# A Cost-Effectiveness Protocol for Flood-Mitigation Plans Based on Leeds’ Boxing Day 2015 Floods

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## Abstract

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## 1. Introduction

## 2. Tool: Flood-Excess Volume (FEV)

## 3. Data: FEV Revisited for the River Aire Boxing Day 2015 Flood

## 4. Main Result: Square-Lake Cost-Effectiveness Protocol for Flood Mitigation

#### 4.1. Background and Existing Non-FEV-Based Information

- (i)
- The basic scheme aims to protect Leeds against flooding events with a 1:200-year return period.
- (ii)
- Natural flood management (NFM) will be used to offset increased flood risk due to climate change. In the upper catchment, NFM will include the re-meandering of the River Aire and its tributaries and the planting of trees to increase tree coverage in the catchment from $7\%$ to $15\%$. Further discussion of NFM and on the flood-mitigation effects of tree planting is deferred to Appendix B.
- (iii)
- Certain constrictions in the river course causing flow stowage at floods, recall Figure 2, will be removed within Leeds. These constrictions include, for example, narrow river passages formed by derelict or abandoned bridge structures. In addition, some river stretches are widened thus giving-room-to-the-river (GRR), with limited quantification.
- (iv)
- Two flood plains, approximately seven miles upstream from Leeds, at Calverley and at Rodley, are considered to enhance flood-water storage, using adjustable weirs. Estimated storages at Calverley and Rodley are respectively ${V}_{Cal1}=1.00\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ and ${V}_{Rod1}=2.20\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$.
- (v)
- Higher flood-defence walls will be used in Leeds, with varying heights at different locations, depending on the inclusion of: (a) only the highest flood-defence walls, higher walls with (b) the Calverley flood-storage area, (c) only the Rodley one or (d) both the Calverley and Rodley flood-storage areas. A breakdown of the height of the defence walls of these four options is given in a table ([21], §3.5.1), each option giving protection against floods with a 1:200-year return period, presumably based on computer simulations of the river hydraulics in such a flood event.
- (vi)
- FASI will be updated to provide increased protection against floods; specifically, up from a 1:100- to a 1:200-year return period. (In [20] it is noted that “These storage areas used in times of flood would achieve reductions in downstream defence heights whilst providing residual benefits to the Phase One scheme.” and that “Crucially, by progressing a Phase Two scheme consisting of Natural Flood Management, floodwater Storage Areas and Removal of Obstructions, it’s expected that the standard of protection of the Phase One area will be uplifted to a 1 in 200 standard of protection, effectively delivering the third stage of the phased approach of the whole programme early.” Given the lack of clarity, we hypothesise that some wall-height increases and/or GRR/flood-plain storage may be required downstream of the Dark Arches.)
- (vii)
- Potential enhancement of flood-storage sites further upstream, involving adjustable weirs as well, at the Cononley Washlands near Skipton and Holden Park near Keighley, about 42 km upstream from Leeds, which both have substantially larger flood-storage volumes, has been dismissed because they were deemed too far away from Leeds and would thus not be able to protect against flooding caused by extreme rainfall nearby Leeds.
- (viii)
- The only costs mentioned are $\pounds 109\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ for the combined case of flood protection with higher flood-defence walls in Leeds and the enhanced Calverley flood-storage area. When also the Rodley area is included, the costs become $\pounds 123\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$, whence the costs for the enhancement of the Rodley flood plains are inferred to be $\pounds 14\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$.

