This report presents a comparison of published discharge values through selected water control structures computed by the USGS to discharge values calculated using theoretical equations and coefficients. The verification of direct discharge measurements adds a level of certainty that is not possible with the theoretical equation approach. However, the personnel, time, equipment, and maintenance required to collect and analyze the data increase cost. This report compares percent differences between the two methods to help evaluate the tradeoff between quality of discharge data and cost.

Selected water control structures were limited to those (1) located in Florida; (2) where traditional calculation methods could be applied; and (3) for which the USGS published discharge values between October 1, 2007, and September 30, 2019. This period was selected for the use of instantaneous values in the discharge calculations. Previous years’ data were mainly daily values, which have traditionally been the USGS-published product.

Special-use structures, such as those with ballasts and (or) parabolic dimensions, were excluded from this comparison for simplicity. A general overview about hydraulic conditions and characteristics is provided for the structures investigated in this study. However, specific details about structure designs and operations are outside the scope of this report.

The flow regimes encountered at water control structures are classified as orifice or weir flow. Orifice flow (eqs. 1, 3, or 4, table 1) occurs when water flows beneath the partially opened gate, but the gate still obstructs flow. Weir flow (eqs. 2, 6, or 7, table 1) occurs when the gate is opened to a height clear of the water, or at least two-thirds of the headwater elevation, and the water flows freely through the opening. Both flow regimes can be classified as free or submerged, depending on the tail-water elevation relative to the headwater and gate opening (Sanders and Feaster, 2004).

Submerged orifice flow occurs when the tail-water elevation rises above the elevation of the opened gate, which obstructs the free flow of water through the gate opening. Submerged orifice flow is calculated using equation 3 or 4, both of which are similar to the free orifice equation but include the addition of a submergence coefficient, C_gt or C_gs. Equation 3 for submerged orifice flow is found in most textbooks, whereas equations 4 and 5 are used by the USGS to simplify the calibration. Submerged weir flow is calculated using equation 6 if conditions dictate that submergence affects flow through the structure. Submerged weir flow occurs when the tail-water elevation rises to a specified value of the ratio of tail-water to headwater elevation, known as the submergence ratio, R_w. Sanders and Feaster (2004) suggest using the submergence ratio default value of 0.6, which is typically used for USGS computations unless otherwise specified. Hulsing (1967) suggests a submergence ratio of 0.5–0.85, depending on the structure type, and Chow (1959) suggests a submergence ratio of 0.80 for a specific Parshall flume structure.

Headwater and tail-water levels at water control structures were collected in accordance with USGS standards (Sauer and Turnipseed, 2010a). Discharge was routinely measured by using various types of equipment, depending on depth, velocity, and environmental conditions, in accordance with USGS standards (Sauer and Turnipseed, 2010b). Discharge coefficients were computed using the relevant equations for each structure depending on hydraulic conditions.

USGS methods of discharge computation through water control structures were followed as outlined in Sanders and Feaster (2004). This publication is tailored to the setup and operation of an outdated USGS program, DAMFLO.2, but the procedures for rating structures, equations, and discharge calculations are the same and currently used to produce instantaneous and daily discharge values published by the USGS. The basic concepts are presented here, and more detailed explanations and computation options can be found in Sanders and Feaster (2004).

The flow coefficients for each flow regime observed at the structure are determined from discharge measurements. For free orifice and free weir flow regimes, a simple back calculation was performed using equation 1 or 2 to determine the flow coefficient value at a range of gate openings and headwater levels by using the measured discharges. Real-time discharge for free orifice flow was calculated using the relation of discharge coefficient to vertical gate opening. For free weir flow, the relation between discharge coefficient to headwater level was used.

Equation 3 for submerged orifice flow is found in most textbooks; however, equation 4 is used by the USGS instead, because it simplifies the development of a calibration relation. The submerged orifice coefficient used in equation 4 was computed from the additional step shown in equation 5, where C_gs is related to the submergence ratio (h₃/h_g) by calibration (Sanders and Feaster, 2004).

The most common submerged weir flow calculation uses equation 6, which includes the addition of a submergence discharge coefficient, C_ws. This coefficient is related to the submergence ratio (h₃/h₁) by calibration (Sanders and Feaster, 2004) and is the method used for submerged weir flow calculation for each structure in this study unless otherwise specified. Another technique used to calculate submerged weir flow utilizes the unit fall method (Kennedy, 1984), whereby the ratio of measured discharge to the square root of fall is related to the tail-water elevation by calibration, as shown in equation 7 (Sanders and Feaster, 2004). The criteria to differentiate free weir flow from submerged weir flow varies by source, which suggests each structure should be analyzed individually after a number of submerged weir flow measurements are made. The USGS uses 0.60 for R_w by default, upon suggestion by Collins (1977), but this can be changed if more detailed knowledge of the structure is known.

