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Passive and active control of diversions to an off-line reservoir for flood stage reduction
Advances in Water Resources 29 (2006) 861–871 www.elsevier.com/locate/advwatres
Passive and active control of diversions to an off-line reservoir for flood stage reduction Brett F. Sanders b
a,*
, John C. Pau b, David A. Jaffe
c
a Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, United States Donald Bren School of Information and Computer Sciences, University of California, Irvine, CA 92697, United States c Pacific Advanced Civil Engineering, Inc., Fountain Valley, CA 92708, United States
Received 24 October 2004; received in revised form 30 June 2005; accepted 14 July 2005 Available online 10 October 2005
Abstract Diversion of excess streamflow to an off-line reservoir is examined as a wave interference problem that can be controlled to reduce the cresting stage of a flood. Flood diversions create depression waves in the stream channel which, superimposed upon the flood, decrease flood stage. In the context of an O(102) km2 coastal watershed in northern California, numerical modeling was performed to compare the performance of three idealized diversion control strategies including passive control, weir control, and gate control. It was found that gate control, which creates a dam-break like flow into an off-line reservoir, can be optimized to accomplish 2–3 times the flood depth reduction of passive control. Capturing 2% of the runoff, for example, the cresting depth is reduced less than 1% with passive control but 2–4% by gate control. This is particularly important in areas with little off-line or over-bank storage, such as urban watersheds. Timing of the gate action is critical. Optimal control requires gate action slightly before peak stage arrives at the diversion structure, by a duration that scales with the capacity of the reservoir. The lead time of flood forecasts appears compatible with the lead time necessary to optimize diversions, and decision support systems should compensate for forecast uncertainty by early gate action. If by poor design or operator error the gate is opened too late, gate control becomes less effective than passive control. 2005 Elsevier Ltd. All rights reserved. Keywords: Flood control; Off-line reservoir; Flood modeling; Diversions
1. Introduction Off-line reservoirs are a structural approach to flood control first considered by Nolte and Schwab [11,12]. The basic idea is to divert or pump excess storm water from streams into detention basins during storms, then return water to the channel later when capacity is available. Nolte and Schwab envisioned this approach for regions of nearly level topography where in-stream reservoirs are not practical. But now that the ecological *
Corresponding author. Tel.: +1 949 824 4327; fax: +1 949 824 3672. E-mail address: [email protected] (B.F. Sanders). 0309-1708/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2005.07.015
consequences of in-stream reservoirs are better appreciated, such as blocking fish migration and trapping sediment, there is renewed motivation to understand how a range of detention basin options including off-line reservoirs can be engineered for both flood control and water quality control purposes. Engineered wetlands are one type of detention basin that are increasingly being constructed to improve water quality adversely impacted by pollutants such as biochemical oxygen demand, metals, nutrients, organics, and sediments [19], though in-line configurations are more common. Larger, storm-water oriented facilities may also be engineered to support land use for agricultural, wildlife habitat, and/or recreational purposes, similar to the Red River
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in Grand Forks, North Dakota where low-lying homes were removed and replaced with recreational areas and campgrounds [10]. Other examples are floodways such as the Red River Floodway in Manitoba [9] and the Yolo Bypass in California [18] but these differ from off-line reservoirs because floodways transmit water downstream. The problem of controlling diversions to mitigate flood stage involves the interaction of three waves: the flood wave and two depression waves generated by the diversion. Under subcritical channel flow conditions, one depression wave moves upstream (regressive wave) and one moves downstream (progressive wave), while both are progressive under supercritical channel flow conditions [16]. A practical consequence of this wave action is a potential to control flood stage both downstream and upstream of an off-line reservoir under subcritical channel flow conditions, but only downstream of a reservoir under supercritical flow conditions. The utility of regressive depression waves is supported anecdotally by reports that during the 1993 floods in the midwest region of the United States, St. Louis was spared from inundation following failure of a river levee downstream along the Mississippi River [1]. This exemplifies a dynamic wave phenomena, meaning that momentum is not simply balanced between gravity and friction but also with inertia. Jaffe and Sanders [7] developed a 2D hydrodynamic model to simulate the effect of levee breaches on flood hydrographs in a rectangular mildly sloped channel, and applied the model to examine the relationship between flood stage and properties of the breach such as the breach length and the capacity of the off-line reservoir. Both regressive- and progressive-wave control of the flood was realized in the model, and the controllability of floods was linked to the duration and magnitude of the flood, the capacity of the off-line reservoir, and the timing of breaches relative to the flood hydrograph. This paper presents a comparative analysis of three idealized, gravity driven, off-line reservoir control scenarios using the reservoir configuration shown in Fig. 1. These scenarios include passive control where the off-line reservoir fills and drains with the rising and falling limbs of the hydrograph, respectively, through an unregulated diversion channel; weir control where diversions are controlled by a weir structure in the diversion channel and commence once flood stage exceeds the height of the weir crest; and gate control where diversions are controlled by a gate structure and commence upon opening the gate. Gate control creates dynamic wave action similar to a levee breach, while flow varies slowly with passive and weir control. These three options span a range of operational complexity. No operational decisions are required for passive control, the weir height must be set for weir control, and the timing of the gate opening must be set for gate control. The lat-
Q
Channel
wd Control Structure
Qd
Off-line Reservoir
Fig. 1. Off-line reservoir configuration relative to channel. Weir and gate control structures are examined in this study as well as passive control. For passive control, there is no control structure in the diversion channel.
ter options would require forecasting models to predict the timing and magnitude of a flood. While in the flash flood limit flood forecasting has been recognized as a tremendous challenge [8], in many cases real-time monitoring and forecasting has proven to save lives and reduce damages through early warning and emergency management systems and by real-time regulation of flood control infrastructure [4]. A key element of these systems in the US is the automated local evaluation in real-time (ALERT) program, developed in the 1970Õs by the National Weather Service, which uses sensors in the field to transmit environmental data to a central computer in real time. The purpose of this paper is to examine the relative performance of passive, weir, and gate diversion strategies to identify potential advantages and liabilities of operational complexity as described above. This is done in the context of an O(102) km2 coastal watershed in California using realistic stream characteristics but, for obvious reasons, a hypothetical off-line reservoir. This study builds upon the work by Jaffe and Sanders [7] by examining the effectiveness of diversions in the context of realistic stream flows, flood durations, and natural channel geometry.
2. Site description A 12.2 km portion of San Francisquito Creek in the San Francisco Bay area of California is the focus of this study, downstream of USGS gage 11 164 500 as shown in Fig. 2. The region experiences a mild wet winter and a warm dry summer as is characteristic of Mediterranean climates. Based on National Weather Service data collected in Palo Alto between 1953 and 2004, the watershed averages 39 cm of rainfall, over 90% of which falls in the months of November through April. High and low temperatures average 26/13 C in August and 14/6 C in January. The creek drains a 95 km2 catchment which extends westward from San Francisco Bay to the ridge of the Santa Cruz Mountains. The upper
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terized by a cross-sectional shape that closely resembles a trapezoid. Due to this shape and for consistency with the hydraulic model used in this study, a four parameter geometric description was adopted involving the bottom width wb, bottom elevation zb, bank width wt, and bank elevation zt, as shown in Fig. 3. These parameters were estimated by a graphical fitting procedure. The channel bed has a concave-up longitudinal profile with a bed slope of roughly 0.5% over the first 2 km, a slope of roughly 0.25% between kilometers 2 and 10, and a nearly horizontal bed slope between kilometers 10 and 12. The bottom width and bank width average roughly 10 and 30 m, respectively, over the first 10 km though these parameters vary locally by as much as 10 m. Bridges cross the creek at several points and the abutments, which restrict flow, are modeled as a rectangular cross-section. Flooding commonly occurs immediately upstream of the abutments.
Fig. 2. Map of San Francisquito Creek and surrounding watershed. Reach used in study is between gage and San Francisco Bay. Relative position of diversion reservoir and monitoring stations are noted.
portion of the catchment is forested, while the lower portion near the bay is developed and prone to flooding. It has been reported that neither the channel nor bridge culverts of San Francisquito Creek are sized to accommodate the 100-year flood [17]. For example, the capacity of the channel has been estimated to be as low as 170 m3/s while the 100-year discharge has been estimated to be 280 m3/s. Channel geometry data, consisting of transects at 283 stations, were obtained from the Santa Clara Valley Water District. The channel bed is earthen and charac-
3. Methods A numerical model was developed to simulate flood propagation through the study reach and the diversion of water to an off-line reservoir. Input to the model includes a flood hydrograph at the upstream boundary and parameters that control the configuration of the diversion scenario, and output includes flood depths along the reach. A watershed runoff model was not used because the vast majority of the drainage area is upstream of the gaging station, and lateral inflow
60
(a)
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40 30 Bank Elevation (zt)
20 10 Bottom Elevation (zb)
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60
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40
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4
6 Distance (km)
8
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Fig. 3. Channel properties along the study reach. Note narrow channel width at 5.6, 7.0, and 8.1 km, where bridges restrict flow.
