Study site and site model
The selected study area is the Lewis Creek channel and adjacent floodplain in the vicinity of the historic Quinlan Covered Bridge in Charlotte, northwestern Vermont (Fig. 2). The study reach is 1025 m long. The upstream drainage area of the river at this location is approximately 180 km2. The Lewis Creek watershed spans the Northern Green Mountain and Champlain Valley biogeophysical regions in northwestern Vermont. The region is characterized by a humid, temperate climate, with mean annual precipitation reported as 1074 mm (Olson 2014) and mean annual temperature recorded as 7.8 °C (NOAA 2016). Mean annual runoff (488 mm) comprises 45 % of the precipitation (USGS 2010), and surface waters drain to Lake Champlain. Land use in the Lewis Creek watershed is estimated as 62 % forested, 26 % agricultural and 8 % developed (Troy et al. 2014).
Mean annual flows in the Lewis Creek are estimated as 3.1 m3/s, based on historic records for a nearby US Geological Survey gaging station located nearly 6.5 180 km downstream with a drainage area of 199 180 km2. The peak flow recorded since 1990 is 133 m3/s on 27 April 2011 (USGS 2010). Regionally, the study reach of the Lewis Creek is located at the transition from a semi-confined valley to a much broader alluvial valley. In the upper half of the study reach the channel is constrained along the north bank by a moderate to steep, forested, valley wall and is vertically disconnected from the floodplain along the south bank which has been cleared and modified to accommodate a gravel road and light density residential development. In the lower half of the reach, the valley setting is more open and the degree of channel incision below the floodplain is less pronounced. Riparian areas are partly forested and partially in hay and lawn.
The Quinlan Bridge span (Fig. 3) is less than the natural bankfull width of the Lewis Creek channel, and the bridge is oriented at a sharp angle to the Lewis Creek. Flows are constricted through the bridge span leading to upstream aggradation and scour of the bridge abutments. Roads in vicinity of the bridge are elevated above the floodplain and both laterally and vertically constrain the channel and floodplain on approach to the bridge. Ice jams regularly cause localized flooding upstream and downstream of the bridge, threaten the integrity of the abutments of this historic bridge, and subject a nearby residential property to inundation and fluvial erosion hazards (SMRC 2010).
In 2010, an analysis of the bridge was contracted to provide recommendations on several alternatives to existing conditions for the purpose of reducing the risk of further damage. Mitigation scenarios considered included lowering adjacent roads, lowering the floodplain, removing berms and realigning the bridge (SMRC 2010). To perform the analysis, a HEC-RAS (Hydrologic Engineering Center-River Analysis System) model was built, calibrated and validated. HEC-RAS is a widely used river and stream modeling software (Goodell 2014) designed and distributed by the US Army Corps of Engineers (USACE.) It supports modeling of many hydraulic structures, including bridges, and simulations of alternatives provide the predicted physical variables needed (such as velocity and stage) to evaluate scour and erosive potential for proposed scenarios. HEC-RAS was used to evaluate and compare multiple scenarios related to encroachment as a proof of concept.
The reach modeled by Milone and Macbroom, Inc. is 1025 m long and drops approximately 5.8 m in elevation through the reach. The model extends from just upstream of the Scott Pond Dam (which operates in a run-of-river mode) to just downstream of the bridge, and is comprised of 13 cross sections. Eight of the 13 cross sections are labeled in Fig. 4. The model geometry shows the cross sections in plan view (Fig. 4a) as well as a cross section of the river along its length (Fig. 4b). Only these Eight cross sections include floodplain access modifications for the proof-of-concept presented in this work, as increasing floodplain access was not physically realistic at all locations. The cross sections are labeled with with XS\(_{1}\) representing the most upstream cross section and XS8 the most downstream. The bridge is between XS\(_{6}\) and XS\(_{7}\).