The method employed by the USGS relies on recurring discharge measurements over a range of gate settings and water levels to verify the flow regimes and ratings. Some structures may only experience certain conditions, such as free weir flow or submergence, in emergency situations. Flow coefficients determined from discharge measurements conducted under these flow regimes may be different from those calculated from theoretical equations or previously used ratings, and rating adjustments may be necessary after these discharge measurements for a more accurate definition.

Theoretical discharges referenced in this report are those produced using selected equations shown in table 1 for each flow regime and weir shape encountered. Values for the discharge coefficients were selected from references, as shown in table 2.

Typical values for the free orifice flow coefficient, C, include the following ranges: 0.5–0.7 (U.S. Army Corps of Engineers [USACE], 2020), 0.55–0.60 (Finnemore and Franzini, 2002), and 0.57–0.71 (Tillis and Swain, 1998). The study done by Tillis and Swain (1998) included three vertical lift gate structures in south Florida under free orifice flow and produced an average free orifice flow coefficient of 0.65, with a range from 0.57 to 0.71. An average value was calculated from the range of these three sources to produce a free orifice flow coefficient of 0.60 for typical vertical lift gates in Florida. Information is limited for radial gates, and the suggested range of 0.6–0.8 is slightly larger than that suggested for vertical lift gates (USACE, 2020). Therefore, an average coefficient of 0.70 was calculated for free orifice flow through radial gates.

Submerged orifice discharge for the theoretical computations was calculated using equation 3, which incorporates the vertical gate opening to calculate the cross-sectional area of the orifice. Values of the submerged orifice flow coefficient, C_gt, are practically the same as for free orifice flow, except when water levels are less than 10 feet (ft) when referenced to the gate sill, C_gt may be slightly less (Finnemore and Franzini, 2002). The Bureau of Reclamation (2001) determined a submerged orifice coefficient for standard rectangular irrigation weirs experimentally to be 0.61 for fully contracted orifices and 0.70 for those with bottom and side contractions suppressed. Based on these two references, the same coefficients were used for submerged orifice flows as free orifice flows.

The structures investigated in this study are classified as sharp, ogee, or broad crested weirs. A free weir flow coefficient, C_w, of 3.2 is used as the default value for sharp crested weirs in USACE (2020). Using the procedures outlined in Rantz and others (1982) for thin plate weirs with a vertical gate face and sill at the streambed produced a free weir coefficient of 3.21 for weirs with side contractions. Because all sharp crested structures included in this study have side contractions, an average of the two coefficients, 3.2, was used. USACE (2020) suggests a range of coefficients of 3.2–4.1 for ogee crested weirs and 2.6–3.1 for broad crested weirs. Therefore, average values of 3.6 for ogee crested weirs and 2.8 for broad crested weirs were used for the theoretical computations for each of these weir shapes.

No sharp crested weirs experienced submerged weir flow regimes during this study. The adjustment of free weir flow for submergence, C_ws, at broad crested weirs was calculated using methods presented in Hulsing (1967) for submerged paved highway embankments. This method uses the submergence ratio to determine C_ws and applies no adjustment for submergence ratios below 0.80. A similar method and diagram are suggested by USACE (2020) to adjust weir flows for submergence. The only submerged weir discharge that occurred at ogee crested weirs in this study was estimated by the USGS and would not be a good comparison for the two methods, so submerged weir flow comparisons for ogee weirs were omitted.

The theoretical discharge was calculated for the same time intervals in which they were published by the USGS, and the same criteria were used to determine the flow regime and submergence to provide a constant comparison between methods. Discharge was computed at typical time intervals of 15 minutes, 30 minutes, and 1 hour, and is herein referred to as instantaneous. The instantaneous discharge values were used to compute a daily mean discharge value, and the comparisons were done using daily mean values. USGS discharge data are available from the USGS National Water Information System database (U.S. Geological Survey, 2021). The published values include all computed discharge values at the station location and may differ from the discharge through the structure because of added flows such as those from lockages or separate channels adjacent to the structure. The discharges used for the comparison and the calculated percent differences are available in a USGS data release (Ryan and Hazelbaker, 2022).