#### 4.1.1. Roadmap of Flood-Mitigation Diagnostic for Numerical Simulations

- A comparison of the flood level and a numerical simulation of a hydraulic/hydrological model (or ensemble thereof) in a three-panel hydrograph, such as the one in Figure 4, augmented with additional curves of the 1:200${}^{+}$-year return-period Boxing Day flood 2015; this is to demonstrate and to communicate confidence in the simulation tool used. (This matches a remark in [20] “A review and update of the development of hydraulic/hydrological models alongside data collected since Boxing Day to inform an options appraisal, and fully assess the extent of a proposed scheme area”, which comparison/development we have not seen to date.)
- A reference simulation (or an ensemble thereof) for a 1:200-year return-period event to establish the corresponding FEV(s) for such an event at the most critical location(s) between Kirkstall and the “Dark Arches” weir, as well as at the Armley River gauge, without any new flood defences. Note the discrepancy between step I and step II in that this simulation in step II is required because LCC chose to protect against 1:200-year return-period floods events rather than the actual higher-magnitude 1:200${}^{+}$-year Boxing-Day flood event.
- Another reference or the base simulation (or an ensemble thereof) including the effects of GRR and river-bed clearances and corresponding FEV(s).
- Simulations (or an ensemble thereof) of various flood-mitigation scenarios and their FEV reductions relative to the reference FEVs in II and III. In these simulations, (infinitely) high walls are used extending current natural or flood walls, in order to determine whether, where and at what height new flood defence walls are required in the different scenarios.
- Square-lake graphs of cost-effectiveness analyses of these various scenarios relative to the reference FEVs in II and III. Alternative, one can decide to iterate and go back to an update, using GRR and/or river-bed clearances, of step III.

#### 4.1.2. Remarks and Simplifying Assumptions for FASII${}^{+}$

- Per (i),
- the Boxing Day flood of 2015 was an extreme flood event with a 1:200${}^{+}$-year return period. FASII, designed to protect against a flood with a 1:200-year return period, does not therefore protect against a future Boxing-Day-type flood in Leeds. A new Boxing-Day-type flood would consequently overtop the higher defence walls proposed in FASII, which will need sluice gates that can channel floodwaters back into the river once river levels are subsiding.
- Per (ii),
- NFM contributes $0\%$ to the basic 1:200-year return-period flood protection without climate-change effects being taken into account (see Appendix B). Given new information (personal communication with EA February 2019), NFM’s contribution will be $(3\pm 2)\%$, used as extra flood mitigation against climate-change uplift, here taken at a $5\%$ maximum. The errors of $2\%$ are our guess of the uncertainty, used here for the purpose of illustration.
- Per (iii),
- removing constrictions and river-bed widening (i.e., GRR) will alter the FEV calculation at Armley. While the FEV (9) of ${V}_{e}=9.34\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ concerns a 1:200${}^{+}$-year return-period event, we will use it as the FEV for FASII${}^{+}$ with its 1:200-year return period, either by ignoring the difference in these FEVs (between a 1:200-year and 1:200${}^{+}$-year event) or by marginally reducing the threshold ${h}_{T}$ in such a way that it precisely compensates for the lowered peak flow. In engineering practice, this could be corrected by calculating the FEV corresponding to a computer-simulated flood in Leeds for the target flood, or an ensemble of target floods, with a 1:200-year return period (cf. steps II and III in Section 4.1.1). FEV has not been used as a tool in FASII, and data from a reference simulation for a 1:200-year return-period flood event are not available. As an illustration, river-bed widening is estimated to occur for river levels above ${z}_{GRR}\approx 1.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ over a width of ${w}_{GRR}=5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ for a transverse bankslope ${b}_{s}=2$ (note that [20] mentions intermittent strips of ${W}_{GRR}=10,20,30\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ and we chose a conservative yet continuous width estimate for the purposes of illustration). The river slope is estimated to be ${S}_{0}=0.0005$ (the river drops about $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ using height contours on the map nearest to the river of $50\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ to $40\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ between Kirkstall Forge and Armley with three weirs of circa $2.