Minimizing Roadway Embankment Damage from Flooding (2016)

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27 CHAPTER FOUR HYDROLOGIC AND HYDRAULIC FACTORS INTRODUCTION Hydrologic and hydraulic factors are extremely important in the design of embankments subjected to flooding. These factors control the volume of water that is likely to flood the embankment’s surroundings, the duration of the flood, and the water surface elevation. These factors affect the design against nearly all failure modes and in particular the over- topping failure mode and the seepage failure mode. How- ever, the selection and estimation of these factors is not a straightforward process. It is fundamental to couple every decision with engineering judgment to consider, to a feasible extent, site variability and constraints. The goal of this chapter is to present hydrologic and hydraulic concepts that could be employed in the design of roadway embankments subjected to flooding. The infor- mation included herein is based on available literature and guided by the survey responses. The following useful con- cepts shed light on some important design considerations. USEFUL CONCEPTS This section presents useful concepts on the selection of the design flood frequency and the impact of the flood frequency selection on such design parameters as discharge and veloc- ity. A brief description of coastal parameters is also included. Design Flood Frequency One of the most important hydrologic factors to be consid- ered in the design of embankments subjected to flooding is the design flood. This design flood is chosen on the basis of the recurrence interval, also called the return period. The 1% chance flood (100-year flood) and the associated river flow dis- charge Q100 are often used. The 100-year flow discharge Q100 (m3/s) is the discharge that has a 1/100 probability of being exceeded in any one year. Another commonly considered flood is the 500-year flood and the 500-year river flow discharge Q500 (m3/s), which has a 1/500 probability of being exceeded in any one year. One simple way to obtain the flow discharge value for these floods is to collect the flow history as a function of time (flow hydrograph, Figure 49). The flow discharge is typically collected through a stream gage placed along a river or regres- sion of stream gage data for ungaged sites. FIGURE 49 Hydrographs representing daily peak flow discharge, velocity, and water depth (Briaud 2013). The following simple graphical method serves as a good illustration of the process. This method (e.g., Chow et al. 1988) consists of obtaining the yearly maximum flow parameters from the hydrograph, ranking them in descending order of intensity, calculating for each flow the probability of exceedance as the rank divided by the total number of observations + 1, then plotting the flow versus the probability of exceedance on a semi log paper such as the one in Figure 50. Once the data are plotted, a linear regression is performed over, say, the first 20 to 30 years of data and extrapolated to the 0.01 probability of exceed- ance for the 100-year flood and to the 0.002 probability of exceedance for the 500-year flood. The return period is the inverse of the probability of exceedance. There are other and more refined ways of obtaining these design floods, but this simple graphical method helps one to understand the process and the meaning of a 100-year flood: a flood that has a 1% chance of being exceeded in any one year. Figure 50 shows the result of an analysis for the hydrograph at the Woodrow Wilson Bridge. As that figure illustrates, the 100-year flood has a flow discharge of 12,600 m3/s and the 500-year flood has a value of 16,600 m3/s.

