HEC 25 - Tidal Hydrology, Hydraulics, and Scour at Bridges

Chapter 3 Basic Tidal Hydraulic Methods

3.1 Introduction

A variety of techniques are available for the hydraulic evaluation of bridges in tidal waterways. These techniques range from simple conceptual approaches to full hydro-dynamic modeling. The basic simplified methods are discussed in HEC-18 (Richardson and Davis 2001) and are summarized in this chapter. More complex dynamic modeling approaches are presented in Chapter 4.

Unlike the analysis of bridge hydraulics for extreme conditions on rivers, where the discharge is known for a given flood frequency, tidal hydraulic methods are used to compute the peak discharge during an astronomical tide, hurricane storm surge or hurricane storm tide. For simple methods (tidal prism, Section 3.2, and orifice approach, Section 3.3) the continuity equation is applied to determine the velocity needed for scour calculations. Complex methods (Chapter 4) use unsteady flow simulation to determine the peak discharge and compute the hydraulic parameters needed for scour calculations. Routing (Section 3.4) is intermediate in complexity, and can generally be replaced with a simple 1-dimensional model.

The applicability and limitations of the various tidal hydraulic approaches to different tidal waterway conditions are also discussed in this chapter. The simple approaches are adequate for many tidal waterways. The simple methods become less suitable as the length, size, or complexity of the waterway increases. Complex methods produce more reliable estimates of tidal hydraulic conditions. The greater effort required to apply the complex methods is justified when the simple methods are overly conservative and result in unwarranted bridge design, construction, and countermeasure costs. The complex methods are even more appropriate when multiple bridges are included in a single model so the costs and benefits can be distributed.

3.2 Tidal Prism

The tidal prism approach (Neill 1973) can be used when the waterway or bridge does not significantly constrict the flow, the water surface is relatively level, and there is little energy loss as flow travels through the reach. As shown in Figure 3.1, this approach is most applicable to estuaries. The basic assumption is that the entire estuary fills and empties simultaneously with the daily tide or storm tide. While this is physically impossible, it is reasonable for small waterways. The assumption results in a simple equation for maximum discharge:

Q_max= π x VOL/t

(3.1)

where:

Q_max = Maximum discharge during a tidal cycle (ft³/s, m³/s)
VOL = Volume of water in the tidal prism between low and high tide (ft³, m³)
T = Tidal period between successive high or low tides (s)

The resulting discharge is often a conservative estimate of peak flow. Figure 3.2 shows the tidal stage and discharge hydrographs resulting from this approach. The maximum discharge is assumed to occur about midway between low and high tides.

Figure 3.1. Estuary (after Neill 1973).

Figure 3.2. Simplified tidal hydrograph (after Neill 1973).

In addition to waterway constriction, other factors influence the applicability of the simplified tidal prism method. If he estuary is long and the limit of tidal action is far inland, then this approach can be overly conservative because the water surface within the tidal storage zone will not be level and it cannot be assumed that the interior tide rises and falls simultaneously with the ocean tide. Figure 3.3 shows typical tide attenuation in an estuary. If there is little difference in both tide height and travel time (time between ocean at estuary high tides) then the tidal prism approach can produce reasonable results. If there is a significant difference in either height or travel time, then the flood prism approach is overly conservative.

Another factor that can greatly influence the accuracy of the tidal prism approach is whether flows are confined primarily to the channel or, especially during a storm surge, inundation of the floodplain occurs. When the floodplain area is inundated, the prism volume increases rapidly with respect to elevation. Because floodplain flow is shallow and vegetation causes high roughness, energy loss can be significant, making the simple prism approach overly conservative. Figure 3.4 shows typical estuary volume versus stage relationships. Case 1 is an estuary that has little floodplain, and Case 2 is an estuary with significant floodplain area. For Case 2, the large floodplain area results in significant curvature in the storage curve. Assuming a storm tide ranges from elevation 0 to 4, the simple prism method is more applicable to Case 1 because approximately half of the prism volume is filled at elevation 2. For Case 2, flows are confined to the channel up to elevation 2 and only 20 percent of the prism volume would actually be filled. For Case 2, use of the simple prism approach results in half of the prism volume is filled at elevation 2. This produces 2.5 times the discharge than could actually occur based on available volume. A slight modification to the simple tidal approach is to solve for the discharge using the following equation:

Q = ΔVOL/Δt

(3.2)

where:

Q = Discharge for the time step(ft³/s,m³/s)
ΔVOL = Incremental chang in stored volume for the time step (ft³,m³)
Δt = Computational time increment (s)

This method uses the storm tide hydrograph to determine a change of elevation for the time increment. The change in volume for that change in elevation is determined from the stage versus storage curve for the estuary. Except for small estuaries, dynamic modeling is more appropriate because higher resistance in the floodplain areas can also make the tidal prism method overly conservative.

3.3 Orifice Approach

The orifice approach is applicable to constricted waterways, such as inlets to bays, where significant energy loss is confined to the inlet channel and the adjoining bay is only partially filled during a tide cycle (Figure 3.5). The maximum velocity (ft/s, m/s) is calculated as:

V_max=C_d(2g ΔH)^½

(3.3)

where:

C_d = Coefficient of discharge
ΔH = Maximum head differential (ft, m)

Figure 3.3. Typical Tides in an Estuary.