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ drops, so the effective drop is $2.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ over circa 5 km, yielding the estimate). The Manning coefficient used is ${C}_{m}=0.04$. Altogether this yields the following rating-curve correction, see also [18],$$\begin{array}{c}\hfill {\tilde{Q}}_{GRR}\left(\overline{h}\right)=Q\left(\overline{h}\right)+\frac{\sqrt{{S}_{0}}}{{C}_{m}}\frac{{\left(\right)}^{(\overline{h}-{z}_{GRR})}}{5/3}{\left(\right)}^{{w}_{GRR}}2/3& ,\end{array}$$$${V}_{eGRR}=8.69\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3},$$
- Per (v),
- building higher walls will also change the rating curve and the FEV at Armley station but we expect the changes to be small given that “overbanking” was reported to be relatively small at Armley [14]. Hence, for simplicity and without loss of generality in establishing our cost-effectiveness protocol, we keep our original FEV of ${V}_{e}=9.34\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ for also the 1:200-year return-period flood event reduced to ${V}_{eGRR}=8.69\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$, cf. (11). We can ignore the influence of FASI in (vi) on FASII${}^{+}$, given that the weir under the railway station acts as a control point, even in flood conditions, cf. Figure 3. Some of the breakdown in FASII is reproduced in Table 3 for two locations with the highest flood-defence walls proposed; Table 3 has moreover been extended with extra information on the percentages gained from, and a comparison with, the FEV fraction alleviated based on ${V}_{e}$, cf. (9), at the Armley station, as well as its GRR-reduction ${V}_{eGRR}$. The reference simulation with walls only for the 1:200-year return-period event should be one in which removal of constrictions and local river-bed widening (GRR) is already included. In FASII, some kind of manual optimisation of weir control must have been included, which led to the reduced wall heights in Table 3. However, information and hydrographs for these simulations have to date been unavailable.
- Per (vii),
- for the Boxing Day flood in 2015, the peak flow at Kildwick was about $163\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}/\mathrm{s}$ with the peak flow at Armley being $344\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}/\mathrm{s}$, cf. Reference [9]. A significant tentative reduction of the discharge at or near Kildwick, found by enhancing the flood-storage capacity at the Cononley Washlands near Skipton upstream of Kildwick as well as the Holden Park flood plain near Keighley, would reduce the inflow and therefore increase the flood resilience in Leeds. For the Boxing Day flood in 2015 with its peak rainfall around Bingley and Bradford downstream of Kildwick, a reduction by $26\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}/\mathrm{s}$ over ${T}_{f}=32\phantom{\rule{3.33333pt}{0ex}}\mathrm{h}$, yielding $26\times 32\times 3600=3.00\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ in volume, would yield a roughly similar reduction at Armley and a lowering of the peak levels by $(5.22-4.96)\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}=0.26\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (obtained by using the Armley river-gauge data). These river levels follow from the available measurements at Armley. (We tacitly assumed here a linear rating curve, for simplicity, but the argument can be updated using the measurements and rating curve at Kildwick.) It is noteworthy that this appreciable lowering of peak levels contrasts with the arguments of LCC in [21], which dismiss upstream storage sites because they are deemed to be too far away. Morever, while the argument discarding the use of flood storage further upstream is used by LCC to dismiss such upstream storage sites, it is inconsistenly not used to similarly discount the much less significant NFM flood-alleviation measures far upstream (see Appendix B).
- Per (viii),
- given the lack of background on further costings of FASII, we will introduce credible figures and create a hypothetical flood-alleviation scheme FASII${}^{+}$ to illustrate our new, generic methodology and protocol for analysing flood-alleviation schemes using FEV analysis. This implies that no inferences other than methodological ones can be drawn from what follows, since, of course, our hypothetical FASII${}^{+}$ does not apply directly to the real FASII.