28 FIGURE 50 Flood frequency curve obtained from a measured flow discharge hydrograph (Briaud 2013). Note that the example in Figure 49 shows a hydrograph for a period of nearly 40 years. However, the hydrograph could be for a much shorter period of time, such as one single flood where the hydrograph may be only 48 hours long and show the increase and decrease of velocity during the flood (Figure 51). This might be called a single storm or flood hydrograph. A storm hydrograph shows how a drainage basin responds to rainfall during a storm event. This type of hydrograph covers a relatively short period, for instance, hours or days. It is useful in the applications that require knowledge of the flow discharge over time. The peak flow discharge is of interest to the designers because the struc- tures are sized accordingly, but the hydrograph provides a higher level of precision. Hydrographs provide useful information about the anticipated flood, such as the duration of overtopping. Many methods are available for the computation of hydro- graphs, and can be found in such references as HDS-2 (McCuen et al. 2002). When sufficient data are avail- able, the computations can be done without difficulties. However, if such data are not available, synthetic methods can be used. The selection and reliability of an adequate method in this case requires a clear understanding of the site conditions and of the existing methods coupled with engineering judgment. An example of a storm hydrograph is shown in Figure 51. Its main elements are the base flow, the time base, the time to peak, the rising and recession limbs, and the peak flow discharge. Flow Discharge Versus Recurrence Interval Note that there is a nonlinear relationship between the flow discharge and the recurrence interval or return period. The following model was found to fit the data well for river flow gages in Texas (Briaud et al. 2009; Figure 52). FIGURE 51 Elements of a flood hydrograph (Source: HDS2). FIGURE 52 Relationship between flow discharge Q and recurrence interval RI (Briaud et al. 2009). (4.1) Where: – Q is the flow discharge, – Q100 is the flow discharge for the 100-year flood, – RI is the recurrence interval also known as the return period, and – and RI100 is the 100-year recurrence interval equal to 100. Equation 4.1 indicates, for example, that for the 500-year flood, the RI ratio is 5 but the flow discharge ratio is 1.52. In other words, the RI is 500% larger but the flow discharge is only 52% larger. Velocity and Water Depth Versus Flow Discharge Once the design flood is selected and the corresponding flow discharge Q is known, it is very useful to obtain the veloc- ity and the water depth that corresponds to each flow value. Let’s start with the relationship between the flow discharge and the velocity. Using Manning’s equation and the defini-

29 tion of the hydraulic radius, the following relationship for a rectangular channel cross section appears reasonable (Bri- aud et al. 2009): (4.2) Where: – v is the velocity, – v100 is the velocity for the 100-year flood, – Q is the discharge corresponding to v, and – Q100 is the discharge for the 100-year flood. For a triangular channel cross section, the same approach gives the following: (4.3) Considering that natural river channels are closer to being rectangular (exponent = 0.4) than triangular (expo- nent = 0.25), an average exponent of 0.35 is selected and the relationship is (4.4) Similarly, the relationship between the water depth and the flow discharge can be found by using Manning’s equa- tion and the definition of the hydraulic radius. For a rectan- gular channel cross section, Briaud et al. (2009) give: (4.5) Where: – y is the water depth, – y100 is the water depth for the 100-year flood, – Q is the discharge corresponding to y, and – Q100 is the discharge for the 100-year flood. For a triangular channel cross section, the same approach gives (4.6) Considering that natural river channels are closer to being rectangular (exponent = 0.6) than triangular (expo- nent = 0.375), an average exponent of 0.525 is selected and the relationship is (4.7) Impact of Recurrence Interval on Velocity and Water Depth Equations 4.1–4.7 can be combined to link the velocity and the water depth to the recurrence interval as follows: (4.8) (4.9) Where: – v is the velocity, – y is the water depth, – Q is the discharge, – RI is the recurrence interval, – v100 is the velocity for the 100-year flood, – y100 is the water depth for the 100-year flood, – Q100 is the discharge for the 100-year flood, and – RI100 is the 100-year recurrence interval. Equations 4.8 and 4.9 convey a very important message: the velocity and the water depth are not very sensitive to the recur- rence interval. For example, if the RI ratio is 5, as in the case of the ratio between the 500-year flood and the 100-year flood, the velocity ratio is 1.16. In other words, when the RI is 500% larger, the velocity is only 16% larger. For the same example, the water depth ratio is 1.25 or the velocity is 25% larger. As another example, if one were to apply a factor of safety equal to 1.5 on the 100-year flood velocity, it would be equivalent to using the 10,000-year flood as an extreme event. This is what is done in the Netherlands. If instead the 500-year flood is considered as the extreme event, the factor of safety on the velocity is 1.16.