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downstream of this point is small in comparison, roughly 2% of the total streamflow [17]. Below is a description of the models used to route the hydrograph through the study reach, to account for off-line diversions, and to measure the outcome associated with each control scenario. 3.1. Routing model Dynamic routing of the flood is accomplished by solving the integral form of the continuity and momentum equations [3]. Based upon a channel reach located between x1 and x2 where flow is evaluated between times t1 and t2, the equations appear as follows, Z x2 Z t2 ½Aðx; t2 Þ Aðx; t1 Þ dx þ ½Qðx2 ; tÞ Qðx1 ; tÞ dt x1 t1 Z t 2 Z x2 ¼ qd ðx; tÞ dx dt ð1Þ t1 x1 Z x2 Z t2 ½Qðx; t2 Þ Qðx; t1 Þ dx þ ½ðI 0 ðx2 ; tÞ þ I 1 ðx2 ; tÞÞ x1
t1
ðI 0 ðx1 ; tÞ þ I 1 ðx1 ; tÞÞ dt Z t 2 Z x2 Z t2 Z x2 dzb ðxÞ dx dt ¼ I 2 ðx; tÞ dx dt gAðx; tÞ dx t1 x1 t1 x1 Z t 2 Z x2 1 sðx; tÞP ðx; tÞ dx dt q t x Z 1 t2 Z 1x2 ud ðx; tÞqd ðx; tÞ dx dt ð2Þ t1
x1
where A is the cross-sectional area, Q is the volumetric discharge, zb is the bed elevation, s is the shear stress at the bed, P is the wetted perimeter, q is the fluid density, qd is the diversion rate per unit length of channel, or lateral outflow, ud is the streamwise component of the lateral outflow velocity, Z hðx;tÞ Z rðx;zÞ=2 uðx; y; z; tÞ2 dy dz I 0 ðx; tÞ ¼ 0
rðx;zÞ=2 2
¼ bQðx; tÞ =Aðx; tÞ Z hðx;tÞ gðhðx; tÞ zÞrðx; zÞ dz I 1 ðx; tÞ ¼ 0 Z hðx;tÞ orðx; zÞ dz I 2 ðx; tÞ ¼ gðhðx; tÞ zÞ ox 0
ð3Þ ð4Þ ð5Þ
where r represents the width of the channel and b is the momentum correction coefficient that has been reported to vary from 1.0 to 1.3 [2]. For the present study b = 1.0. Using a trapezoid to model each channel cross-section, the model cannot account for the stair-stepping of the top-width above the channel bank. Instead, the model assumes that, even above the bank, the top-width increases linearly according to the side-slope of the channel cross-section. This may lead to biases in the predictions including: an overprediction of peak flood depth, and an overprediction of depth reductions result-
ing from diversions. The closer the modeled cross-section captures the true top-width, the lesser these biases will be. When results are examined in the relative sense, as is done in this study, this may bias upwards the magnitude of predicted flood depth reductions caused by diversions. However, the ranking of control options, in terms of their flood fighting effectiveness, will not be biased. Additionally, this formulation does not account for transmission losses due to stream bed infiltration or lateral inflow. These processes are important components of flood forecasting models [5], but are ignored in the present study because results are only interpreted in a relative sense. That is, lateral inflow or infiltration are not expected to change because a small fraction of the flood runoff is diverted off-line. 3.2. Diversion models Three types of diversion strategies are examined using the reservoir configuration shown in Fig. 1 including passive control, weir control, and gate control. For the passive control scenario, it assumed that the channel and off-line reservoir with surface area A are hydraulically connected at all times by a short rectangular channel (as long as the width of the river levee) with a width wd and a bottom elevation equal to that of the reservoir and channel thalweg, zd. Exchange of water is controlled by the difference in energy between the channel and reservoir. During the rising limb of a flood the reservoir fills while it drains during the falling limb. For the weir control scenario, the configuration is identical to the passive control scenario except that a sharp crested weir spans the diversion channel with an adjustable crest height zw. It is assumed that the reservoir is empty prior to a flood, and begins to fill when flood stage exceeds the weir crest. Therefore, the flood control potential of the reservoir can be focused on the peak of the event. It is entirely possible that the crest height could be adjusted in real-time during a storm event for optimal control purposes, but for simplicity it is assumed constant throughout the flood. For the gate control scenario, the configuration is identical to the passive control scenario except that a gate spans the diversion channel and can be either open or closed. It is assumed that the reservoir is empty and the gate is closed prior to a flood, and that the gate is opened sometime during the flood. Opening the gate causes a dam-break flow into the off-line reservoir, so the key parameter is the time at which the gate is opened, tg. Let the diversion rate be defined as the lateral outflow over the length of the breach as follows, Z Qd ðtÞ ¼ qd ðx; tÞ dx ð6Þ wd
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where qd is defined in Eq. (1), and assume that flow exits through a short channel normal to the stream so ud in Eq. (2) equals zero, as shown in Fig. 1. Also, let the volume of water in the reservoir be, Z t Qd ðsÞ ds ð7Þ VðtÞ ¼ Vo þ to
where Vo corresponds to the volume in the reservoir at the beginning of the flood, to. The water elevation in the reservoir is given by, gr(t) = hr(t) + zd where hr ðtÞ ¼ VðtÞ=A, and the water elevation in the river channel is given by, gc(t) = h(xd,t) + zd. For the passive control scenario, Vo is computed based on the pre-flood stream stage and the diversion rate is modeled as, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gc ðtÞ þ gr ðtÞ 2zd Qd ðtÞ ¼ sðtÞ wd 2gjgc ðtÞ gr ðtÞj 2
signðaÞ ¼
1
if a P 0
1
if a < 0
Dt QðtÞ A
gr ðt þ Dt=2Þ ¼ gr ðtÞ þ
ð9Þ
Dt gr ðt þ DtÞ ¼ gr ðtÞ þ Qðt þ Dt=2Þ Corrector A
and
was used to numerically integrate the dynamic routing equations [13,15]. This model is suited to a wide range of Froude numbers, has very low dissipation properties, and can accurately simulate discontinuous flow arising from hydraulic jumps. This feature is ideal for the present study because the sudden opening of a gate generates waves which propagate away from the diversion point. Since the model does not solve an energy equation, it does not model explicitly account for energy losses at bridge crossings and other points of constriction. Nevertheless, limited analysis has indicated that energy is properly dissipated by the model [13]. The method uses a predictor–corrector time-stepping approach that involves advancing the solution from t to t + Dt/2 in a predictor step, and from t to t + Dt in a corrector step. This approach is also used to advance the solution in the reservoir, i.e.,
ð8Þ where sðtÞ ¼ signðgc ðtÞ gr ðtÞÞ
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ð10Þ
For the weir control scenario, Vo ¼ 0 and the diversion rate is modeled as, 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 > > wd 2gðgc ðtÞ zw Þ if gc ðtÞ > zw > gr ðtÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > gc ðtÞþgr ðtÞ2zw > > wd 2gjgc ðtÞ gr ðtÞj < sðtÞ 2 Qd ðtÞ ¼ if gc ðtÞ > zw ; gr ðtÞ > zw > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > 2gðgr ðtÞ zw Þ3 if gr ðtÞ > zw > gc ðtÞ w > d > > : 0 if gc ðtÞ < zw ; gr ðtÞ < zw ð11Þ For the gate control scenario, Vo ¼ 0, Qd(t) = 0 while the gate is closed, and Qd is computed using Eq. (8) after the gate is opened which corresponds to t P tg. These models of the diversion rate are motivated by energy arguments as described by [2]. In practice, each of these relations for the diversion rate should be scaled by a coefficient of discharge that is calibrated based on experimental data. Here we seek trends in the relative performance of different control alternatives, and for this purpose a precise characterize of the discharge coefficient is not necessary. 3.3. Numerical method The dynamic routing model, reservoir model, and diversion model were solved simultaneously to predict the effect of off-line diversions on flood stage. A total variation diminishing finite volume numerical method
Predictor ð12Þ
hence the diversion rate, Q(t) is computed based on gr(t) and gc(t) while Q(t + D/2) is computed based on gr(t + D/2) and gc(t + D/2). As is described by Sanders [13], the predictor step associated with the numerical solution of the routing equations updates the momentum equation in terms of the velocity V = Q/A. For stability purposes, an implicit discretization is used for the lateral outflow term. To apply the model, the study reach was uniformly discretized by 1000 computational cells and each of the four channel geometry parameters was linearly interpolated using the nearest survey points. Linear interpolation of channel geometry is preferable to higher-order interpolants such as cubic splines and piece-wise Hermitian polynomials [14]. Convergence of the numerical model at this resolution has been verified [15]. Numerical integration was performed using a time step Dt = 0.5 s. 3.4. Flood control scenarios Simulations were performed using one of four triangular hydrographs with duration td set to either 6, 12, 24, or 48 h. Each hydrograph is characterized by a baseflow of 10 m3/s, a peak flow Qp = 280 m3/s which is comparable to the 100-year event [6], and a rising and falling limb of equal duration. At the downstream boundary, the water level was set to mean sea level. In the model, a hypothetical off-line reservoir was set along the creek roughly 3 km from the upstream boundary, in cell 246 of 1000 as shown in Fig. 1. The reservoir was positioned upstream of areas susceptible to flooding, as the creek bed is relatively steep which prohibits control from downstream. The lower portion
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widen. To indicate the performance of each control scenario in absolute terms, the maximum depth at each monitoring station hmax was recorded. Using four different hydrographs, nine different values of the reservoir area A, five different values of the control structure width wd, four different weir control options, six different gate control options, as well as the passive control option and the do-nothing scenarios, a total of 4 · 9 · 5 · (4 + 6 + 1) + 4 = 1984 simulations were performed. Simulations were carried out by executing Fortran code on roughly 200 nodes of the medium performance computing (MPC) cluster at the University of California, Irvine.