#### 4.2. Scenarios with Available Flood-Storage Volume on Flood Plains

#### 4.3. FEV-Based Cost-Effectiveness Analysis

- –
- Calverley’s flood-storage enhancements gained by building an adjustable weir for use in extreme flood events will cost $\pounds 10\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ (including both a flood-warning system and compensation against loss of farming income over a certain period), given that the (slightly larger) Rodley flood-plain enhancement, also including an adjustable weir, costs $\pounds 14\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$. This is approximately (see Table 3) $\pounds 10\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/7\%=\pounds 1.43\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/1\%$ (i.e., $\pounds 1.43\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ per one percent of flood protection). The maximum storage of $11\%$ under optimal weir operation leads to additional storage of circa $4\%$ (11%–7%) used against climate-change uplift or more generally against occurrence of higher-magnitude events.
- –
- Rodley’s flood-plain enhancement is approximately $\pounds 14\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/11\%=\pounds 1.27\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/1\%$ (i.e., £$1.27\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ per one percent of flood protection). The maximum storage of $24\%$ under optimal weir operation leads to additional storage of circa $13\%$ (≈24%–11%) used against climate-change uplift.
- –
- Calverley and Rodley’s flood-plain enhancements, in combination, are approximately $\pounds 24\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/13\%=\pounds 1.85\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/1\%$. It is noteworthy that, given the results of the hydraulic simulations and the figures in [21], together with the data in Table 3, the discrepancy of circa $6\%$ between the sum of the independent reductions and the total reduction of the combined sites remains unexplained to date. One possible explanation for the discrepancy is that the volumes of ${V}_{Cal1}=1.0\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ and ${V}_{Rod1}=2.2\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ were total and not available flood volumes and another one is that the wall-heights and corresponding flood-volume reductions in the simulations performed are suboptimal because the adjustable weirs have been operated suboptimally, with the maximum percentages of FEV-reduction unreached. We will therefore consider the following variable ranges of the respective FEV fractions: for Rodley $[11,25]\%$; Calverley $[7,11]\%$; and, Calverley and Rodley $[13,34]\%$, in which only the lower bounds will be used for the basic flood-mitigation scheme while any additional volume is used to offset climate-change uplift. The maximum storage of $34\%$ under optimal operation of both weirs leads to additional storage of circa $21\%$ (34%–13%) used against climate-change uplift.
- –
- NFM has been updated to include leaky dams as well as tree planting at a total cost of $\pounds 8\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$. The costs towards flood mitigation are budgeted at $\pounds 4\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$, while another $\pounds 4\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ concerns compensation costs allocated to nature restoration and leisure activity in the area. NFM will be used to offset climate change uptake at $(3\pm 2)\%$, with the error estimate by necessity having been guessed pending further ensemble simulations by the EA to date (personal communication 2019 with the EA).
- –
- FASI updates to a 1:200-year flood protection are costed at $\pounds 14\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$.
- –
- Constrictions to be removed as well as river-bed widening or GRR are costed at $\pounds 10\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ for a $7\%$ FEV reduction, so $\pounds 1.43\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/\%$, cf. (10) and (11). The river stretch between Kirkstall and the railway station is quite urbanised with old and new build-up relatively close to the river. Nonetheless, there are ample new opportunities to invest intermittently in river-bed widening, at and upstream of the Armley Museum and upstream of the Dark Arches, by converting abandoned lots in either low-lying park- and/or wetland and via modern land development in which the houses, offices or businesses are raised on one-to-two-metre mounts for flood protection. The soil for these mounts can come from parking lots, roads and berms that have been lowered by excavation. The latter exemplifies responsible building in flood plains for modern cities. Given the intermittent nature of these adaptations, we have heuristically adapted the rating curve, leading to the aforementioned $7\%$ reduction of the FEV.
- –
- 85 beaver colonies in the Aire catchment headwaters. In [23] is reported enhanced flood-water storage of circa ${V}_{b}=1100\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ behind 13 beaver dams in a beaver colony spanning $200\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. With 85 beaver colonies in the headwater of the Aire catchment, a best estimate then yields a $85\times {V}_{b}/{V}_{e}=85\times 1100/(9.34\times {10}^{6})\approx 1\%$ reduction of the FEV [24,25,26]. Given the uncertainties in flood protection, we estimate the contribution to be $[0,1]\%$ for $\pounds 1\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$, exclusively used for climate-change uplift. Additional benefits are wildlife enhancement and water-quality increase. The colonies would be managed in parallel, to avoid flood waves when dams collapse, and are not fenced in.
- –
- Higher defence-wall costs are deduced from the above costings and the stated total cost of FASII of $\pounds 109\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$, for higher walls, constriction removal and the Calverley flood storage inclusive; each has a respective relative percentage of flood protection of $86\%$, $7\%$ and $7\%$, hence the costs of the case with $100\%$ higher flood-defence walls is estimated to be $\pounds (109-10-10-14-10)\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/0.86=\pounds 75.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ (i.e., total costs minus cost constriction removal, FASI update, weir construction and “old” NFM costs). This then yields $\pounds 75.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/86\%=\pounds 0.88\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/1\%$ (i.e., costs per one percent) of flood protection by higher flood-defence walls.
- –
- Upstream flood-plain storage at Cononley Washlands and Holden Park, enhanced, total costs taken at $\pounds 35\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ (inclusive); this figure is based on FASI, which includes two very advanced adjustable weirs, higher defence walls and removal of a spit of land between the River Aire and the Aire-Calder navigation, and which costed $\pounds 50\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$, see Reference [27]. Hence, this yields $\pounds 35\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/45\%=\pounds 0.78\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}/1\%$.

