chow as 1.85 and 2.0, respectively, and correspond to a spillway with a vertical face.

The shape of the portion of the spillway surface downstream from the crest is of interest to the authors, and is relevant to this work. The coordinates along the Wappapello Ogee spillway surface, available to the authors from original design drawings (U.S. Engineer Office Memphis, Tennessee 1944), were mostly located downstream from the crest. The concept of a design discharge Q_{D}, as described in Equation 3, the curve fit for design discharge coefficient C_{D} of Equation 4, and the WES Profile Shapes of Equation 5, were all combined to develop an estimate for the design energy head H_{D} of the Wappapello Dam Ogee spillway. This process is described in Section 4.4 where these results are used to support computational hydraulic modeling with HEC-RAS. Wappapello Dam was designed in 1938, before the WES Profile Shapes were available for precise specification of the spillway surface. However, results obtained by the HEC-RAS hydraulic modeling confirmed that the Ogee shape of the spillway surface is in fact closely approximated by the WES Profile Shapes.

## Hydraulic Modeling and Results

### Overview of Computational Hydraulic Modeling

An overview of the computational hydraulic modeling performed in this study is provided in Table 1. It has been developed to assist the reader in following the detailed descriptions given in this article. The study began before the 2011 event but entered a second phase when that event provided additional data for evaluation. Prior to the 2011 event the only available limited-use Ogee spillway discharge data was that from the 1938 Rating Curve. Based on the 1938 Rating Curve discharges, and a user-specified spillway discharge coefficient value of C = 3.0 (constant), the pre-2011 study culminated in a verified HEC-RAS computational hydraulic model. It was used to generate inundation mapping of areas immediately downstream of the spillway, and to predict the lake elevation at which USACE facilities would be approached by floodwaters. The 2011 event provided fresh data and a sense of urgency to further the study. The post-2011 study culminated in a verified HEC-RAS model which incorporates improved spillway discharge computations based on a variable discharge coefficient. This discharge coefficient is computed internally by HEC-RAS during program execution, based on hydraulic parameters updated during each time step. The post-2011 HEC-RAS model was used to assess the effects of increased spillway approach depth.

### Pre-2011 Event Hydraulic Modeling

As stated previously, before the spillway was overtopped in 2011 a HEC-RAS computational hydraulic model was developed. A graphical representation of the HEC-RAS hydraulic model including boundary conditions, as viewed in the Geometry Editor, is shown in Figure 4. This model was successfully verified, and several modeling scenarios were generated. The model consisted of the main stem of the St. Francis River starting from Wappapello Dam and extending downstream 520.3 m (1707 ft) to a confluence, and from there 1306 m (4284 ft) further downstream to where it terminated at a location just below County Highway 517 bridge. A USGS gaging station is located at this bridge. The model also included a channel starting in Wappapello Lake, passing downstream through the limited-use Ogee spillway, and forming a 660.5 m (2167 ft) long tributary ending at the confluence described above. This channel is the spillway channel of Figure 1. The main stem was modeled by 27 cross sections, seven upstream and 20 downstream of the confluence, respectively. The tributary, requiring higher resolution, was modeled with 48 cross sections. Stationing and elevations were obtained from the National Elevation Dataset (NED) via ArcMap and the HEC-GeoRAS toolbar, and supplemented by additional sources. Thus, the HEC-RAS model includes one main stem with one tributary – requiring a total of three boundary conditions: two upstream and one downstream.

The bottom friction was specified utilizing the widely accepted Manning's n value. Selection and assignment of values for Manning's n was accomplished by a polygonal *landuse* feature class available via HEC-GeoRAS in ArcMap. This feature class requires the user to construct polygons that cover the entire flow domain. These polygonal areas are constructed such that they define or enclose areas of similar land use. An attribute table containing numerical values of Manning's n, indexed by land use, was constructed. Values of bottom friction were specified by assignment from a second table cross referenced by land use type. Therefore, all areas of a similar land use received the same numerical assignment of Manning's n.

The HEC-GeoRAS toolbar allows the user to execute a routine that assigns Manning's n to extents along each cross section using these polygonal areas and corresponding attribute tables. Therefore, the user is able to incorporate a spatially varying bottom friction based on expected, or reasonable, values associated with known land use. Manning's n values ranged from 0.02 to 0.035 for most of the main channel. Values as high as 0.12 were assigned to rip-rapped areas. All values originated by closely following guidance supplied in the HEC-RAS User's Manual (Hydrologic Engineering Center 2010b).