of the watershed is heavily developed, so siting an offline reservoir would be practically impossible, but in less-developed watersheds this would not necessarily be the case. The area of the reservoir in the model was set to take on a value of either 1 · 103, 2 · 103, 5 · 103, 1 · 104, 2 · 104, 5 · 104, 1 · 105, 2 · 105, or 5 · 105 m2. The width of the diversion channel wd was set to take on a value of either 1, 2, 4, 8, or 10 m. For weir control, the weir crest height zw was set to either 1, 2, 4, or 5 m above the channel bed. For gate control, the gate was opened at one of six instances either before or after the peak stage. For the 6 h event, the gate was opened at tg = 1.5, 2.0, 2.5, 3, 3.5, or 4 h. For the 12 h event, the gate was opened at tg = 3, 4, 5, 6, 7, or 8 h. For the 24 h event, the gate was opened at tg = 9, 10, 11, 12, 13, or 14 h. For the 48 h event, the gate was opened at tg = 18, 20, 22, 24, 26, or 28 h. For each simulation, the peak depth at four hypothetical monitoring stations downstream of the diversion point was recorded to serve as a performance index. These stations correspond to cell 459, 569, 653, and 809 out of 1000, or a distance of 5.58, 6.93, 7.95, and 9.85 km from the upstream boundary, respectively. Stations 459, 569 and 653 are positioned just upstream of bridges, where flow is constricted and flooding has historically been problematic, and station 809 is positioned just upstream of where the channel begins to
4. Results and discussion Model simulation results for gate control and passive control are shown in Fig. 4, where the maximum depth hmax at STA 653 is plotted versus off-line area, A. Similar results were obtained for the other stations, so these are not reported. The clearest trend is that hmax decreases with increasing A under the passive control scenario, which is the expected response [7]. Scatter in the data is due to the sensitivity of the depth to other model parameters, i.e., tg and wd. Gate control results show it is possible to reduce peak flood stage further than by pas-
7.2
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Max. Depth at STA 653
6 hr flood
12 hr flood
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Fig. 4. Maximum depth hmax at STA 653 plotted versus A for passive control and gate control. For each value of A, hmax from passive control varies depending on the diversion channel width wb, while hmax from gate control varies depending on wb and the timing of the gate action tg.
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sive control, for the same off-line area. In some cases, particularly when the off-line area is relatively small, 2–3 times the flood stage reduction results from gate control versus passive control. As the off-line area becomes larger, however, the added advantage of gate control over passive control becomes negligible. These results also show that as the flood becomes longer, passive control scenarios become less sensitive to the width of the control structure, wd which restricts flow. For example, with the 6 h flood the effectiveness of passive controls varies considerably based on the diversion width wd, while there is little variability with the 48 h flood. This trend also applies to gate control, as illustrated in Fig. 5 where maximum flood depth at STA 653 is contoured as a function of gate timing tg and diversion channel width wd based on an off-line reservoir with area A ¼ 105 m2 . In the limit of the 6 h flood, maximum depth is clearly sensitive to both tg and wg; while for the 48 h flood maximum depth is much more sensitive to tg than wg. Fig. 5 illustrates the importance of optimizing the configuration of the gate diversion, particularly the timing of the gate action. For example, no flood stage reduction is accomplished if the gate is opened too late. This highlights an engineering challenge: correctly configuring the diversion to enhance, not inhibit, flood stage reduction. Model results for weir control and passive control are presented in Fig. 6, and show that weir control is not as
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effective as gate control for flood stage reduction. There appears to be a minor advantage to weir control over passive control when the off-line area is very small, in this case less than 1 · 104 m2; otherwise, there is little benefit. For this reason, weir control results are not examined further. The preceding results for passive control and gate control collapse onto a single set of axes by non-dimensionalizing the flood depth and flood plain area as follows, h ¼
hmax h0max
and
A ¼
Ah0max 1 t Q 2 d p
ð13Þ
where h0max represents the peak flood depth in the absence of a diversion. Physically, A corresponds to the ratio of the off-line storage capacity to the total flood runoff, so A ¼ 0 implies that no runoff is diverted while A ¼ 1 implies that the entire hydrograph was diverted. h indicates peak flood depth at the monitoring station relative to what would occur in the absence of control. For example, h = 1 implies no flood depth reduction compared to the do-nothing scenario. Fig. 7 shows the passive control and gate control results in terms of A versus h . For passive control, h is only dependent upon A when A < 0:1; while for A > 0:5, h is nearly independent of A . The line h ¼ 1 0:3A was manually fitted to the envelope of
Fig. 5. Contours of maximum depth hmax in meters at STA 653 for A ¼ 1 105 m2 plotted versus the diversion channel width wb and the timing of the gate action tg.
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Max. Depth at STA 653
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x 10
Fig. 6. Maximum depth hmax at STA 653 plotted versus reservoir area A for passive control and weir control. For each value of A, hmax from passive control varies depending on the diversion channel width wb, while hmax from weir control varies depending on wb and weir height zw.
passive control results for A < 0:2 (Fig. 7a), and presented in comparison to the gate control results in Fig. 7b. This figure illustrates that gate control is capable of achieving 2–3 times the flood stage reduction of passive control when A < 0:2, but for A > 0:2 the potential for improved stage reduction diminishes with increasing A . These results also show that the effectiveness of off-line diversions is not limited by flood duration, only that longer duration floods require a proportionally larger reservoir to accomplish the same level of flood stage reduction as a shorter flood. Whether gate control performs better than passive control depends on the values of wd and tg, which correspond to the diversion width and the timing of the gate action. (Recall Fig. 5, which illustrates the sensitivity of flood stage to these parameters.) Fig. 7b shows there are many results both above and below the line corresponding to the envelope of gate control results. To determine the conditions under which gate control enhances (not inhibits) flood stage reduction, the values of wd and tg leading to the smallest values of hmax for each value of A were identified. Fig. 8a and b presents these results as dimensionless gate width w = wd/W and dimensionless lead time dt = 12(tr tg)/td, respectively, versus dimensionless off-line area, where W is a reference channel width set to 10 m and tr is the duration of the rising limb of the hydrograph. This figure shows that the opti-
mal width and optimal lead time increase with flood plain area for 0 6 A 6 0.05. The finite set of data points limits the precision with which optimal parameters can be identified, but the general trend is clearly evident. For A > 0:05, w = 1 and dt = 1 appear optimal. The significance of the latter result is not clear because w = 1 represents a limit of the parameter range used in this study. Fig. 8c presents h resulting from optimal values of w and dt , highlighting the advantage of optimal gate control over passive control. For example, if the reservoir can capture roughly 2% of the flood (A ¼ 0:02), flood depth is reduced less than 1% by passive control but 2–4% by gate control. Based on a peak flood depth of 7 m, this corresponds to a 7 and 14–28 cm reduction, respectively. The problem of optimizing diversions can be interpreted from a wave interference perspective. Taking the flood to be a positive wave, and assuming the diversion creates two depression (or negative) waves consistent with one-dimensional shallow-water hydrodynamics, the challenge is for the negative waves to be of sufficient amplitude and duration to cancel out or at least mitigate peak stages of the flood. The amplitude of the negative waves is controlled by the diversion rate, or wb in the case of gate control; while the duration of the wave controlled by both the off-line storage and the diversion rate, A and wb, respectively, in the case of gate control. Assuming
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6 hr 12 hr 24 hr 48 hr
0.98 0.96 0.94
*
h , Passive Control
1
h*=1−0.3A*
0.92
(a) 0.9
0
0.5
1
1.5
A* 1 6 hr 12 hr 24 hr 48 hr
0.96 *
h =1−0.3A*
0.94
*
h , Gate Control
0.98
0.92
(b) 0.9
0
0.05
0.1
0.15
0.2
0.25
A*
Fig. 7. Dimensionless peak flood stage h at STA 653 versus dimensionless reservoir area A for passive control (a) and gate control (b) scenarios. A can be interpreted as the fraction of the total flood runoff that is captured by the reservoir. For passive control note that for A < 0:1, flood stage reduction is only dependent upon A ; while for A > 0:5 flood stage reduction becomes nearly independent of A . The line h ¼ 1 0:3A is fitted to the envelope of passive control results (a), and presented along with the gate control results (b). Note that gate control results fall both above and below the line h ¼ 1 0:3A , highlighting the potential to improve, but the risk to impair, flood mitigation efforts compared to passive control.