#### 4.3.1. Illustration of Protocol on Hypothetical Scenarios

- – S1:
- the circa $7\%$ extra Calverley flood storage and reduced higher defence walls with GRR and extra NFM;
- – S2:
- the circa $11\%$ extra Rodley flood storage and further-reduced higher walls with GRR and extra NFM;
- – S3:
- the circa $13\%$ extra Calverley and Rodley flood-storage and even-more-reduced higher walls with GRR and extra NFM;
- – S4:
- the extra Cononley Washlands and Holden Park $45\%$ flood-storage sites and the most-reduced higher walls with GRR and extra NFM; these replace the Rodley and Calverley (combined) sites, the latter which are most expensive per percentage storage gained.

#### 4.3.2. Limitations of FEV-Based Approach

- Zeroth-order approach: Three-dimensional flood dynamics is reduced to the analysis of FEV at or near the most critical point along a river where flooding starts. Generally river hydraulics are modelled in a one- or two-dimensional manner: it is therefore best to consider the zeroth-order FEV-analysis as a diagnostic at the worst spot.
- Retention: Only the averaged and cumulative effects of retention measures far upstream of the point of FEV-analysis are considered. Spatio-temporal considerations en route to the most critical point of flooding are thus ignored.
- Costs and effectiveness: Only effectiveness is considered here but not benefits, which would require a full economic analysis of damages saved and/or costs incurred, for example, for urban planning. In addition, maintenance aspects should be included more prominently.

## 5. Summary and Discussion

- FEV can be used to diagnose detailed hydraulic-flow calculations a posteriori; rather than using a measured flood hydrograph, one can first compute a reference-flood hydrograph and an associated FEV (or a range of such volumes for a range of thresholds), and then express calculations of scenarios with various flood-mitigation measures relative to this reference-flood hydrograph as ($100\%$) reductions of the associated FEV; such an approach can also be explored in a probabilistic manner by using ensemble calculations for a distribution of reference-flood hydrographs with different return periods, with FEVs calculated and compared at various critical spatial locations—see the roadmap in Section 4.1.1;
- FEV can be used as a complementary means of classifying flood events; flood hydrographs can be narrow, high and low-volume or broad, relatively high and high-volume, each with vastly different FEVs; for flood mitigation it is meaningful to reclassify return periods for river floods with sufficiently high peak levels in terms of FEV rather than in terms of only river-peak levels; and, this will be meaningful for only floods with peaks surpassing certain threshold levels ${h}_{T}$;
- FEV can play a central role in defining a new protocol to optimise the assessment of flood-mitigation scenarios, including a cost-effectiveness analysis; it may prove beneficial in certifying such a protocol in flood-mitigation handbooks. It is interesting to note that in 2019 a modified scenario S1 was chosen by LCC as flood-alleviation plan, including higher walls augmented with some wetland creation around Kirkstall, GRR between Kirkstall weir and the Dark Arches weirs at Leeds’ railroad station (cf. Figure 1) as well as enhanced dynamics flood-storage at Calverley [20]. NFM measures planned are expected to offer circa $5\%$ extra flood mitigation against climate-change uplift. Enhanced flood-plain storage far upstream of Leeds, cf. S4, has been dismissed; these have already provided some flood reduction for Leeds and upstream communities, cf. remarks in [30]. Enhanced storage at Rodley, close to Leeds, cf. scenarios S2 and S3, has been dismissed; the (formerly) optimal location for a dynamic weir at the narrow end of Rodley valley was moreover occupied by dwellings built in 2019.
- Finally, the square-lake cost-effectiveness protocol lends itself well to “gamification” [31,32], that is, integrating game elements into science education. This is recognised as a powerful way to engage and to inform those who may not be scientifically-literate, such as the general public and policy-makers, but nonetheless seek and would benefit from improved understanding. In particular, the concept and subsequent gamification of the so-called “stabilization wedges” [33] has proved successful in climate science. It is vital that both citizens and politicians understand how to “investigate, evaluate, and comprehend science content, processes, and products” [31], including flood-mitigation; inspired by the stabilization wedges, we intend to develop a game to communicate the efficacy and costs of various flood-mitigation scenarios, not only to inform the public but also to encourage evidence-based decision-making.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FEV | Flood-excess volume |