The limited-use spillway was modeled as a 226 m (740 ft) long Ogee spillway with a constant discharge coefficient of 3.0. The discharge from this spillway crosses over Missouri State Highway T only 90.8 m (298 ft) downstream from the structure. The highway was modeled as a broad-crested weir.

The HEC-RAS unsteady flow option was used in order to accurately calculate the discharge over the Ogee spillway. Analysis and observation revealed that subcritical flow occurs at the downstream terminus of the model, and that the backwater flowline has no influence on the value of discharge over the spillway. Topographic data were used to determine a downstream boundary normal depth slope of 0.000025.

Prior to 2011 the limited-use spillway had been overtopped only once, for a few days during March-April 1945. The authors decided to utilize the few existing 1945 hydraulic field measurements to simulate the 1945 event as a modeling scenario, for purposes of inundation mapping. Thus, essentially no field data existed with which to verify the model over a range of discharges. The authors opted instead to use the 1938 Rating Curve. This approach allowed greater flexibility by having access to spillway discharges corresponding to a continuous series of hypothetical but realistic lake elevations. In this way a realistic verification simulation was developed, featuring four representative lake elevations to use as input to the HEC-RAS model.

The boundary condition upstream of the spillway was set to a lake elevation hydrograph featuring periods of steady flow at each of the four selected lake elevations. The upstream boundary condition of the main stem of the St. Francis River (tunnel discharge) was set to an inflow discharge hydrograph. This unsteady hydrograph was specified according to a reasonable approximation of the established water control plan for Wappapello Dam. The plan calls for a maximum tunnel discharge of 283.18 cms (10,000 cfs). In the unlikely event of spillway overtopping, the plan calls for maintaining a combined total of 283.18 cms (10,000 cfs) from the tunnel and spillway. As the lake elevation rises and the spillway discharge increases, the tunnel discharge should be lessened. Discharge through the tunnel should be discontinued when the spillway discharge reaches 283.18 cms (10,000 cfs). The discharge hydrograph specified at the upstream boundary of the main stem of the St. Francis River approximated this water control plan.

All HEC-RAS computational simulations were run with bottom elevations *fixed*. Thus, options within HEC-RAS designed to accommodate erosion and deposition were not used.

In this manner it was possible to run the HEC-RAS model and adjust only the discharge coefficient of the Ogee spillway. Table 2 shows discharge over the spillway computed by HEC-RAS compared against that read from the 1938 Rating Curve. Trial-and-error revealed that a discharge coefficient of 3.0 in the HEC-RAS model yielded results comparable to the 1938 Rating Curve. The percentage differences in HEC-RAS discharge versus those read from the 1938 Rating Curve were at most 3.19 percent. The HEC-RAS model was considered to be verified with respect to spillway flow based on the results shown in Table 2.

Moreover, each of the four selected lake elevations of Table 2 is associated with an implied discharge coefficient C. These four implied discharge coefficients were estimated using simplifying assumptions, and were found to range from a minimum of 2.59 to a maximum of 2.74. They represent approximations of C based entirely on the physical model study (U.S. Waterways Experiment Station 1938a and b) and as such are completely independent from HEC-RAS. These approximations for C were found later after all HEC-RAS simulations were run, and serve to confirm the general range of values associated with this particular spillway.

The verified HEC-RAS model was then set up to simulate two unsteady flow modeling scenarios. The first of these was the 1945 overtopping event. This scenario was prepared by following basically the same procedure outlined above with the exception that the lake elevation hydrograph of April-May 1945 was used. Among the many results from this modeling scenario included inundation mapping which showed flooded areas downstream of the spillway.

The second modeling scenario was developed via a trial-and-error approach. It involved developing a stair-stepped (hypothetical) lake elevation hydrograph which reached an elevation high enough to indicate incipient flooding of the USACE facilities located downstream of the spillway. Again the HEC-RAS model was run the same as for the verification simulation with the exception that the hypothetical lake elevation hydrograph was substituted. The hypothetical modeling scenario was essentially a simple lake step elevation hydrograph characterized by a peak elevation of 132.9 m (405.0 ft) [NAVD 1988].

The inundation mapping results from these two modeling scenarios are shown in Figure 5. The lighter colored cross-hatching depicts inundation predicted by HEC-RAS modeling of the 1945 event. The darker colored cross-hatching depicts inundation predicted by HEC-RAS modeling of the hypothetical event. Figure 5 shows these two superimposed flooded areas at their respective maximum elevation. Note that the inundation mapping indicates water backing up a ravine and approaching the Wappapello Lake Management Office facility. Figure 5 shows that the lighter cross-hatching does not reach the facility, but the darker cross-hatching does. Thus, HEC-RAS modeling predicts a lake elevation of 132.9 m (405.0 ft) [NAVD 1988] is required for downstream flooding to reach the Wappapello Lake Management Office, and that the water will approach that location from the northeast along the axis of the ravine.