that A is relatively small, which it must be if there is an inundation concern in the first place, then there is a tradeoff between the amplitude and duration of the negative wave. In one extreme, the diversion might lower flood stage by 10% or more, but only for a few minutes, after which flood stage rises again to its previous level as if there was no control at all. In the other extreme, the diversion might be sustained for the whole duration of the flood, but at a rate so slow that flood stage is never lowered by more than a fraction of a percent. Hence, optimal control exists somewhere in between these limits, and it involves a diversion rate that is large enough to lower flood stage by a few percent and a duration long enough to encompass the cresting period of the flood. Whether or not the wave action is dynamic in this application is not clear, for in relatively steep channels the momentum balance in the streamwise direction is less sensitive to inertial effects than in mild channels. Model predictions of maximum depth shown in Figs. 4 and 5 reveal that peak depth, in the absence of control, varies slightly depending upon the event duration so the flood is certainly not a kinematic wave. For example, peak depth at STA 653 is predicted to be roughly 6.9 m for the 6 h event and 7.0 m for the 48 h event.
Work by Jaffe and Sanders [7] suggested that off-line diversions could be engineered to crop the peak stage of relatively short duration floods via either regressive or progressive depression waves, but results of the San Francisquito application indicate longer floods in steep channels are equally controllable using a proportionally larger off-line reservoir. This suggests that it is the amplitude and duration of the depression waves that are most important for control by progressive depression waves in steep channels; whether the flood is a dynamic wave or simply driven by gravity and friction has less bearing on the controllability of flood stage. The problem of engineering a diversion for optimal flood mitigation remains a great challenge. Here, we benefit from a priori knowledge of the flood wave, while in practice this must be forecast in real-time. We also assume that the reservoir is initially empty before the flood, which may not necessarily be the case in the event of successive storms which yield multi-modal hydrographs. Nevertheless, results of this study suggest that gate control is as or more effective than passive control so long as the diversion commences before peak flood stage reaches the diversion point. Here we considered a triangular hydrograph with symmetrical rising and falling limbs, and the optimal lead time was found to
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(a) Opt. w
*
1.5 1 0.5 0
0
0.02
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0.02
0.04
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2
(b) Opt. δ t
*
1.5 1 0.5 0
0
1 Passive Control Opt. h
*
h*=1−0.3A*
0.95
(c) 0.9
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
A*
Fig. 8. Values of dimensionless diversion width w (a) and dimensionless lead time dt (b) that minimize the dimensionless peak flood stage h at STA 653 for each value of dimensionless off-line area A . Values of h at STA 653 resulting from optimal values of w and dt are presented in (c), along with the line that corresponds to envelope of passive control results.
be roughly tr/12; this may be different depending upon the shape of the hydrograph which highlights the need for watershed- and event-specific engineering. In the limit that diversion commences too soon, gate control simply performs as well as passive control; whereas in the limit that diversions commence too late, passive control performs better. Hence, it is critical that diversions be properly timed. Real-time monitoring and forecasting of flood conditions is becoming increasingly common and according to a report by the National Hydrologic Warning Council (NHWC), there is strong evidence that such systems save lives and reduce flood damages [4]. Since the effectiveness of gate control hinges on the lead time of the gate action, a key question is whether the lead time of forecasts is greater than the lead time for gate action. According to the NHWC, ‘‘crest-stage forecasts can be made a few hours in advance for cities and communities along streams draining small basins, but they can be made two weeks or more in advance . . . along rivers draining large basins’’ [4]. Taking San Francisquito Creek as a ‘‘small basin’’, it would appear that the lead time for forecasts is sufficient to support optimal gate action. However, to error on the side of caution, compensating for uncertainty in flood hydrograph predictions, decision
support systems should be biased towards diversions that begin earlier than optimal, by an amount that scales with the degree of uncertainty.
5. Summary and conclusions Communities must consider a wide variety of structural and non-structural measures to mitigate flood risks. Among structural measures, off-line reservoirs are an attractive alternative to in-line reservoirs for many environmental reasons, mainly because the former does not create a barrier to sediment transport and fish migration or permanently flood riparian habitat. Based on conditions in a O(102) km2 watershed in northern California, active control of diversions to an off-line reservoir creates an opportunity to at least double the flood depth reduction accomplished by passive control. The active control scenario involves use of a gate which remains closed until slightly before peak flood stage reaches the diversion structure, at which point it opens creating a dam-break like flow into an off-line storage area that quickly lowers river stage. For this control to be effective, the gate action must be optimized relative to the flood hydrograph. In this
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study, a triangular hydrograph was used and the optimal time to open the gate was slightly before peak flood stage reaches the diversion structure, by a duration that increases with the volumetric capacity of the reservoir. The added benefit of active control over passive control is maximum when the reservoir captures less than 20% of the flood volume. As off-line storage increases beyond this value, the benefit of active control over passive control diminishes. Finally, floods with relatively short and long periods are equally controllable by an off-line diversion, though the latter requires a proportionally larger reservoir to accomplish the same level of control. Accurate forecasts of flood hydrographs are needed to identify and implement optimal diversion strategies. The lead time typical of operational forecasting systems appears sufficient to make decisions regarding gate control actions, but with little time to spare. Poorly designed or implemented active control scenarios may cause more flooding than would otherwise occur if passive control were adopted. This highlights the risk of operational complexity in the context of flood controllability. In circumstances where the potential rewards outweigh the risk, advanced flood forecasting and decision support systems are needed to optimize the placement, configuration, and operation of flood control infrastructure such as diversion basins, pumps, gates, weirs and other control facilities. These systems should error on the side of an early gate action to compensate for uncertainty in flood forecasts. Hydrodynamic models should be validated before being used to engineer diversion strategies. Validation should focus on the skill with which the model accurately routes flood waves through the watershed and accurately predicts flood stage, and the ability of the model to resolve the wave action resulting from dambreak type diversions into off-line reservoirs. In smaller watersheds data such as these are usually not available and in such cases monitoring programs should first be instituted.
Acknowledgements This work was supported by a grant from the National Science Foundation (CMS-9984579), whose support is gratefully acknowledged.
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References [1] Bulkley J. Personnel communication to B. Sanders, 1997. [2] Chow VT. Open-channel hydraulics. McGraw-Hill; 1959. 680 p. [3] Cunge JA, Holly Jr FM, Verwey A. Practical aspects of computational River Hydraulics. Pitman; 1980. [4] EASPE, Inc. Use and benefits of the national weather service river and flood forecasts. Report prepared for the National Hydrologic Warning Council, 2002. Available from: http://www.udfcd.org/ Nhwc/. [5] El-Hames AS, Richards KS. An integrated, physically based model for arid region flash flood prediction capable of simulating dynamic transmission loss. Hydrol Process 1998;12:1219–32. [6] Jaffe DA. Levee breaches for flood reduction. PhD Dissertation, Environmental Engineering Graduate Program, University of California, Irvine, 2002. [7] Jaffe, Sanders. Engineered levee breaches for flood mitigation. J Hydraul Eng 2001;127(6):471–9. [8] Gupta H, Sorooshian S, Gao X, Imam B, Hsu K-L, Bastidas L, et al. The challenge of predicting flash floods from thunderstorm rainfall. Philos Trans R Soc Lond A 2002;360:1363–71. [9] Haque CE. Risk assessment, emergency preparedness and response to hazards: the case of the 1997 Red River valley flood, Canada. Natural Hazards 2000;21:225–45. [10] Hansen B. Army Corps of Engineers develops Red River Valley flood plan. Civil Eng 2004;74(11):22–3. [11] Nolte BH, Schwab GO. Pumped storage, an alternative to channelization. Ohio Rep Res Dev Agric, Home Econom, Nat Resour 1971;56(5):70–1. [12] Nolte BH, Schwab GO. Streamflow regulation with pumped storage reservoirs. Trans ASAE 1974;17(3):440–2. [13] Sanders BF. High-resolution and non-oscillatory solution of the St. Venant equations in non-rectangular and non-prismatic channels. J Hydraul Res 2001;39(3):321–30 (For information on how the model accounts for energy losses, see Discussion and Reply by Authors. J Hydraul Res, 41(6), 668–72.). [14] Sanders BF, Chrysikopoulos CV. Longitudinal interpolation of parameters characterizing channel geometry by piece-wise polynomial and universal kriging methods: effect on flow modeling. Adv Water Res 2004;27:1061–73. [15] Sanders BF, Jaffe DA, Chu AK. Discretization of integral equations describing flow in nonprismatic channels with uneven beds. J Hydraul Eng 2003;129(3):235–44. [16] Sanders BF, Katopodes ND. Active flood hazard mitigation. I: Bidirectional wave control. J Hydraul Eng 1999;125(10):1057–70. [17] San Francisquito Creek Coordinated Resources Management and Planning Flood Control and Erosion Task Force. Reconnaissance Investigation Report of San Francisquito Creek, Peninsula Conservation Center, Palo Alto, CA, 1998. [18] Schemel LE, Cox MH, Hager SW, Sommer TR. Hydrology and chemistry of floodwaters in the Yolo Bypass, Sacramento River system, California, during 2000. US Geol Surv, Water Res Inv Rep 02-4202. [19] Ward AD, Trimble SW. Environmental hydrology. Lewis Publishers; 2003. 475 p.