EA | Environment Agency |

UK | United Kindom |

FASI | Leeds’ flood alleviation Scheme I |

FASII | Leeds’ flood alleviation Scheme II |

GRR | Giving-room-to-the-river |

LCC | Leeds City Council |

NFM | Natural Flood Management |

HW | Higher walls |

FPS | Flood-plain storage |

## Appendix A. Propagation of Error Due to Rating-Curve Uncertainty

**Figure A1.**Flood duration as an implicit function of threshold discharge, ${T}_{f}={T}_{f}\left({Q}_{T}\right)$ (black line), for the Boxing Day flood at Armley gauge station. The red line shows a linear approximation local to ${Q}_{T}=219\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}/\mathrm{s}$, the gradient of which is used to approximate $\partial {T}_{f}/\partial {Q}_{T}$ in the FEV error calculation (A3). In this case, $\partial {T}_{f}/\partial {Q}_{T}\approx -564\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$.

## Appendix B. Bounds on Flood-Mitigation by Tree Planting

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**Figure 1.**Environment Agency (EA) flood-risk maps of the sectors Kildwick (top left in upper map) to Apperley Bridge (yellow marker in both maps) and Apperley Bridge to Leeds (bottom right in lower map). Available online at https://flood-map-for-planning.service.gov.uk, for example, search (easting, northing) = (419,052, 437,681) for Apperley Bridge. Extra flood storage may be available at locations 1, 2, 3 and 4 on the lower map, respectively corresponding to the top-left, top-right, bottom-left and bottom-right photographs in Figure 11.

**Figure 2.**Boxing Day 2015 flooding levels of the River Aire near Apperley Bridge (river near the yellow marker in Figure 1; the old bridge just upstream of the modern A658-road bridge).

**Top**: the filled flood plain upstream of the bridge constriction at 09:56:04 on 26 December 2015.

**Bottom**: the flow under Apperley Bridge at circa 09:58:30 on the same date, showing the dangerous proximity to choking, akin to experiments in [11]. Photo and video-grab courtesy of O.B.

**Figure 3.**

**Top left**: plaques at Leeds’ Armley Mills Museum indicate 1866 and 2015 flood-peak levels, the latter associated with the peak level of $5.22$ m at the Armley river-level gauge. Craig Duguid (Fluid Dynamics’ Centre for Doctoral Training, Leeds) provides a sense of scale.

**Top right**: high level and supercritical flow in the “Dark Arches” underneath Leeds railway station at 13:36:04 on 27 December 2015; the high-water mark on the wall outlines the peak level, at 13:36:36 on 27 December 2015.

**Bottom left**: subcritical flow emanating from these arches at 13:29:42 on 27 December 2015.

**Bottom right**: entrance and top of the weir at these arches on 19 February 2018 with subcritical inflow. Photos and video-grabs courtesy of O.B.

**Figure 4.**The rating curve including errors (solid line, top-left panel) and its linear approximation (dashed line) are displayed as well as river levels (lower-left panel) and peak-flow rates including errors (top-right panel) of the River Aire at Armley, Leeds, around Boxing Day 2015. Dotted lines indicate a chosen threshold, the corresponding discharge and a mean river level. The flood-excess volume (FEV) is found by determining the blue-shaded “area”, as in (2), under the discharge curve $Q\left(t\right)=Q\left(\overline{h}\right)=Q\left(\right)open="("\; close=")">\overline{h}\left(t\right)$ displayed on the vertical axis as a function of time t on the horizontal axis. It involves the in-situ river level $\overline{h}=\overline{h}\left(t\right)$ as a function of time t above a threshold discharge ${Q}_{T}=Q\left({h}_{T}\right)$. Given a rating curve, ${Q}_{T}$ is fixed once a chosen threshold river level, here ${h}_{T}=3.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, has been provided. The flood duration for threshold ${h}_{T}$ is ${T}_{f}=32\phantom{\rule{3.33333pt}{0ex}}\mathrm{h}$. The rectangle represents the mean FEV, ${V}_{e}\approx (\overline{Q}\left({h}_{T}\right)-{Q}_{T}){T}_{f}=9.34\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}/\mathrm{s}$ with the hatched “area” defined in (3). The dashed-line correction to the rating curve concerns giving-room-to-the-river (GRR) or river-bed widening. It leads to a higher threshold ${Q}_{TGRR}$ with a reduced FEV for the same river-level threshold ${h}_{T}$. The grey shading denotes the error in Q due to rating-curve uncertainty (not shown for the GRR curve). Data from EA.