Figure 3.4. Typical storage relationships for tidal prism method.

Figure 3.5. Bay and inlet (after Neill 1973).

The discharge coefficient includes entrance and exit losses and friction loss in the channel (van de Kreeke 1967). This is analogous to treating the inlet as if it were a culvert. The discharge coefficient cannot exceed 1.0 and for many practical applications can be assumed to equal 0.8. The discharge coefficient can be estimated by

C_d = (1/R)^1/2 (3.4)

where:

(3.5)

and

R = Coefficient of resistance
K = 1.486 in English units and 1.0 in SI units
K_o = Velocity head loss coefficient on the ocean side taken as 1.0 if the velocity goes to 0
K_b = Velocity head loss coefficient on the bay side taken as 1.0 if the velocity goes to 0
n = Inlet channel Manning's n
L_c = Length of inlet (ft, m)
h_c = Average depth of flow in the inlet at the mean tide elevation (ft, m)

As shown in Figure 3.6, this approach is most applicable to daily tides where the head differential can be measured. For storm tide discharge computations, the head differential would need to be estimated, which in practical application can be difficult. Once the maximum velocity is estimated, the maximum discharge is then calculated from the continuity equation (Q=VA). The peak discharge occurs for the maximum head differential, which can occur at elevations other than mid-tide, but is often assumed to occur at mid-tide. Figure 3.6 illustrates why the tidal prism approach is not at all applicable to inlets. This is because the tide range and timing are so different between the ocean and bay.

Figure 3.6. Typical Tides in a Bay.

Another issue with inlets is the condition where degradation does not produce a reduction in velocity. In rivers, channel degradation and contraction scour reduce flow velocity as the channel enlarges. In inlets to large bays, however, flow velocity may not decrease as the inlet degrades and waterway area increases. This is because inlet velocity is related to the head differential between the ocean and bay (Equation 3.3). The discharge coefficient can actually increase as the inlet depth increases. If the bay is large, the corresponding increase in discharge may not significantly affect the bay water surface level and the head differential, DH, may not decrease. Therefore, inlet velocity can remain relatively constant as the inlet degrades and the degradation can continue unabated. Inlet degradation may be limited by an erosion resistant layer but, if no resistant layer is present, the inlet will continue enlarge until, eventually, the tidal prism is satisfied and velocities actually do decrease.

3.4 Routing Approach

To overcome the limitations of the tidal prism approach (assumed zero energy loss) and orifice approach (unknown head differential during a storm surge), routing methods can be applied. A method developed by Chang et al. (1994) is presented in HEC-18 (Richardson and Davis 2001). The method combines the orifice equation, the storage relationship for the bay, and the surge hydrograph to calculate flows through a constricted waterway such as the inlet to a bay (Figure 3.5). The ocean storm tide is input as a function of time, the bay storage is related to bay water surface elevation, and the discharge through the waterway is calculated from the head differential between the ocean and the bay. The iterative solution technique is presented in HEC-18 using a spreadsheet or simple computer program. A similar approach is used in the ACES-Inlet model (USACE 1992). In ACES, the bay is treated as a storage area and the flow in the inlet is computed for the input storm surge hydrograph. The ACES-Inlet model can also be used to compute flow in two inlets connected to a single bay.

Routing methods can have varying levels of refinement. They can, for example, include upland runoff as a separate inflow. They can also account for overtopping flow over approach roadway embankments and barrier islands. If the bay is formed by a barrier island, overtopping would act as relief for the bridge. When the ebb tide drops below the overtopping elevation, unless a breach or new inlet forms, the entire bay may have to drain through the existing inlet.

In the case of large or long bays, the routing method can also be conservative because, as with the simple prism method, the bay is assumed to fill as a level pool. The level-pool assumption is an approximation of the actual flow conditions. In large bays the error this assumption causes can be significant. For small bays the level-pool assumption is not overly conservative. In most cases where routing is considered, a 1-dimensional model, such as HEC-RAS or UNET, can be substituted with a similar level of effort.

3.5 Method Selection

Based on the assumptions and limitations incorporated into the previously described approaches, selection of the appropriate method is dependent on the characteristics of the tidal waterway or crossing. Figure 3.7 provides general guidance on how these factors influence the selection process. Dynamic modeling should always provide the most reliable results, although the effort required for dynamic modeling is not warranted for some bridges over tidal waterways. In Figure 3.7, each factor (in capital letters) increases in amount or significance from left to right. As the factor increases in significance, the recommended analysis approach changes from simple techniques such as tidal prism or orifice to increasing degrees of sophistication, such as 1- or 2-D hydrodynamic.

As an example of the information in Figure 3.7, the second item in Figure 3.7 illustrates the modeling choices for floodplain size. An estuary with little or no floodplain would be adequately modeled with tidal prism. As floodplain size increases, 1- or 2-D modeling would be more appropriate. For very wide floodplains, 2-D modeling is recommended. The seventh item in Figure 3.7 shows that if wind effects are very important, 2-D modeling is needed as wind stresses are better simulated in a 2-D model. All other approaches (tidal prism, orifices, routing, and most 1-D models) do not incorporate wind effects, and are applicable when wind is not an important.

Figure 3.7. Factors influencing the selection of a tidal modeling approach.

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