Passive and active control of diversions to an off-line reservoir for flood stage reduction Brett F. Sanders b
a,*
, John C. Pau b, David A. Jaffe
c
a Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, United States Donald Bren School of Information and Computer Sciences, University of California, Irvine, CA 92697, United States c Pacific Advanced Civil Engineering, Inc., Fountain Valley, CA 92708, United States
Received 24 October 2004; received in revised form 30 June 2005; accepted 14 July 2005 Available online 10 October 2005
Abstract Diversion of excess streamflow to an off-line reservoir is examined as a wave interference problem that can be controlled to reduce the cresting stage of a flood. Flood diversions create depression waves in the stream channel which, superimposed upon the flood, decrease flood stage. In the context of an O(102) km2 coastal watershed in northern California, numerical modeling was performed to compare the performance of three idealized diversion control strategies including passive control, weir control, and gate control. It was found that gate control, which creates a dam-break like flow into an off-line reservoir, can be optimized to accomplish 2–3 times the flood depth reduction of passive control. Capturing 2% of the runoff, for example, the cresting depth is reduced less than 1% with passive control but 2–4% by gate control. This is particularly important in areas with little off-line or over-bank storage, such as urban watersheds. Timing of the gate action is critical. Optimal control requires gate action slightly before peak stage arrives at the diversion structure, by a duration that scales with the capacity of the reservoir. The lead time of flood forecasts appears compatible with the lead time necessary to optimize diversions, and decision support systems should compensate for forecast uncertainty by early gate action. If by poor design or operator error the gate is opened too late, gate control becomes less effective than passive control. 2005 Elsevier Ltd. All rights reserved. Keywords: Flood control; Off-line reservoir; Flood modeling; Diversions
1. Introduction Off-line reservoirs are a structural approach to flood control first considered by Nolte and Schwab [11,12]. The basic idea is to divert or pump excess storm water from streams into detention basins during storms, then return water to the channel later when capacity is available. Nolte and Schwab envisioned this approach for regions of nearly level topography where in-stream reservoirs are not practical. But now that the ecological *
Corresponding author. Tel.: +1 949 824 4327; fax: +1 949 824 3672. E-mail address: [email protected] (B.F. Sanders). 0309-1708/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2005.07.015
consequences of in-stream reservoirs are better appreciated, such as blocking fish migration and trapping sediment, there is renewed motivation to understand how a range of detention basin options including off-line reservoirs can be engineered for both flood control and water quality control purposes. Engineered wetlands are one type of detention basin that are increasingly being constructed to improve water quality adversely impacted by pollutants such as biochemical oxygen demand, metals, nutrients, organics, and sediments [19], though in-line configurations are more common. Larger, storm-water oriented facilities may also be engineered to support land use for agricultural, wildlife habitat, and/or recreational purposes, similar to the Red River
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in Grand Forks, North Dakota where low-lying homes were removed and replaced with recreational areas and campgrounds [10]. Other examples are floodways such as the Red River Floodway in Manitoba [9] and the Yolo Bypass in California [18] but these differ from off-line reservoirs because floodways transmit water downstream. The problem of controlling diversions to mitigate flood stage involves the interaction of three waves: the flood wave and two depression waves generated by the diversion. Under subcritical channel flow conditions, one depression wave moves upstream (regressive wave) and one moves downstream (progressive wave), while both are progressive under supercritical channel flow conditions [16]. A practical consequence of this wave action is a potential to control flood stage both downstream and upstream of an off-line reservoir under subcritical channel flow conditions, but only downstream of a reservoir under supercritical flow conditions. The utility of regressive depression waves is supported anecdotally by reports that during the 1993 floods in the midwest region of the United States, St. Louis was spared from inundation following failure of a river levee downstream along the Mississippi River [1]. This exemplifies a dynamic wave phenomena, meaning that momentum is not simply balanced between gravity and friction but also with inertia. Jaffe and Sanders [7] developed a 2D hydrodynamic model to simulate the effect of levee breaches on flood hydrographs in a rectangular mildly sloped channel, and applied the model to examine the relationship between flood stage and properties of the breach such as the breach length and the capacity of the off-line reservoir. Both regressive- and progressive-wave control of the flood was realized in the model, and the controllability of floods was linked to the duration and magnitude of the flood, the capacity of the off-line reservoir, and the timing of breaches relative to the flood hydrograph. This paper presents a comparative analysis of three idealized, gravity driven, off-line reservoir control scenarios using the reservoir configuration shown in Fig. 1. These scenarios include passive control where the off-line reservoir fills and drains with the rising and falling limbs of the hydrograph, respectively, through an unregulated diversion channel; weir control where diversions are controlled by a weir structure in the diversion channel and commence once flood stage exceeds the height of the weir crest; and gate control where diversions are controlled by a gate structure and commence upon opening the gate. Gate control creates dynamic wave action similar to a levee breach, while flow varies slowly with passive and weir control. These three options span a range of operational complexity. No operational decisions are required for passive control, the weir height must be set for weir control, and the timing of the gate opening must be set for gate control. The lat-
Q
Channel
wd Control Structure
Qd
Off-line Reservoir
Fig. 1. Off-line reservoir configuration relative to channel. Weir and gate control structures are examined in this study as well as passive control. For passive control, there is no control structure in the diversion channel.
ter options would require forecasting models to predict the timing and magnitude of a flood. While in the flash flood limit flood forecasting has been recognized as a tremendous challenge [8], in many cases real-time monitoring and forecasting has proven to save lives and reduce damages through early warning and emergency management systems and by real-time regulation of flood control infrastructure [4]. A key element of these systems in the US is the automated local evaluation in real-time (ALERT) program, developed in the 1970Õs by the National Weather Service, which uses sensors in the field to transmit environmental data to a central computer in real time. The purpose of this paper is to examine the relative performance of passive, weir, and gate diversion strategies to identify potential advantages and liabilities of operational complexity as described above. This is done in the context of an O(102) km2 coastal watershed in California using realistic stream characteristics but, for obvious reasons, a hypothetical off-line reservoir. This study builds upon the work by Jaffe and Sanders [7] by examining the effectiveness of diversions in the context of realistic stream flows, flood durations, and natural channel geometry.
2. Site description A 12.2 km portion of San Francisquito Creek in the San Francisco Bay area of California is the focus of this study, downstream of USGS gage 11 164 500 as shown in Fig. 2. The region experiences a mild wet winter and a warm dry summer as is characteristic of Mediterranean climates. Based on National Weather Service data collected in Palo Alto between 1953 and 2004, the watershed averages 39 cm of rainfall, over 90% of which falls in the months of November through April. High and low temperatures average 26/13 C in August and 14/6 C in January. The creek drains a 95 km2 catchment which extends westward from San Francisco Bay to the ridge of the Santa Cruz Mountains. The upper
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terized by a cross-sectional shape that closely resembles a trapezoid. Due to this shape and for consistency with the hydraulic model used in this study, a four parameter geometric description was adopted involving the bottom width wb, bottom elevation zb, bank width wt, and bank elevation zt, as shown in Fig. 3. These parameters were estimated by a graphical fitting procedure. The channel bed has a concave-up longitudinal profile with a bed slope of roughly 0.5% over the first 2 km, a slope of roughly 0.25% between kilometers 2 and 10, and a nearly horizontal bed slope between kilometers 10 and 12. The bottom width and bank width average roughly 10 and 30 m, respectively, over the first 10 km though these parameters vary locally by as much as 10 m. Bridges cross the creek at several points and the abutments, which restrict flow, are modeled as a rectangular cross-section. Flooding commonly occurs immediately upstream of the abutments.