30 Coastal Parameters The parameters to be considered in a coastal embankment subject to hydrodynamic forces depend on the nature of the flow. Two failure modes that cause surficial erosion were identified in chapter two: overtopping failure mode with three flow mechanisms (“Overtopping” in chapter two) and wave erosion (“Wave Erosion” in chapter two). The main parameters required for coastal analysis and design are namely the height of surge above the roadway crest for overtopping: Case I, the design run-up wave elevation for Case II, and the design significant wave height for Case III. To design for wave action on the seaward slope of the embankment, the design wave height and wave run-up elevation are required. This will be further explained in this chapter. The selection of a design wave, a design water level, or a recurrence interval is, as in the case of river embankments, a very critical issue. Generally, recurrence intervals between 25 years and 50 years (4% and 2% yearly probability of exceedance) are adopted in coastal projects (Douglas and Krolak 2008; FHWA 2014). The recurrence interval affects the design wave height and, therefore, the riprap stone size and the extent of armoring (Douglas and Krolak 2008). As a result, the adoption of a risk-based approach in the selection of a “design recurrence interval” with consideration given to the design life of the structure is important. This will be further discussed in chapter seven. HYDROLOGICAL METHODS AND CONSIDERATIONS Riverine Hydrology Estimating peak discharges for various recurrence intervals is a challenging task. Selecting an adequate method depends on whether the site is gaged or ungaged. Gaged sites are located at or in the vicinity of a gaging station. If continu- ous streamflow records exist for a sufficient period of time, statistical methods are applied to determine the peak dis- charge. Ungaged sites are sites that are not at or near a gaging station and do not have relevant streamflow record. For this purpose, deterministic methods are available. The accuracy of these methods is dependent on the availability of relevant data. Because data stations are generally unevenly distrib- uted, the calculated values are only estimations of the prob- able values. The methods currently used based on the survey results are described here. Hydrological Means and Methods If available for the site, gage data are preferred for the basis of estimating peak discharges. The key issue is to have suf- ficient recorded data for a continuous period of time. Oth- erwise, existing studies (including flooding case examples, FEMA studies, and regression equations) can be used as well as rainfall-runoff models. Gaged Sites Based on current DOT practice, a number of methods are used to obtain the peak flows in a river. The designer selects the most appropriate method while considering the limita- tions of each method coupled with engineering judgment. The methods used to obtain the peak flow in a river are listed from the most commonly used to the least commonly used: • Direct Available Data can be obtained through the U.S. Geological Survey (USGS) Streamflow Information Program. This database includes gage data for gaged sites and is available online (http:// water.usgs.gov/nsip/). Other sources of informa- tion are listed in the Federal Lands Highway manual (FHWA 2014). This manual includes, in addition to gage data, such state-specific studies as peak-flow frequency estimates for gaged sites and magnitudes of flood flows for selected annual exceedance prob- abilities. When statistical analysis of raw data are used, the guidelines in Bulletin 17B (USGS 1982) are followed; in particular, the log Pearson Type III distribution is adopted. • Regional regression equations have been developed for almost all states to estimate peak-streamflow frequency for ungaged sites in natural basins. This involves using peak streamflow frequency data from gaging stations in natural basins. For some areas, urban regression equations have been developed and are being used. • FEMA Flood Insurance Studies are often available where highway facilities encroach upon established or planned regulatory floodplains. A flood frequency curve approved by FEMA for the site may be avail- able. These studies may indicate if an embankment is subject to flooding and the likely recurrence inter- val of the peak flow. The information provided by these studies, however, is generally used for regu- latory compliance and not for design. Additionally, local flood studies would be available and may be used for calibration. Ungaged Sites and Regression Equations For ungaged sites, the peak flow is most commonly esti- mated by one of the following methods: USGS regression equations, state regression equations, the rational method, and FEMA Flood Insurance Studies. The rational method is commonly used for areas less than 200 acres to estimate peak flows from urban, rural, or combined areas. It is carried out in accordance with the methods presented in such rel- evant references as McCuen et al. (2002), HDS-2 and Brown et al. (2009), and HEC-22.