### 2011 Overtopping Event

Work reported thus far was assembled and presented in a poster at a regional conference (Nail and Kopsky 2011) in April 2011. Almost immediately after this date a series of storm fronts passed through southeast Missouri causing high soil moisture levels. Another round of very heavy rains followed, delivering approximately 12.7 cm (5.00 inches) in 24 hours, and totaled about 63.5 cm (25.0 inches) in one week, at Greenville, Missouri, a short distance upstream from Wappapello Lake. These precipitation events caused the lake to overtop the limited-use Ogee spillway early in the morning of 28 April 2011 at 2:00 AM CDT. The lake crested about a week later at 121.90 m (399.94 ft) [NAVD 1988] at 10:00 AM CDT on 3 May, which saw an elevation head of just over five feet at the spillway. Overtopping continued until the lake elevation finally dropped below the spillway crest early in the afternoon of 11 May. The limited-use spillway, which had not been overtopped since 1945, experienced two weeks of flow. The integrity of the dam and spillway were never compromised, and no lives were lost. However, damage from downstream flooding was very extensive. Missouri Highway T, just below the spillway, was completely eroded away almost immediately after the overtopping event began on 28 April. Additional perspective of the 2011 overtopping event can be gained by viewing the short video available on line (U. S. Army Corps of Engineers, St. Louis District, 2011, Accessed 18 March 2014, http://www.youtube.com, TeamSaintLouis Channel, *2011 Wappapello Lake Flood Fight, May 2, 2011*.). This dramatic overtopping event focused attention on the spillway, its performance, and the flooding downstream.

### Post-2011 Event Analysis and Modeling

The authors developed a HEC-RAS model of the 2011 overtopping by extensively modifying the HEC-RAS model that was developed prior to the 2011 event. Detailed post-2011 event topography data were obtained for the area of the channel, immediately downstream from the spillway, and extending nearly to the confluence with the main stem of the St. Francis River. The new topographic data was needed to reflect the severe channel erosion that occurred during the 2011 flood, which included the exposure of bedrock in some places. Advanced capabilities of ArcMap were utilized to splice these high resolution elevations into an existing raster file. This splicing was done so as to have one continuous raster file with sufficient data available for HEC-GeoRAS to interpolate new stationing and bottom elevations for export into HEC-RAS.

The revised HEC-RAS model covered the same geographical extents, and had the same number of cross sections within the main stem of the St. Francis River, as the model developed prior to the 2011 overtopping event. However, a total of 58 cross sections were utilized to define the spillway channel, extending from the confluence with the main stem upstream to Wappapello Lake. The bottom friction was specified in exactly the same manner as in the pre-2011 event model. A post-2011 event aerial photograph was spliced into existing aerial photography to aid in revisions to the polygonal feature class specification of bottom friction. The boundary conditions were specified in the same way as in the pre-2011 event HEC-RAS model described earlier in the article.

Spillway performance in the 2011 flood was assessed using established techniques outlined in USACE literature, as found in the EM (USACE 1992). The first concern was to determine if the spillway crest had been partially submerged by backwater during the flood, resulting in less spillway flow than the designers expected for the given lake level. Figure 6 is a graph developed using the USACE techniques found on Plate 3-5 of the EM. As is clearly visible from this graph, at values of h_{d}/H greater than about 0.4 the discharge coefficient is essentially unaffected. Note that a value of h_{d}/H equal to 0.5 corresponds to a downstream water surface elevation located half way between the spillway crest and the upstream lake elevation. Although no measurements were available, it was clear from video and photographic images recorded during the 2011 overtopping event, that the downstream water surface elevation never reached the point where h_{d}/H was equal to 0.5. Therefore, according to USACE guidance, spillway flow during the 2011 event was not diminished by backwater.