Fig. 2. Map of San Francisquito Creek and surrounding watershed. Reach used in study is between gage and San Francisco Bay. Relative position of diversion reservoir and monitoring stations are noted.
portion of the catchment is forested, while the lower portion near the bay is developed and prone to flooding. It has been reported that neither the channel nor bridge culverts of San Francisquito Creek are sized to accommodate the 100-year flood [17]. For example, the capacity of the channel has been estimated to be as low as 170 m3/s while the 100-year discharge has been estimated to be 280 m3/s. Channel geometry data, consisting of transects at 283 stations, were obtained from the Santa Clara Valley Water District. The channel bed is earthen and charac-
3. Methods A numerical model was developed to simulate flood propagation through the study reach and the diversion of water to an off-line reservoir. Input to the model includes a flood hydrograph at the upstream boundary and parameters that control the configuration of the diversion scenario, and output includes flood depths along the reach. A watershed runoff model was not used because the vast majority of the drainage area is upstream of the gaging station, and lateral inflow
60
(a)
50 Elevation (m)
40 30 Bank Elevation (zt)
20 10 Bottom Elevation (zb)
0 −10
0
2
4
6
8
10
12
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Width (m)
60
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40
Bottom Width (wb)
20
0
0
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4
6 Distance (km)
8
10
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Fig. 3. Channel properties along the study reach. Note narrow channel width at 5.6, 7.0, and 8.1 km, where bridges restrict flow.
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downstream of this point is small in comparison, roughly 2% of the total streamflow [17]. Below is a description of the models used to route the hydrograph through the study reach, to account for off-line diversions, and to measure the outcome associated with each control scenario. 3.1. Routing model Dynamic routing of the flood is accomplished by solving the integral form of the continuity and momentum equations [3]. Based upon a channel reach located between x1 and x2 where flow is evaluated between times t1 and t2, the equations appear as follows, Z x2 Z t2 ½Aðx; t2 Þ Aðx; t1 Þ dx þ ½Qðx2 ; tÞ Qðx1 ; tÞ dt x1 t1 Z t 2 Z x2 ¼ qd ðx; tÞ dx dt ð1Þ t1 x1 Z x2 Z t2 ½Qðx; t2 Þ Qðx; t1 Þ dx þ ½ðI 0 ðx2 ; tÞ þ I 1 ðx2 ; tÞÞ x1
t1
ðI 0 ðx1 ; tÞ þ I 1 ðx1 ; tÞÞ dt Z t 2 Z x2 Z t2 Z x2 dzb ðxÞ dx dt ¼ I 2 ðx; tÞ dx dt gAðx; tÞ dx t1 x1 t1 x1 Z t 2 Z x2 1 sðx; tÞP ðx; tÞ dx dt q t x Z 1 t2 Z 1x2 ud ðx; tÞqd ðx; tÞ dx dt ð2Þ t1
x1
where A is the cross-sectional area, Q is the volumetric discharge, zb is the bed elevation, s is the shear stress at the bed, P is the wetted perimeter, q is the fluid density, qd is the diversion rate per unit length of channel, or lateral outflow, ud is the streamwise component of the lateral outflow velocity, Z hðx;tÞ Z rðx;zÞ=2 uðx; y; z; tÞ2 dy dz I 0 ðx; tÞ ¼ 0
rðx;zÞ=2 2
¼ bQðx; tÞ =Aðx; tÞ Z hðx;tÞ gðhðx; tÞ zÞrðx; zÞ dz I 1 ðx; tÞ ¼ 0 Z hðx;tÞ orðx; zÞ dz I 2 ðx; tÞ ¼ gðhðx; tÞ zÞ ox 0
ð3Þ ð4Þ ð5Þ
where r represents the width of the channel and b is the momentum correction coefficient that has been reported to vary from 1.0 to 1.3 [2]. For the present study b = 1.0. Using a trapezoid to model each channel cross-section, the model cannot account for the stair-stepping of the top-width above the channel bank. Instead, the model assumes that, even above the bank, the top-width increases linearly according to the side-slope of the channel cross-section. This may lead to biases in the predictions including: an overprediction of peak flood depth, and an overprediction of depth reductions result-
ing from diversions. The closer the modeled cross-section captures the true top-width, the lesser these biases will be. When results are examined in the relative sense, as is done in this study, this may bias upwards the magnitude of predicted flood depth reductions caused by diversions. However, the ranking of control options, in terms of their flood fighting effectiveness, will not be biased. Additionally, this formulation does not account for transmission losses due to stream bed infiltration or lateral inflow. These processes are important components of flood forecasting models [5], but are ignored in the present study because results are only interpreted in a relative sense. That is, lateral inflow or infiltration are not expected to change because a small fraction of the flood runoff is diverted off-line. 3.2. Diversion models Three types of diversion strategies are examined using the reservoir configuration shown in Fig. 1 including passive control, weir control, and gate control. For the passive control scenario, it assumed that the channel and off-line reservoir with surface area A are hydraulically connected at all times by a short rectangular channel (as long as the width of the river levee) with a width wd and a bottom elevation equal to that of the reservoir and channel thalweg, zd. Exchange of water is controlled by the difference in energy between the channel and reservoir. During the rising limb of a flood the reservoir fills while it drains during the falling limb. For the weir control scenario, the configuration is identical to the passive control scenario except that a sharp crested weir spans the diversion channel with an adjustable crest height zw. It is assumed that the reservoir is empty prior to a flood, and begins to fill when flood stage exceeds the weir crest. Therefore, the flood control potential of the reservoir can be focused on the peak of the event. It is entirely possible that the crest height could be adjusted in real-time during a storm event for optimal control purposes, but for simplicity it is assumed constant throughout the flood. For the gate control scenario, the configuration is identical to the passive control scenario except that a gate spans the diversion channel and can be either open or closed. It is assumed that the reservoir is empty and the gate is closed prior to a flood, and that the gate is opened sometime during the flood. Opening the gate causes a dam-break flow into the off-line reservoir, so the key parameter is the time at which the gate is opened, tg. Let the diversion rate be defined as the lateral outflow over the length of the breach as follows, Z Qd ðtÞ ¼ qd ðx; tÞ dx ð6Þ wd
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where qd is defined in Eq. (1), and assume that flow exits through a short channel normal to the stream so ud in Eq. (2) equals zero, as shown in Fig. 1. Also, let the volume of water in the reservoir be, Z t Qd ðsÞ ds ð7Þ VðtÞ ¼ Vo þ to
where Vo corresponds to the volume in the reservoir at the beginning of the flood, to. The water elevation in the reservoir is given by, gr(t) = hr(t) + zd where hr ðtÞ ¼ VðtÞ=A, and the water elevation in the river channel is given by, gc(t) = h(xd,t) + zd. For the passive control scenario, Vo is computed based on the pre-flood stream stage and the diversion rate is modeled as, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gc ðtÞ þ gr ðtÞ 2zd Qd ðtÞ ¼ sðtÞ wd 2gjgc ðtÞ gr ðtÞj 2
signðaÞ ¼
1
if a P 0
1
if a < 0
Dt QðtÞ A
gr ðt þ Dt=2Þ ¼ gr ðtÞ þ
ð9Þ
Dt gr ðt þ DtÞ ¼ gr ðtÞ þ Qðt þ Dt=2Þ Corrector A
and
was used to numerically integrate the dynamic routing equations [13,15]. This model is suited to a wide range of Froude numbers, has very low dissipation properties, and can accurately simulate discontinuous flow arising from hydraulic jumps. This feature is ideal for the present study because the sudden opening of a gate generates waves which propagate away from the diversion point. Since the model does not solve an energy equation, it does not model explicitly account for energy losses at bridge crossings and other points of constriction. Nevertheless, limited analysis has indicated that energy is properly dissipated by the model [13]. The method uses a predictor–corrector time-stepping approach that involves advancing the solution from t to t + Dt/2 in a predictor step, and from t to t + Dt in a corrector step. This approach is also used to advance the solution in the reservoir, i.e.,
ð8Þ where sðtÞ ¼ signðgc ðtÞ gr ðtÞÞ
865
ð10Þ