To avoid erroneous comparisons, the discharge values from both methods were not compared if (1) headwater, tail water (if submerged), or gate opening data were missing; (2) USGS-published values were estimated; (3) discharge values were changed from rated conditions because of debris or construction; or (4) the flow regime changed during the day. Differences calculated in the comparison still occurred from structure age and weathering, the combination of flow through multiple gates, and flow that did not approach the structure at a completely perpendicular angle, all of which were not accounted for in the theoretical approach. The final comparison used the percent difference calculated from both daily mean discharges (eq. 8). To avoid large percent differences resulting from small actual differences, no comparisons were done with discharge below 10 cubic feet per second (ft³/s). The period considered for this study includes calculated discharge values between October 1, 2007, and September 30, 2019.

Histograms were constructed using a range of percent differences as a simple and meaningful way to view how well the two datasets compare. They were not constructed to show the symmetry or shape of the percent differences but to display meaningful ranges that would occur had the chosen coefficients and theoretical ratings been used instead of those used to compute USGS discharge.

The water control structures investigated in this study were limited to (1) structures having vertical lift and radial gate operations; (2) structures whose discharge data were computed by the USGS using conventional methods; and (3) structures located in Florida (fig. 1). Basic details about each structure, including compared flow regimes, are shown in table 3.

Daily mean discharge values were compared for both submerged orifice and submerged weir flow regimes at Cherry Lake Outlet. Of the 590 values compared, the theoretical discharge was an average 28.6 percent lower than the USGS daily mean discharge, and the average absolute difference was 36.4 percent. The most common range of daily percent differences was between 30 and 40 percent (fig. 2). Theoretical daily mean discharge values ranged from 14.7 to 580 ft³/s. There was a reversal of percent differences from positive to negative as theoretical discharge increased, and discharges greater than 500 ft³/s had percent differences near −50 percent (fig. 3).

Discharge compared for Apopka Beauclair Canal was either free or submerged orifice flow. Some free weir flow occurred, but not enough values were available to compute and compare a daily mean theoretical discharge. The theoretical daily mean discharge ranged from 11.3 to 627 ft³/s, and the most common range of daily percent differences was between −20 and 0 percent for the 382 values compared (fig. 4). The theoretical discharge was an average 26.5 percent higher than the USGS discharge, and the average absolute difference was 27.3 percent. The percent difference increased (became more negative) as theoretical discharge increased, and percent differences larger than −50 percent occurred for theoretical discharges greater than 350 ft³/s (fig. 5).

Free orifice flow through the Moss Bluff structure was compared for 495 daily mean values. The theoretical discharge ranged from 15.8 to 2,300 ft³/s, the average percent difference between theoretical and USGS discharge was 7.35 percent, and the average absolute difference was 7.49 percent. The most common range of daily percent differences was between 10 and 15 percent (fig. 6). Mostly constant percent differences, either near 0 or 12 percent, were calculated for theoretical discharges above 100 ft³/s (fig. 7). This occurred because the same theoretical rating is used to compute the USGS values, and the coefficient was later changed to fit the discharge measurements.

Discharge was compared at Rodman Dam for 4,110 daily mean values, all of which were classified as free orifice flow. The theoretical discharge ranged from 17.6 to 5,130 ft³/s, and the average percent difference between theoretical and USGS discharge was 18.5 percent. The average absolute difference was also 18.5 percent because all percent differences were positive. The most common range of daily percent differences was between 10 and 20 percent (fig. 8). Percent difference decreased as discharge increased, and theoretical discharges greater than 1,000 ft³/s were within 20 percent of the USGS discharge (fig. 9).

Only 34 daily mean discharge values were compared at S–12–A under submerged orifice conditions because of limited gate openings for complete days. The theoretical daily mean discharge ranged from 38.3 to 206 ft³/s and was lower than the USGS values by an average of 22.2 percent. The average absolute difference was also 22.2 percent because all percent differences were positive. All percent differences except three fell between 20 and 25 percent (fig. 10), and the percent differences remained below 30 percent for the entire range of theoretical discharges (fig. 11).

Discharge at S–12–B was compared for 141 daily mean values all under submerged orifice conditions. The theoretical daily mean discharge ranged from 23.2 to 292 ft³/s and was lower than the USGS values by an average of 20.1 percent, which also matches the average absolute difference percent difference. The most common range of daily percent differences was between 15 and 20 percent (fig. 12), and the percent differences remained near 20 percent for theoretical discharges greater than 75 ft³/s (fig. 13).