31 Regressions equations give the peak flow QT in a river for a given return period T. The equations are based on databases of peak flow obtained from gaged stations. Each peak flow value is associated with several parameters that likely influ- ence the peak flow. Some of these parameters are the drain- age area A of the basin contributing to the flow, the mean elevation in the river E, the percent C of bedrock underlying the basin, the percent U of urban area within the basin, and the percent S of storage within the basin. Each influencing factor may be multiplied by a weighing coefficient and a method such as the general least squares method to develop the regression equation. This equation may be of the form (4.10) Where a, b, c, d, e, and f are the regression coefficients that depend on T. Indeed, Equation 4.10 would be different for the 2-year flow (Q2), the 5-year flow (Q5), the 20-year flow (Q20), the 100-year flow (Q100), and the 500-year flow (Q500). Some of the simplest equations are of the form (4.11) These equations are often developed on a regional basis to improve the precision and decrease the scatter. They are extremely useful as they give QT, which are fundamental design parameters. However, during the design process it is very important to keep the significant scatter in mind, which is associated with the estimate of QT. Also important is to realize that the scatter increases as the return period increases. For example, the precision for Q100 is much poorer than the precision for Q10. Figure 53 shows an example of the scatter that can be expected with some of the most effective regression equations. Note that both scales are log scales indicating that the scatter is very large. This is why it is always much more reliable if at all possible to use the flow QT given by analyzing a gage station on the river. FIGURE 53 Example of comparison between predicted peak flow from regression equation and measured peak flow from a gage station data analysis (Asquith and Slade 1997). Hydrology Software for Gaged and Ungaged Sites The software used by the surveyed DOTs for hydrologic purposes include the following: USGS StreamStats, USACE HEC-HMS, USDA NRCS WinTR-55, WMS, and USACE HEC-SSP. A brief description of the capabilities of each soft- ware package is included in Table 5. TABLE 5 HYDROLOGY SOFTWARE PACKAGES ADOPTED IN CURRENT PRACTICE Software Description Stream- Stats Web-based Geographic Information System (GIS) application used to create streamflow statistics for gaged and ungaged sites. HEC-HMS USACE Hydrologic Modeling System simulates the com- plete hydrologic processes of dendritic watershed systems. It includes hydrologic analysis procedures such as event infil- tration, unit hydrographs, and hydrologic routing in addition to a range of advanced capabilities and tools. WinTR-55 Tool used to perform accurate hydrological analysis of small watershed systems. WMS Watershed Modeling System is a complete program for developing watershed computer simulations. WMS sup- ports lumped parameter, regression, and 2-D hydrologic watershed modeling, and can be used to model both water quantity and water quality. HydroCAD Computer-aided design tool used to model storm water run- off. It provides commonly used drainage calculations that include the rational method, SCS, NRCS, SBUH, etc. HEC-SSP USACE statistical software package used to perform sta- tistical analyses of hydrologic data. The current version of HEC-SSP performs flood flow frequency analysis is based on Bulletin 17B, Guidelines for Determining Flood Flow Frequency (1982). Another technique to obtain information on the flow is to use the complete system of USGS gages in a state, associate each gage with a longitude and latitude coordinate, and develop an interpolation technique for a location within the area defined by the closest three gages. This is what is done with the free- ware TAMU-FLOW for some states (Texas and Massachusetts at present). The location of the 3,116 USGS gages in Texas and neighboring states is shown in Figure 54. Figure 55 shows the map of recurrence interval for any location in Texas devel- oped based on the analysis of 744 gage records and automated in TAMU-FLOOD. The precision of this technique can be assessed by comparing predicted and measured velocities at the location of an existing gage (Figure 56). This figure shows the frequency distribution for the difference between the predicted and observed velocity ratio Vmo/V100 where Vmo is the maxi- mum observed velocity and V100 the velocity corresponding to the 100-year flood. As shown in the figure, the mean difference is 3.2% and the standard deviation is 20.8%, which is very sat- isfactory (Figure 57). Other Sources of Information The following are other sources of information useful for hydrologic calculations:

32 • Soil type information: NRCS Web Soil Survey Web Application (http://websoilsurvey.nrcs.usda.gov/) • Precipitation information: National Weather Service (http://hdsc.nws.noaa.gov/hdsc/pfds/) and local climate centers. FIGURE 54 Mapping of recurrence interval by using the gages in a state (Briaud et al. 2009), location of the flow gages used for mapping Texas and neighboring states. FIGURE 55 Mapping of recurrence interval by using the gages in a state (Briaud et al. 2009), maximum recurrence interval RImo map for Texas for 1920 to 2006. Additional Considerations A number of important considerations are highlighted herein when carrying out hydrologic calculations: • Data considerations: gage data might not be sufficient, or variations might require combined statistical analy- sis as recommended in Bulletin 17B (USGS 1982). • Evaluation of flood records: Bulletin 17B explains the importance of evaluating the adequacy and applicabil- ity of the flood records that constitute the basis for the flood frequency analysis. In the analysis process, essen- tial considerations are described and include the cli- matic trends, randomness of events, watershed changes, mixed populations, and reliability of flow estimates. • Use of storm (flood) hydrograph versus peak flow: the peak flow is the commonly used hydrological design parameter. However, for applications in which the duration of an event or the volume of the discharge are useful (for instance, the duration of an overtopping event), storm hydrographs would be used. FIGURE 56 Interpolation scheme and precision of the predictions using TAMU flood (Briaud et al. 2009), interpolation scheme among four neighboring flow gages and verification. FIGURE 57 Interpolation scheme and precision of the predictions using TAMU flood (Briaud et al. 2009), predicted versus measured Vmo/V100. Coastal Environments Limited sources for coastal hydrology were provided through the survey responses. The main source of coastal informa- tion for tides and currents is the National Oceanic and Atmo- spheric Administration (http://tidesandcurrents.noaa.gov/).