For the weir control scenario, Vo ¼ 0 and the diversion rate is modeled as, 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 > > wd 2gðgc ðtÞ zw Þ if gc ðtÞ > zw > gr ðtÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > gc ðtÞþgr ðtÞ2zw > > wd 2gjgc ðtÞ gr ðtÞj < sðtÞ 2 Qd ðtÞ ¼ if gc ðtÞ > zw ; gr ðtÞ > zw > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > 2gðgr ðtÞ zw Þ3 if gr ðtÞ > zw > gc ðtÞ w > d > > : 0 if gc ðtÞ < zw ; gr ðtÞ < zw ð11Þ For the gate control scenario, Vo ¼ 0, Qd(t) = 0 while the gate is closed, and Qd is computed using Eq. (8) after the gate is opened which corresponds to t P tg. These models of the diversion rate are motivated by energy arguments as described by [2]. In practice, each of these relations for the diversion rate should be scaled by a coefficient of discharge that is calibrated based on experimental data. Here we seek trends in the relative performance of different control alternatives, and for this purpose a precise characterize of the discharge coefficient is not necessary. 3.3. Numerical method The dynamic routing model, reservoir model, and diversion model were solved simultaneously to predict the effect of off-line diversions on flood stage. A total variation diminishing finite volume numerical method
Predictor ð12Þ
hence the diversion rate, Q(t) is computed based on gr(t) and gc(t) while Q(t + D/2) is computed based on gr(t + D/2) and gc(t + D/2). As is described by Sanders [13], the predictor step associated with the numerical solution of the routing equations updates the momentum equation in terms of the velocity V = Q/A. For stability purposes, an implicit discretization is used for the lateral outflow term. To apply the model, the study reach was uniformly discretized by 1000 computational cells and each of the four channel geometry parameters was linearly interpolated using the nearest survey points. Linear interpolation of channel geometry is preferable to higher-order interpolants such as cubic splines and piece-wise Hermitian polynomials [14]. Convergence of the numerical model at this resolution has been verified [15]. Numerical integration was performed using a time step Dt = 0.5 s. 3.4. Flood control scenarios Simulations were performed using one of four triangular hydrographs with duration td set to either 6, 12, 24, or 48 h. Each hydrograph is characterized by a baseflow of 10 m3/s, a peak flow Qp = 280 m3/s which is comparable to the 100-year event [6], and a rising and falling limb of equal duration. At the downstream boundary, the water level was set to mean sea level. In the model, a hypothetical off-line reservoir was set along the creek roughly 3 km from the upstream boundary, in cell 246 of 1000 as shown in Fig. 1. The reservoir was positioned upstream of areas susceptible to flooding, as the creek bed is relatively steep which prohibits control from downstream. The lower portion
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widen. To indicate the performance of each control scenario in absolute terms, the maximum depth at each monitoring station hmax was recorded. Using four different hydrographs, nine different values of the reservoir area A, five different values of the control structure width wd, four different weir control options, six different gate control options, as well as the passive control option and the do-nothing scenarios, a total of 4 · 9 · 5 · (4 + 6 + 1) + 4 = 1984 simulations were performed. Simulations were carried out by executing Fortran code on roughly 200 nodes of the medium performance computing (MPC) cluster at the University of California, Irvine.
of the watershed is heavily developed, so siting an offline reservoir would be practically impossible, but in less-developed watersheds this would not necessarily be the case. The area of the reservoir in the model was set to take on a value of either 1 · 103, 2 · 103, 5 · 103, 1 · 104, 2 · 104, 5 · 104, 1 · 105, 2 · 105, or 5 · 105 m2. The width of the diversion channel wd was set to take on a value of either 1, 2, 4, 8, or 10 m. For weir control, the weir crest height zw was set to either 1, 2, 4, or 5 m above the channel bed. For gate control, the gate was opened at one of six instances either before or after the peak stage. For the 6 h event, the gate was opened at tg = 1.5, 2.0, 2.5, 3, 3.5, or 4 h. For the 12 h event, the gate was opened at tg = 3, 4, 5, 6, 7, or 8 h. For the 24 h event, the gate was opened at tg = 9, 10, 11, 12, 13, or 14 h. For the 48 h event, the gate was opened at tg = 18, 20, 22, 24, 26, or 28 h. For each simulation, the peak depth at four hypothetical monitoring stations downstream of the diversion point was recorded to serve as a performance index. These stations correspond to cell 459, 569, 653, and 809 out of 1000, or a distance of 5.58, 6.93, 7.95, and 9.85 km from the upstream boundary, respectively. Stations 459, 569 and 653 are positioned just upstream of bridges, where flow is constricted and flooding has historically been problematic, and station 809 is positioned just upstream of where the channel begins to
4. Results and discussion Model simulation results for gate control and passive control are shown in Fig. 4, where the maximum depth hmax at STA 653 is plotted versus off-line area, A. Similar results were obtained for the other stations, so these are not reported. The clearest trend is that hmax decreases with increasing A under the passive control scenario, which is the expected response [7]. Scatter in the data is due to the sensitivity of the depth to other model parameters, i.e., tg and wd. Gate control results show it is possible to reduce peak flood stage further than by pas-
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Max. Depth at STA 653
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Fig. 4. Maximum depth hmax at STA 653 plotted versus A for passive control and gate control. For each value of A, hmax from passive control varies depending on the diversion channel width wb, while hmax from gate control varies depending on wb and the timing of the gate action tg.
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sive control, for the same off-line area. In some cases, particularly when the off-line area is relatively small, 2–3 times the flood stage reduction results from gate control versus passive control. As the off-line area becomes larger, however, the added advantage of gate control over passive control becomes negligible. These results also show that as the flood becomes longer, passive control scenarios become less sensitive to the width of the control structure, wd which restricts flow. For example, with the 6 h flood the effectiveness of passive controls varies considerably based on the diversion width wd, while there is little variability with the 48 h flood. This trend also applies to gate control, as illustrated in Fig. 5 where maximum flood depth at STA 653 is contoured as a function of gate timing tg and diversion channel width wd based on an off-line reservoir with area A ¼ 105 m2 . In the limit of the 6 h flood, maximum depth is clearly sensitive to both tg and wg; while for the 48 h flood maximum depth is much more sensitive to tg than wg. Fig. 5 illustrates the importance of optimizing the configuration of the gate diversion, particularly the timing of the gate action. For example, no flood stage reduction is accomplished if the gate is opened too late. This highlights an engineering challenge: correctly configuring the diversion to enhance, not inhibit, flood stage reduction. Model results for weir control and passive control are presented in Fig. 6, and show that weir control is not as
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effective as gate control for flood stage reduction. There appears to be a minor advantage to weir control over passive control when the off-line area is very small, in this case less than 1 · 104 m2; otherwise, there is little benefit. For this reason, weir control results are not examined further. The preceding results for passive control and gate control collapse onto a single set of axes by non-dimensionalizing the flood depth and flood plain area as follows, h ¼
hmax h0max
and
A ¼
Ah0max 1 t Q 2 d p
ð13Þ
where h0max represents the peak flood depth in the absence of a diversion. Physically, A corresponds to the ratio of the off-line storage capacity to the total flood runoff, so A ¼ 0 implies that no runoff is diverted while A ¼ 1 implies that the entire hydrograph was diverted. h indicates peak flood depth at the monitoring station relative to what would occur in the absence of control. For example, h = 1 implies no flood depth reduction compared to the do-nothing scenario. Fig. 7 shows the passive control and gate control results in terms of A versus h . For passive control, h is only dependent upon A when A < 0:1; while for A > 0:5, h is nearly independent of A . The line h ¼ 1 0:3A was manually fitted to the envelope of
Fig. 5. Contours of maximum depth hmax in meters at STA 653 for A ¼ 1 105 m2 plotted versus the diversion channel width wb and the timing of the gate action tg.