A significant improvement in the post-2011 event HEC-RAS model was incorporation of a variable discharge coefficient for the limited-use Ogee spillway. Routines internal to HEC-RAS incorporate technical information provided in the User's Manual (Hydrologic Engineering Center 2010b) and Hydraulic Reference Manual (Hydrologic Engineering Center 2010a), which include extensive references. These routines enable HEC-RAS to determine a discharge coefficient using an algorithm which incorporates spillway characteristics. These identifying characteristics include dimensions and design energy head, H_{D}, as described earlier. The authors became aware that the design energy head of this particular spillway was unknown. In order to proceed with computational hydraulic modeling with HEC-RAS, utilizing a variable discharge coefficient, the design energy head had to be determined. A six-step method of estimating H_{D} was developed. The design energy head was estimated by proceeding as follows:

- 1.
The WES Profile Shape of Equation 5, corresponding to n = 1.85 and K = 2.0, was plotted (downstream from the crest), as shown in Figure 7.

- 2.
Coordinates of the spillway surface were added to the graph obtained in step 1) above, and depicted as triangular symbols in Figure 7.

- 3.
A value of elevation head H

_{d}approximately equal to 5.8 m (19 ft) was read from the graph of Figure 7 by selecting the line which most closely approximates the surface formed by the triangular symbols.

The next three steps involved a trial-and-error process.

- 4.
An initial estimate for design energy head H

_{D}was made. - 5.
The value of H

_{D}was used to determine P/H_{D}in Equation 4, which yields a value of design discharge coefficient C_{D}. - 6.
Equation 3 was used to calculate the design discharge Q

_{D}. This value of Q_{D}was then used with the design elevation head H_{d}(previously obtained from step 3), along with approach depth P, and spillway length L, to calculate the approach velocity. A new value for the design energy head H_{D}could be calculated and compared with the initial estimate of step 4. If the values agreed within a specified tolerance then calculations were stopped. If values did not agree within the tolerance then the estimate for H_{D}was adjusted and the process repeated from step 5.

This procedure yielded a design energy head of 7.160 m (23.49 ft). This value of H_{D} was entered into HEC-RAS, which allowed the software to calculate a unique discharge coefficient for each lake elevation. The authors confirmed their estimate of design energy head H_{D} by obtaining a second estimate of design elevation head H_{d} using Buehler's published curves (Buehler 1954). This second estimate ultimately resulted in an equivalent design energy head. Lake elevations reached during the 2011 event were well below that required to reach the design energy head. If the design energy head on the spillway was exceeded the nappe would have separated from it, giving rise to decreasing pressures and cavitation. Detailed measurements were not recorded, but video and photographic images of the nappe during the 2011 event confirmed that it did not separate from the spillway. Finally, observations of the spillway surface after the 2011 event revealed no indications of damage due to cavitation.

For post-2011 event HEC-RAS modeling it was decided to utilize the 2011 overtopping event for verification. Some pertinent results of the post-2011 event HEC-RAS verification simulation are shown in Figure 8. This figure shows HEC-RAS computed discharges versus USGS field-measured discharges at the County Highway 517 bridge (site of USGS Gage 0739500). The triangular symbols represent USGS field measurements (volumetric flow rate in cms or cfs). These field-measured discharges should not be confused with values of discharge obtained from a stage-discharge rating curve. The smooth solid line on Figure 8 is the HEC-RAS computed discharges. The greatest difference was 11.8 percent, but this difference did not occur until 2:00 PM CDT on 6 May (late on day 35 at the last USGS field measurement shown on Figure 8). By that time the entire overtopping event was nearly finished and discharges had returned to nearly the same values recorded before the event began. Some of the difference arises as a slight lead or lag in time. Overall the agreement is excellent and, based on the results shown in Figure 8, the HEC-RAS model was considered to be verified and capable of modeling scenarios similar to the April-May 2011 overtopping event. The channel reach between the Ogee spillway and the County Highway 517 bridge is short and does not have much storage compared to the maximum spillway flow during the 2011 event. Also, the flow through the Ogee spillway changed slowly. Therefore, routing effects between the spillway and the County Highway 517 bridge were trivial and the USGS-measured flows were accurate measures of spillway flow. Values of the discharge coefficient C computed internally by HEC-RAS varied from a minimum of 2.49 to a maximum of 2.74, with typical values near 2.65.

During the 2011 overtopping event, the approach depth to the spillway was relatively shallow. The earlier-referenced post-2011 event topography data was used to determine the spillway approach depth, which was an averaged value of 0.6319 m (2.073 ft). The authors carefully considered how to utilize the verified post-2011 event HEC-RAS computational hydraulic model to assess the effect, if any, of approach depth on discharge. Hence, the authors elected to develop a series of modeling scenarios for the 2011 overtopping event with varying approach depths. This scheme allowed HEC-RAS to internally adjust the coefficient of discharge for detrimental effects caused by vertical accelerations and bottom friction, caused by relatively shallow (fast moving) water immediately upstream of the spillway.