33 Additional information sources include the following: • Jones et al. (2005), Wave Run-Up and Overtopping • USACE (2008), Coastal Engineering Manual, EM 1110-2-1100 • USACE (2012), Hurricane and Storm Damage Risk Reduction System Guidelines • Office of Federal Lands Highway (2014), Project Development and Design Manual (PDDM; FHWA 2014). HYDRAULIC METHODS AND CONSIDERATIONS As explained in chapter two (“Failure Modes” section), overtopping mechanisms and wave action cause surficial erosion. Overtopping leads to the initiation of back erosion mechanisms on the downstream (landward) slope. Wave action leads to damage on the slope that it impacts. Once the hydrologic data are collected, mainly for the design flood with a chosen return period, the hydraulic work can start. This work consists of, among other things, cal- culating the water elevation and water velocity that corre- sponds to the design flood. Modeling Surface Water Level (For Coastal and Riverine Environments) Surface water level modeling is an important aspect of design. One-dimensional and two-dimensional modeling packages are available for this purpose. One-dimensional models are used to compute the average depth and velocity for open channel flow. Two-dimensional models are used to compute the water surface profile, the depth, and the veloc- ity across the channel. Further information can be found in many references such as HDS-6 and PDDM. Based on DOT current practice, the most commonly used hydraulic engineering software packages are HEC-RAS and HY-8 (Table 6). FLOW DISCHARGE AND VELOCITY EQUATIONS Design equations are included herein for the purpose of discharge and velocity calculations. Discharge calculations yield the overtopping height (and vice versa) and contrib- ute in damage calculations (chapter six). The design veloc- ities computed here can be compared with the permissible velocity of the embankment material or embankment pro- tection system. This will be discussed further in chapters six, nine, and 10. The design equations included here were obtained from a review of the available literature. They give insights to the importance of the input parameters in embankment design. As mentioned before, only two modes of failure will be discussed here (overtopping and wave erosion). Design equations for the failure modes will be discussed in chapter six. TABLE 6 HYDRAULIC SOFTWARE PACKAGES FOR COASTAL AND RIVERINE EMBANKMENTS Software Description HEC- RAS Hydrologic Engineering Centers River Analysis System is used to perform one-dimensional and, recently, two-dimensional steady flow; unsteady flow; sediment transport/mobile bed computations; and water temperature modeling. HY-8 HY-8 automates culvert hydraulic computations using a number of essential features that make culvert analysis and design easier. CASE I: FLOOD OR STORM SURGE OVERTOPPING The flood or surge discharge for an embankment overtopped by water can be calculated using equations for broad crested weirs. Equations are presented herein for riverine and coastal surge. Riverine Overtopping Depending on the type of flow (free flow or submerged), the following equations can be used: – For the free-flow case (low tailwater), the water falls while following the slope contour, creating the following relationship: (4.12) Where: (4.13) Where: – q is the discharge per unit width ft2/s (m2/s), – H is the total head (static + velocity) above the roadway crest in ft (m), – h is the headwater height above the roadway crest in ft (m), – v water velocity in ft/s (m/s), – g is the gravitational acceleration ft/s2 (m/s2), and – C is an experimentally determined discharge coeffi- cient for free flow. Note that the velocity head can be ignored (applicable for the riverine case) if the approach velocity v is assumed to be very low.