Flow magnitudes for various return periods were calculated by Milone and MacBroom using USGS streamflow gaging data from Lewis Creek, Station #04282780 (USGS 2010), and regression equations (Olson 2002). Normal depth was used as a downstream boundary condition based on the original survey. The analysis of alternatives was primarily done using steady-state simulations, but a sediment transport analysis was performed to investigate the potential impact of erosion and sedimentation for the proposed alternatives. The latter requires a quasi-unsteady analysis in HEC-RAS in which a transient event is modeled using a series of steady flows.
For steady flow simulations in HEC-RAS, stage and flow are calculated using energy losses between user-defined cross sections. For transient simulations, it solves the full 1-D St. Venant equations; HEC-RAS version 4.1, used for the Quinlan model, provides support only for 1-D modeling. The recently released version 5.0 provides support for 2-D flow modeling, however our data do not support 2-D simulations. In this work, transient simulations were used with an upstream hydrograph as a boundary condition. The hydrograph was constructed by scaling the quasi-unsteady hydrograph built by Milone and MacBroom for the sediment transport model so that peak flow corresponded with the design (50-year) flow. HEC-RAS routes this flow through the reach and provides hydraulic variables at the bridge for a given scenario.
Scour prediction
Models such as HEC-RAS provide the means to predict physical variables, such as flow, stage or velocity. These variables, in turn, can be used in empirical scour equations as described by the Federal Highway Administration (FWHA) in HEC-18 (Arneson et al. 2012).
Scour predictions were calculated in post-processing using the results of HEC-RAS simulations. The following contraction scour equation is selected for this work, and is one of many outlined by the FWHA in HEC-18 (Arneson et al. 2012):
$$\begin{aligned} Y_{s}=4Y_{0}\left( \frac{V_{0}}{\sqrt{gY_{0}}}\right) ^{\frac{1}{3}}(0.55)K_{1}K_{2}, \end{aligned}$$
(1)
where \(Y_{s}\) is the scour depth [m], \(Y_{0}\) is the water elevation at the bridge [m], \(V_{0}\) is the flow velocity [m/s], g gravitational acceleration [m/s\(^{2}\)], and \(K_{1}\) and \(K_{2}\) are the skew and abutment coefficients, respectively.
Generally, the scour equations are overly conservative (Sheppard et al. 2014). However, for the purposes of evaluating bridge scour relative to a number of proposed scenarios, referred to here as relative scour risk, it is reasonable to interpret higher contraction scour values as corresponding to increased scour risk. While our results used the contraction scour equation, the methodology and the subsequent interpretation of the results would not change if a different scour equation was selected. As these equations are empirical, their validity is constrained to the range of data used to derive them.
When combined with the HEC-RAS model developed by Milone and MacBroom, Inc., Eq. (1) provides the needed hydraulic parameters, and enables scenarios to be evaluated and compared on the basis of bridge scour risk.
Differential evolution (DE) optimization and HEC-RAS modifications
This design challenge can be formulated as a multi-objective optimization problem. To demonstrate the application of a method for evaluating the location-dependent sensitivity of bridge scour to floodplain access and constriction, the Quinlan Bridge HEC-RAS model geometry was modified. The modified geometry represents this stretch of the Lewis Creek as having the maximum amount of floodplain access possible. The design flood was initially (and artificially) constricted entirely to the channel, thus providing no floodplain access up or downstream of the bridge. This is a noteworthy departure from current standard engineering methods and research, as the modified model does not reflect any proposed or hypothetical scenario. Optimization with DE was then used to find the most efficient removal of encroachments to minimize bridge scour at the Quinlan Bridge. To efficiently mitigate scour risk, different magnitudes of encroachment removal may be needed depending on the location; scour sensitivity to floodplain access can be inferred from these optimal encroachment removal values and locations ranked by their impact on scour. Locations that require more extensive encroachment removal to reduce scour are more salient.