Discharge at S–12–C was compared for 316 daily mean values, all under submerged orifice conditions. The theoretical daily mean discharge ranged from 25.0 to 440 ft³/s and was lower than the USGS values by an average of 25.8 percent, which also matches the average absolute difference percent difference. The most common range of daily percent differences was between 25 and 30 percent (fig. 14), but theoretical discharges greater than 300 ft³/s typically had percent differences between 15 and 25 percent (fig. 15).

S–12–D had the most discharge data and largest daily mean discharge of the four S–12 structures compared. The theoretical daily mean discharge was compared for 732 daily mean values that ranged from 14.1 to 662 ft³/s. These theoretical discharge values were lower than the USGS values by an average of 21.2 percent, which also matches the average absolute difference percent difference. The most common range of daily percent differences was between 20 and 25 percent (fig. 16), and that range included the entire range of daily mean theoretical discharges (fig. 17).

Daily mean discharge at S–79 was compared for 2,613 values under submerged orifice conditions. The theoretical daily mean discharges ranged from 20.0 to 14,700 ft³/s, and the comparison produced an average difference of 4.96 percent. The average absolute difference was 5.71 percent. Most theoretical discharges plotted within 20 percent of the USGS discharge (fig. 18). Figure 19 shows a strong positive relationship between the percent difference and theoretical daily mean discharge, where the theoretical discharge plots at least 10 percent low for values above 8,000 ft³/s.

Free orifice and free weir flow were compared for 762 values at Cow Pen Slough, and the theoretical discharge was an average 22.8 percent higher than the USGS discharge. The average absolute difference was 24.6 percent. The theoretical daily mean discharges ranged from 10.5 to 1,360 ft³/s, and approximately half of the theoretical discharges plotted within ±20 percent of the USGS discharge (fig. 20). The percent differences showed an increasing curvilinear relationship with discharge, and mostly positive percent differences were calculated at theoretical discharges greater than 300 ft³/s (fig. 21).

Daily mean discharge at S–551 was compared for 349 values under submerged orifice conditions. The theoretical daily mean discharge ranged from 15.7 to 904 ft³/s, and the comparison indicated an average difference of 14.5 percent, which also matches the average absolute difference percent difference. The theoretical discharge plotted below the USGS discharge for all values compared in the study, and a range of 10–15 percent was the most common (fig. 22). The percent difference showed a constant relationship between 10 and 20 percent as theoretical discharge increased greater than 400 ft³/s (fig. 23).

Daily mean discharge at the Inglis Dam structure was compared for 1,018 values under free orifice conditions. The theoretical daily mean discharge ranged from 12.5 to 3,560 ft³/s, and the comparison produced an average difference of 9.80 percent. The average absolute difference was 10.6 percent. Approximately 75 percent of the theoretical daily mean discharges plotted between 10 and 20 percent lower than the USGS daily mean discharge (fig. 24). The percent difference showed a constant relationship between 0 and 20 percent for all theoretical discharges greater than 1,000 ft³/s (fig. 25).

The flow regimes for the Bypass Channel structure were classified as either free orifice or free weir for the 3,899 values compared. The theoretical daily mean discharge ranged from 176 to 1,510 ft³/s, and the comparison indicated an average difference of 8.20 percent. The average absolute difference was 8.50 percent. The range of 0–5 percent included the most daily mean percent differences, and approximately 99 percent of the theoretical discharges were within ±20 percent of the USGS discharges (fig. 26). The percent difference slightly decreased as theoretical discharge increased (fig. 27).

An overall comparison of the theoretical discharge to the USGS discharge for all structures combined was calculated using the average of absolute values of the daily percent differences from each structure (table 4). The average of the absolute values of the percent differences was 18.7 percent. This average represents the overall difference between theoretical discharge and USGS discharge for all 15,441 values.

The daily mean percent differences were compared for each structure in the study (fig. 28). The only negative mean and median percent differences, which indicate that the theoretical discharge is larger than the USGS discharge, were for 02237700 Apopka Beauclair and 02299710 Cow Pen Slough, which also had the largest range of percent differences.

A comparison of all theoretical and USGS discharge values for all structures shows a curvilinear relationship that begins at approximately 1,500 ft³/s (fig. 29). A good fit between the two methods would plot near the one-to-one line. However, the theoretical discharge plotted farther below the line and included more scatter as USGS discharge increased. This figure shows that even though the overall average of the absolute values of the percent differences is 18.7 percent for all structures included in the study, greater percent differences and less confidence may occur for large discharges.

The free orifice flow regime had the largest number of discharge values compared, followed by submerged orifice, free weir, and submerged weir, which only compared 59 values (table 5). Free orifice flow was compared at six structures, submerged orifice at eight structures, free weir at two structures, and submerged weir at one structure.