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Max. Depth at STA 653
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6 5
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x 10
Fig. 6. Maximum depth hmax at STA 653 plotted versus reservoir area A for passive control and weir control. For each value of A, hmax from passive control varies depending on the diversion channel width wb, while hmax from weir control varies depending on wb and weir height zw.
passive control results for A < 0:2 (Fig. 7a), and presented in comparison to the gate control results in Fig. 7b. This figure illustrates that gate control is capable of achieving 2–3 times the flood stage reduction of passive control when A < 0:2, but for A > 0:2 the potential for improved stage reduction diminishes with increasing A . These results also show that the effectiveness of off-line diversions is not limited by flood duration, only that longer duration floods require a proportionally larger reservoir to accomplish the same level of flood stage reduction as a shorter flood. Whether gate control performs better than passive control depends on the values of wd and tg, which correspond to the diversion width and the timing of the gate action. (Recall Fig. 5, which illustrates the sensitivity of flood stage to these parameters.) Fig. 7b shows there are many results both above and below the line corresponding to the envelope of gate control results. To determine the conditions under which gate control enhances (not inhibits) flood stage reduction, the values of wd and tg leading to the smallest values of hmax for each value of A were identified. Fig. 8a and b presents these results as dimensionless gate width w = wd/W and dimensionless lead time dt = 12(tr tg)/td, respectively, versus dimensionless off-line area, where W is a reference channel width set to 10 m and tr is the duration of the rising limb of the hydrograph. This figure shows that the opti-
mal width and optimal lead time increase with flood plain area for 0 6 A 6 0.05. The finite set of data points limits the precision with which optimal parameters can be identified, but the general trend is clearly evident. For A > 0:05, w = 1 and dt = 1 appear optimal. The significance of the latter result is not clear because w = 1 represents a limit of the parameter range used in this study. Fig. 8c presents h resulting from optimal values of w and dt , highlighting the advantage of optimal gate control over passive control. For example, if the reservoir can capture roughly 2% of the flood (A ¼ 0:02), flood depth is reduced less than 1% by passive control but 2–4% by gate control. Based on a peak flood depth of 7 m, this corresponds to a 7 and 14–28 cm reduction, respectively. The problem of optimizing diversions can be interpreted from a wave interference perspective. Taking the flood to be a positive wave, and assuming the diversion creates two depression (or negative) waves consistent with one-dimensional shallow-water hydrodynamics, the challenge is for the negative waves to be of sufficient amplitude and duration to cancel out or at least mitigate peak stages of the flood. The amplitude of the negative waves is controlled by the diversion rate, or wb in the case of gate control; while the duration of the wave controlled by both the off-line storage and the diversion rate, A and wb, respectively, in the case of gate control. Assuming
B.F. Sanders et al. / Advances in Water Resources 29 (2006) 861–871
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6 hr 12 hr 24 hr 48 hr
0.98 0.96 0.94
*
h , Passive Control
1
h*=1−0.3A*
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0
0.5
1
1.5
A* 1 6 hr 12 hr 24 hr 48 hr
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h =1−0.3A*
0.94
*
h , Gate Control
0.98
0.92
(b) 0.9
0
0.05
0.1
0.15
0.2
0.25
A*
Fig. 7. Dimensionless peak flood stage h at STA 653 versus dimensionless reservoir area A for passive control (a) and gate control (b) scenarios. A can be interpreted as the fraction of the total flood runoff that is captured by the reservoir. For passive control note that for A < 0:1, flood stage reduction is only dependent upon A ; while for A > 0:5 flood stage reduction becomes nearly independent of A . The line h ¼ 1 0:3A is fitted to the envelope of passive control results (a), and presented along with the gate control results (b). Note that gate control results fall both above and below the line h ¼ 1 0:3A , highlighting the potential to improve, but the risk to impair, flood mitigation efforts compared to passive control.
that A is relatively small, which it must be if there is an inundation concern in the first place, then there is a tradeoff between the amplitude and duration of the negative wave. In one extreme, the diversion might lower flood stage by 10% or more, but only for a few minutes, after which flood stage rises again to its previous level as if there was no control at all. In the other extreme, the diversion might be sustained for the whole duration of the flood, but at a rate so slow that flood stage is never lowered by more than a fraction of a percent. Hence, optimal control exists somewhere in between these limits, and it involves a diversion rate that is large enough to lower flood stage by a few percent and a duration long enough to encompass the cresting period of the flood. Whether or not the wave action is dynamic in this application is not clear, for in relatively steep channels the momentum balance in the streamwise direction is less sensitive to inertial effects than in mild channels. Model predictions of maximum depth shown in Figs. 4 and 5 reveal that peak depth, in the absence of control, varies slightly depending upon the event duration so the flood is certainly not a kinematic wave. For example, peak depth at STA 653 is predicted to be roughly 6.9 m for the 6 h event and 7.0 m for the 48 h event.
Work by Jaffe and Sanders [7] suggested that off-line diversions could be engineered to crop the peak stage of relatively short duration floods via either regressive or progressive depression waves, but results of the San Francisquito application indicate longer floods in steep channels are equally controllable using a proportionally larger off-line reservoir. This suggests that it is the amplitude and duration of the depression waves that are most important for control by progressive depression waves in steep channels; whether the flood is a dynamic wave or simply driven by gravity and friction has less bearing on the controllability of flood stage. The problem of engineering a diversion for optimal flood mitigation remains a great challenge. Here, we benefit from a priori knowledge of the flood wave, while in practice this must be forecast in real-time. We also assume that the reservoir is initially empty before the flood, which may not necessarily be the case in the event of successive storms which yield multi-modal hydrographs. Nevertheless, results of this study suggest that gate control is as or more effective than passive control so long as the diversion commences before peak flood stage reaches the diversion point. Here we considered a triangular hydrograph with symmetrical rising and falling limbs, and the optimal lead time was found to
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B.F. Sanders et al. / Advances in Water Resources 29 (2006) 861–871 2 6 hr 12 hr 24 hr 48 hr
(a) Opt. w
*
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*
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0
1 Passive Control Opt. h
*
h*=1−0.3A*
0.95
(c) 0.9
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
A*
Fig. 8. Values of dimensionless diversion width w (a) and dimensionless lead time dt (b) that minimize the dimensionless peak flood stage h at STA 653 for each value of dimensionless off-line area A . Values of h at STA 653 resulting from optimal values of w and dt are presented in (c), along with the line that corresponds to envelope of passive control results.
be roughly tr/12; this may be different depending upon the shape of the hydrograph which highlights the need for watershed- and event-specific engineering. In the limit that diversion commences too soon, gate control simply performs as well as passive control; whereas in the limit that diversions commence too late, passive control performs better. Hence, it is critical that diversions be properly timed. Real-time monitoring and forecasting of flood conditions is becoming increasingly common and according to a report by the National Hydrologic Warning Council (NHWC), there is strong evidence that such systems save lives and reduce flood damages [4]. Since the effectiveness of gate control hinges on the lead time of the gate action, a key question is whether the lead time of forecasts is greater than the lead time for gate action. According to the NHWC, ‘‘crest-stage forecasts can be made a few hours in advance for cities and communities along streams draining small basins, but they can be made two weeks or more in advance . . . along rivers draining large basins’’ [4]. Taking San Francisquito Creek as a ‘‘small basin’’, it would appear that the lead time for forecasts is sufficient to support optimal gate action. However, to error on the side of caution, compensating for uncertainty in flood hydrograph predictions, decision
support systems should be biased towards diversions that begin earlier than optimal, by an amount that scales with the degree of uncertainty.
5. Summary and conclusions Communities must consider a wide variety of structural and non-structural measures to mitigate flood risks. Among structural measures, off-line reservoirs are an attractive alternative to in-line reservoirs for many environmental reasons, mainly because the former does not create a barrier to sediment transport and fish migration or permanently flood riparian habitat. Based on conditions in a O(102) km2 watershed in northern California, active control of diversions to an off-line reservoir creates an opportunity to at least double the flood depth reduction accomplished by passive control. The active control scenario involves use of a gate which remains closed until slightly before peak flood stage reaches the diversion structure, at which point it opens creating a dam-break like flow into an off-line storage area that quickly lowers river stage. For this control to be effective, the gate action must be optimized relative to the flood hydrograph. In this
B.F. Sanders et al. / Advances in Water Resources 29 (2006) 861–871
study, a triangular hydrograph was used and the optimal time to open the gate was slightly before peak flood stage reaches the diversion structure, by a duration that increases with the volumetric capacity of the reservoir. The added benefit of active control over passive control is maximum when the reservoir captures less than 20% of the flood volume. As off-line storage increases beyond this value, the benefit of active control over passive control diminishes. Finally, floods with relatively short and long periods are equally controllable by an off-line diversion, though the latter requires a proportionally larger reservoir to accomplish the same level of control. Accurate forecasts of flood hydrographs are needed to identify and implement optimal diversion strategies. The lead time typical of operational forecasting systems appears sufficient to make decisions regarding gate control actions, but with little time to spare. Poorly designed or implemented active control scenarios may cause more flooding than would otherwise occur if passive control were adopted. This highlights the risk of operational complexity in the context of flood controllability. In circumstances where the potential rewards outweigh the risk, advanced flood forecasting and decision support systems are needed to optimize the placement, configuration, and operation of flood control infrastructure such as diversion basins, pumps, gates, weirs and other control facilities. These systems should error on the side of an early gate action to compensate for uncertainty in flood forecasts. Hydrodynamic models should be validated before being used to engineer diversion strategies. Validation should focus on the skill with which the model accurately routes flood waves through the watershed and accurately predicts flood stage, and the ability of the model to resolve the wave action resulting from dambreak type diversions into off-line reservoirs. In smaller watersheds data such as these are usually not available and in such cases monitoring programs should first be instituted.
Acknowledgements This work was supported by a grant from the National Science Foundation (CMS-9984579), whose support is gratefully acknowledged.
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