34 • For the submerged flow case (high tailwater), the dif- ference in elevation between the upstream water level and the tailwater level is not large, the embankment is submerged, and the modified equation is (4.14) Where: – L is the length of the inundated roadway, and – Cs is a submergence coefficient. Values of C and Cs Yarnel and Nagler (1930) were the first to present charts for the determination of discharge coefficients for overtopping flow over railway and roadway embankments. These charts were later modified by a USGS memo (“Computation of Discharge over Highway Embankments,” March 16, 1955). A compilation and analysis of vital information on broad-crested weirs was then prepared by Tracy (1957). Kindsvater (1964) presented a detailed study that discusses the theoretical and experimental basis for computing the peak discharge from post-flood field observations. The references for where to find the C and Cs val- ues associated with each study is shown in Table 7. TABLE 7 DISCHARGE COEFFICIENTS FROM VARIOUS STUDIES Reference Computation of Discharge Coefficient C and Submergence Factor Cs Yarnel and Nagler (1930) Charts for C and Cs USGS Memo (1955) Charts for C and Cs Kindsvater (1964) Values for C and Cs Bradley (1973) Charts (Figure 58) for C and Cs Powledge et al. (1989) C ranges between 1.60 and 2.15 in Metric Units (2.9 and 3.9 in English Units) Typical value of 1.9 in Metric Units (3.0 in English Units) that is used for level-crested structures Cs from Bradley (1973) Richardson (2001) Charts (Figure 59) The significant and insignificant factors in the discharge characteristics, based on Kindsvater (1964), are shown in Table 8. A similar equation for the flow discharge was presented by Petersen (1986) as follows: (4.15) Where: – Qo is the overtopping discharge in m 3/s (ft3/s), – Cr is the overtopping discharge coefficient, – HWr is the flow depth above the roadway in m (ft), – Kt is the submergence factor, – Ls is the length of the roadway crest along the roadway in m (ft), and – Ku = 1.0 (English units) and Ku = 0.552 (SI units) (Rich- ardson 2001). FIGURE 58 Discharge coefficients for overtopping flow over roadway embankment (after Bradley 1973). TABLE 8 SIGNIFICANT AND INSIGNIFICANT FACTORS INFLUENCING THE OVERTOPPING DISCHARGE Factors Significant Insignificant Embankment Slope _ Insignificant except for its effect on the roller on the downstream side Crest Width To some extent, as it gov- erns the head loss and the boundary layer thickness at the control section _ Crest Roughness To some extent, as it gov- erns the head loss and the boundary layer thickness at the control section _ Embankment Height _ Insignificant Pavement Cross Slope and Shoulder Slope _ Insignificant

35 Coastal Surge Overtopping Coefficient Based on Hughes (2008), it appears appropriate to plug a discharge coefficient value Cf of 0.5443 into the following equation: (4.16) The reasoning presented is that Cf ≤ 0.5443 was pre- sented in Chen and Anderson (1987) nomograms, with this coefficient being a function of the upstream head over the crest to the crest width (H/w). The decrease of Cf is insig- nificant until the head becomes less than 0.5 ft. As a result, “it seems appropriate to be slightly conservative” (Hughes 2008) (Figures 59 and 60; Table 8). FIGURE 59 Discharge coefficients for overtopping flow over roadway embankments (after Petersen 1986). FIGURE 60 Variation of downstream slope velocity as a function of surge height for different Manning coefficients (after Hughes 2008). Velocity Calculations Manning’s equation for flow resistance is presented as follows: (4.17) Where: – v is the velocity in m/s, – R is the hydraulic radius in m, – Sf = sin θ where θ is the slope angle of the river bottom, and – n is Manning’s coefficient (s.m-0.33) ranging from 0.01 for a smooth clay surface to 0.03 for a gravelly surface to 0.05 for a boulder surface. Because a wide channel with steady, uniform flow is assumed, R becomes the flow depth. By substituting for the depth of flow = q/v and Sf = sinθ (θ being the angle of the river bottom slope), the following equation is obtained: (4.19) Where: – q is the flow per unit length of embankment, and – n is Manning’s coefficient. Figures 60 and 61 show the variation of downstream slope velocity and flow thickness, respectively, as a function of increased surge height over the coastal embankment crest. These figures were prepared based on Hughes’s (2008) physical model. FIGURE 61 Variation of the downstream flow thickness as a function of surge height (after Hughes 2008).