Three modeling scenarios were constructed by keeping all specifications of the April-May 2011 overtopping event the same, with the exception of approach depths. The cross sections immediately upstream from the spillway were edited so that all bottom elevations, along the entire length of the spillway, were the same. These three modeling scenarios featured approach depths set to 0.6096 m (2.000 ft), 1.524 m (5.000 ft), and 3.048 m (10.00 ft), respectively. Figure 9 shows the HEC-RAS computed discharges for these three modeling scenarios, compared against USGS field measurements, at the County Road 517 bridge (USGS Gage 0739500). Also shown are the HEC-RAS computed discharges from the post-2011 event verification simulation, which appeared in Figure 8. Values of the discharge coefficient C computed internally by HEC-RAS varied from a minimum of 2.64 to a maximum of 3.27, with typical values near 3.00.

The results displayed in Figure 9 show consistent increases in discharge as the approach depth is made uniform along the 226 m (740) foot length of the spillway, and increased. A maximum increase in discharge of 12.2 percent was predicted for an approach depth of P = 3.048 m (10.00 ft). This value is indicative of a potential increase in spillway performance, obtainable if the depth immediately upstream of it were greater. This trend is well known and documented in the EM (U. S. Army Corps of Engineers 1992) and explained by Sturm (Sturm 2001) and Chow (Chow 1959). This analysis of approach depth leads to the conclusion that, during a flood event exceeding that of 2011, the existing shallow approach depth will hinder outflow early on. Thus, lake levels will rise closer to the top of the earth dam than would be the case if the approach depth were greater.

## Closure

The one-dimensional HEC-RAS multi-purpose open channel flow modeling software was successfully used, with Arc-Map and HEC-GeoRAS, to simulate flow over the Wappapello Dam limited-use Ogee spillway. Prior to the April-May 2011 storm event, during which the limited-use spillway was overtopped for the first time since 1945, a HEC-RAS model was developed, verified, and meaningful inundation mapping results obtained. These results included the extent of inundation that could be expected during a hypothetical overtopping event. HEC-RAS computational hydraulic modeling predicted a lake elevation of 132.9 m (405.0 ft) [NAVD 1988] would be required for the resulting floodwaters overtopping the spillway to reach the Wappapello Lake Management Office. Spillway performance during the 2011 event was analyzed afterwards. Results indicated that submergence of the spillway from downstream did not occur. A technique was employed which successfully estimated the design energy head of 7.160 m (23.49 ft) for this particular spillway. HEC-RAS modeling developed after the 2011 event incorporated this estimated design energy head, allowing the spillway discharge coefficient to vary with discharge. This post-2011 event HEC-RAS model was verified to actual USGS discharge measurements, and used to generate several modeling scenarios which included varying approach depths to the spillway. Results indicated that, while the spillway did perform as designed, the performance is limited by the shallow approach depth.

## Disclaimer

The views contained herein are those of the authors and are not necessarily those of the St. Louis District, USACE. This paper summarizes some of the work performed under two separate work arrangements between the St. Louis District, USACE, and The University of Tennessee at Martin, both known as a Challenge Partnership Agreement. No potential conflicts of interest are reported by the authors.

## Acknowledgments

The authors wish to gratefully acknowledge assistance provided by numerous personnel at the Wappapello Lake Management Office, at the main office of the St. Louis District, USACE, and at The University of Tennessee at Martin. The work underlying this paper would not have been possible without their assistance.

## Notes on Contributors

Gregory H. Nail is currently an Associate Professor in the Engineering Department at The University of Tennessee at Martin, after having started on faculty there in 2002. From 1991 through 2002 he worked as a Research Hydraulic Engineer for the Coastal and Hydraulics Laboratory of the Vicksburg Site, Engineer Research and Development Center (known prior to 1999 as Waterways Experiment Station), of the USACE. Dr. Nail received the Ph. D. from Texas A&M University in 1991, in Mechanical Engineering, with specialization in Fluid Mechanics. He is author or coauthor of 13 scientific papers and technical reports.

Raymond J. Kopsky, Jr., is currently a Civil Engineer (Hydraulics) with the St. Louis District, USACE. He has been employed with the St. Louis District since 1986, and also worked with the District during 1982 while still an undergraduate student. Mr. Kopsky earned an M. S. in Civil Engineering from the Missouri University of Science and Technology (formerly known as The University of Missouri at Rolla) in 1985. His area of specialization is in open channel hydraulics, engineering hydrology, and related topics. He is author or coauthor of several scientific papers and technical reports.