Once the modifications to the HEC-RAS model were implemented, a DE optimization algorithm was wrapped around the model to impose floodplain constriction, enable HEC-RAS simulations, and post-process the calculated contraction scour results without using the graphical interface. Floodplain area was modified and HEC-RAS simulations then used to predict scour. Python code was written to provide this functionality using the HEC-RAS API [Application Program Interface (Goodell 2014)] and the ability to read and write to the HEC-RAS text files. In this work, the DE implementation in the Python library, SciPy, based on the description given by Storn and Price (1997), was wrapped around the combined HEC-RAS/cost function framework.
DE has several parameters that are user-defined. For this application, the crossover fraction was set to 0.7 and the mutation factor sampled from a uniform distribution in (0.5, 1) every generation. The population size was ten. Because DE is a stochastic method, optimization was repeated using random restarts to verify consistent convergence. For each of three values of the weighting parameter \(\beta\) from Eq. (4), batch runs of ten random restarts were performed.
Removal of encroachments on both the left and right side of the channel (facing downstream) was defined along eight cross sections for a total of 16 variables. These variables are defined over a range from 0 to 1, with 0 indicating no floodplain access (full constriction) and 1 indicating full floodplain access (no constriction). This is shown graphically in Fig. 5, with \(\vec{x}\) being a vector whose components represent flood access corresponding to the left or right side of a particular cross section.
Cost function
Construction of the cost function is key, particularly when multiple objectives are involved or when constraints are being enforced using penalty terms, to ensure that solutions meet the constraints and specifications of the real-world problem. A cost function was constructed to combine and weight the two competing objectives (floodplain access and bridge scour) into a scalar value as follows:
$$\begin{aligned} f(floodplain\ access,scour) &=floodplain\ access_{sum}^{2} \\ & \quad +(scour-scour_{min})^{2}. \end{aligned}$$
(2)
with \(floodplain\ access\) a vector of unitless floodplain area values, scour the scour in meters, and \(scour_{min}\) the user-defined threshold scour value. An optimal solution is one with low floodplain access (i.e. few built encroachments) and reduced bridge scour. These objectives are inversely correlated, so the trade-offs between them are defined by a set of pareto optimal (non-dominated) solutions. The cost function weights and combines these objectives into a scalar function to be minimized. Written with more succinct notation, Eq. (2) becomes:
$$\begin{aligned} f(Y_{s}, \vec{x})=\left( \sum _{i}\vec{x}_{i}\right) ^{2}+(Y_{s}-Y_{s\_min})^{2} \end{aligned}$$
(3)
The cost function is equal to the sum of the squares of the floodplain access parameters (\(\vec{x}_{i}\), where i indexes location) and the amount of bridge scour (\(Y_{s}\)) over baseline scour (\(Y_{s\_min}\)) as determined by a simulation with fully open floodplains. It is a function of the entire set of floodplain access parameters encoded in \(\vec{x}\) and the scour, which is an implicit function of \(\vec{x}\), since the level of scour depends on the hydraulic behavior given a specified floodplain access scenario.
If the goal were to perform design optimization and identify a single floodplain design that maximizes encroachment along the eight selected channel locations while minimizing scour at the bridge, rather than evaluate sensitivity of individual locations along the channel, weighting parameters could be added to each term in Eq. (3) to define the trade-offs between the two stakeholder objectives. For the purposes of performing a sensitivity analysis, weights that determine the relative importance of objectives are not necessary because the optimal values of floodplain access will be evaluated relative to one another. In other words, they will be used to rank locations according to sensitivity and their absolute values will not be considered. To test this assumption, Eq. (3) was modified with a weighting factor, \(\beta\), as follows:
$$\begin{aligned} f(Y_{s},\vec{x})=\left( \sum _{i}\vec{x}_{i}\right) ^{2}+\beta (Y_{s}-Y_{s\_min})^{2}. \end{aligned}$$
(4)
Larger values of \(\beta\) implicitly place greater weight on scour reduction, while values closer to zero weight minimization of floodplain access more heavily. Optimization was performed using values of \(\beta\) that relatively weight the two objectives over two orders of magnitude.