36 CASE II: WAVE OVERTOPPING Advances on the determination of overtopping flow param- eters have been made based on small-scale and large-scale experiments on levees carried out in Europe (Schüttrumpf et al. 2002; van Gent 2002; Schüttrumpf and Oumeraci 2005). The differences between van Gent’s work (the Netherlands) and Schüttrumpf’s work (Germany) in 2002 were later rec- onciled to the possible extent through a joint paper. A sum- mary of the work carried out is presented by Hughes (2008). The main design equations are included herein. FIGURE 62 Wave overtopping (after Schüttrumpf and Oumeraci 2005). The velocity and flow depth associated with coastal wave overtopping (Figure 62) can be determined at the following three locations over the embankment (levees): – Point A: The crest edge on the upstream side (uA, hA) – Point B: The crest edge on the downstream side (uB, hB) – Point SB: The slope bottom of the landward slope (usb, hsb) The three key parameters used as input for the flow velocity and flow depth calculations are shown in Table 9. TABLE 9 INPUT PARAMETERS FOR VELOCITY AND DEPTH CALCULATIONS FOR WAVE OVERTOPPING Rc Levee Freeboard Ru2% The run-up elevation exceeded by only 2% of the waves, estimated using the run-up formulas of de Waal and van der Meer (1992) or Hughes (2004) fF Friction factor that accounts for frictional energy loss as the overtopping wave travels across the crest and down the protected-side slope Determination of Input Design Parameter Ru2% Significant Wave Height Hmo This is the primary measure of energy in a sea state and is equal to the average height of the one-third-highest waves. Wave Run-Up Ru,2% The height of the 2% wave run-up can be estimated based on the following equation (Douglas and Krolak 2008): (4.20) Where: – Ru,2% is the run-up level exceeded by 2% of the run-ups in an irregular sea, – Hs is the significant wave height near the toe of the slope, and – r is a roughness coefficient (r = 0.55 for stone revet- ments) and ϑop is the surf similarity parameter defined as: (4.21) Where: – θ is the slope angle (Figure 63), – Hs is the significant wave height, – Tp is the wave peak period, and – g is the gravitational acceleration. FIGURE 63 Wave run-up sketch (after Douglas and Krolak 2008). Equations 4.22 through 4.31 can be used to estimate the wave overtopping peak velocity and associated flow depth over an embankment that is exceeded by only 2% of the incoming waves. Crest Edge on the Upstream Side (Point A in Figure 62) (4.22) Where CAh2%=C2 tanθ (4.23)

37 Where: – hA2% is the peak flow depth exceeded by 2% of the waves, – Hs is the significant wave height [= Hmo], – CAh2% is an empirical depth coefficient (Table 10), – C2 is a constant and θ is the flood-side slope angle, – uA2% is the depth-averaged peak velocity exceeded by 2% of the waves. – g is the gravitational acceleration, and – CAu2% is an empirical velocity coefficient (Table 10). TABLE 10 SUMMARY OF EMPIRICAL COEFFICIENTS FOR FLOW PARAMETERS Empirical Coefficient Schüttrumpf van Gent CAh2% (depth) 0.331,2 and 0.224 0.152,3 CAu2% (velocity) 1.551 and 1.372 1.302,3 CAh50% (depth) 0.171,4 _ CAu50% (velocity) 0.941,4 _ Modified after Hughes (2008) 1 Schüttrumpf et al. (2002) 2 Schüttrumpf and van Gent (2003) 3 van Gent (2002) 4 Schüttrumpf and Oumeraci (2005) According to Hughes (2008), the recommended coeffi- cient values from Table 10 are CAh2% = 0.22 [which is the most recent value provided by Schüttrumpf and Oumeraci (2005)] and CAu2% = 1.55 (1.37 was believed to be a typo). In conclusion, the significant parameters in determina- tion of case a. flow parameters are the flood-side slope and the wave parameters used in the estimation of Ru,2% . Crest Edge on the Upstream Side (Point B in Figure 62) As the overtopping waves flow across the embankment crest, their height and velocity decrease as a result of the surface friction. The following equations can be used to estimate the height and velocity at any location along the crest width: (4.24) Where: – L is the crest width, – xc is the distance along the crest from the flood-side edge (Figure 61), and – C3 is an empirical coefficient (Table 11). TABLE 11 C3 VALUES BASED ON LITERATURE Reference (based on 2% exceedance levels) C3 Value Schüttrumpf et al. (2002) 0.89 for TMA1 spectra 1.11 for natural spectra Schüttrumpf and van Gent (2003) 0.402 and 0.89 Schüttrumpf and Oumeraci (2005) 0.753 for irregular and regular waves 1 TMA spectrum: a spectrum created by combining the first three letters of the three wave data sets (Texel, MARSEN, and ARSLOE). 2 Used if van Gent’s (2001) method for estimation of wave run-up was used. 3 Used if de Waal and van der Meer (1992) or Hughes (2004) was used. The velocity associated with hB2% can be calculated as follows: Where CAh2%=C2 tanθ (4.25) (4.26) Where: – fF is Fanning factor for the embankment surface, – hB2% is the flow depth at that location of the crest previ- ously obtained through Equation 4.24, – n is Manning’s coefficient, – h is flow depth in meters, and – uA2% is calculated at point A. Some fF values based on experimental results from the avail- able literature are presented in Table 12. TABLE 12 fF VALUES BASED ON EXPERIMENTAL RESULTS FROM AVAILABLE LITERATURE Reference Surface Material Fanning Friction Factor fF Schüttrumpf et al. (2002) Wood fiberboard 0.0058 Bare compacted clay surface 0.01 Schüttrumpf and Oumeraci (2005) Range of values on the protected side slope 0.02 (smooth slopes) Cornett and Mansard (1994) Rough revetments and rubble mound slopes 0.1–0.6 Hughes (2008) Grass-covered slopes ≈ 0.01

38 No published values of Fanning’s coefficient exist for armored alternatives. A fist attempt to approximate the Fanning factor was presented by Henderson (1966). In this approximation, a knowledge of Manning’s coefficient is required for a particular armoring product or surface slope. This equation should be used with caution because it has not yet been proven. Slope Bottom of the Landward Slope (Point sb in Figure 62) According to Hughes (2008), two theoretical expressions exist to calculate the velocity at the bottom of the downstream slope. Both formulas give relatively close answers. Schüttrumpf and Oumeraci (2005) presented an iterative solution. An explicit one was presented by van Gent (2002) and is shown here: (4.27) With: (4.28) (4.29) (4.30) Where: – α is the angle of the downstream slope, – sb is the distance down the slope from the crest edge, – hB2% is the flow depth at the crest edge, and – uB2% is the flow velocity at the crest edge. The flow thickness down the slope hsb2% can be estimated using the following equation: (4.31) Case III: Wave and Surge Overtopping Hughes (2008) presented this equation to calculate the dis- charge associated with this combined case of wave and surge overtopping: (4.32) Where: – Qws is the average combined wave and surge discharge, – Hmo is the energy-based significant wave height, and – Rc is the freeboard, which is negative for this formula. Figure 64 shows a plot of the data of the combined aver- age discharge versus relative freeboard. This equation was derived based on total of 27 experiments covering a range of three storm surge elevations exceeding the levee crest and nine irregular wave conditions, and it is to be used strictly for waves shoaling on a 1:4.25 levee flood-side slope. If the slope was milder or steeper, different results would be expected because the seaward slope affects the waves that overtop the levee. FIGURE 64 Dimensionless combined average discharge versus relative freeboard (after Hughes 2008) SUMMARY This chapter presented an overview of the hydrologic and geologic factors that play a role in the design of roadway embankments in riverine and coastal environments during flooding. The impact of a selected recurrence interval on the velocity and water depth was also assessed. The next chapter discusses the geological and geotechnical factors involved in roadway embankment design.