**Figure 5.**Flooding starts at Kirkstall on 26 December 2015 at 12:18 with the Armley river level recorded at $4.16$ m, which motivates one to take a threshold river level at ${h}_{T}=3.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Photo courtesy Jessica Worsey.

**Figure 6.**By using Gaugemap (http://www.gaugemap.co.uk) at the Armley station, one can quickly estimate the river level of the mean area indicated by the top rectangle above ${h}_{T}=3.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, which yields a mean height at the arrow of about $4.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$; by using the online time slider, one can also also find the duration ${T}_{f}\approx 32\phantom{\rule{3.33333pt}{0ex}}\mathrm{h}$ for which the flood level was above ${h}_{T}$.

**Figure 8.**River Aire flood-excess volumes ${V}_{e}$ (left axis, crosses) and equivalent square-lake sizes (right axis, circles) for various threshold levels ${h}_{T}$, from the Armley data of the Boxing Day 2015 flood.

**Figure 9.**Top: FEV expressed as a square $2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$-deep lake of side-length 2161 m. Bottom: the FEV lake, shallow relative to its side length, viewed from the top. Basic scenario S0 is depicted with higher flood-defence walls and National Flood Management (NFM) giving respectively $100\%$ and $0\%$ protection (see Appendix B) against the desired 1:200-year return-period flood event. Indicated in the square lake are the relative size of each flood-mitigation measure, expressed both as lake area and the length of double-ended arrows, and its costs in UK pounds sterling. Split and total costs are indicated in absolute terms and per percentage of the total flood mitigation gained. Total costs (excluding NFM), here equalling the protection offered by using only higher flood-defence walls, are indicated underneath the double-ended arrow spanning the width of the entire flood lake.

**Figure 10.**Scenarios S1 to S4 with contribution of each flood-mitigation measure proportional to the length of the (inverted) double-headed arrows. Contributions to the square flood-excess lake (of the same capacity as the FEV) are: higher walls with wall-heights indicated (HW—mauve); river-bed widening (GRR—red); and (enhanced) flood-plain storage (FPS—green). The uncertain, slanted parts of FPS, natural flood management (NFM—pink) and beaver colonies (small sliver—blue) are used as extra flood mitigation to offset climate-change effects, beyond the 2161 $\mathrm{m}$ side, with the mean denoted as the vertical dashed line. Above or alongside the double arrow the total percentage of protection for each flood-mitigation measure is given; below the double arrow costs and relative cost per $1\%$ of flood mitigation are stated. Total costs are indicated under the double arrow spanning the lake width.

**Figure 11.**Evidence of extra/limited storage capacity above and beyond the Boxing Day 2015 flood levels for the River Aire; 1. top left: extra storage capacity downstream of Apperley Bridge both upstream of the Victorian railway bridge and downstream (26 December 2015 at 15:27:33); 2. top right: no extra storage capacity at Kirkstall Forge, where a shallow cut has been added to the sharp bend to alleviate flood risk on the visible train tracks; 3. bottom left: extra storage on the Rodley nature reserve flood plain (27 December 2015 at 12:15:05); and, 4. bottom right: downtown flooding just upstream of the station where the Leeds-Liverpool canal could be used as bypass of the “Dark Arches” hydraulic control point under Leeds’ train station (27 December 2015 at circa 13:20:53), shown in Figure 3. Photos and video-grab courtesy of O.B.

**Table 1.**The latest [14] coefficients ${C}_{j},{a}_{j},{b}_{j}$ and limb thresholds ${h}_{0}=0.2$ and ${h}_{j}$ for $j=1,2,3$ for the rating curve at the river-level gauge station at Armley in Leeds.

j | ${\mathit{h}}_{\mathit{j}}$ | ${\mathit{C}}_{\mathit{j}}$ | ${\mathit{a}}_{\mathit{j}}$ | ${\mathit{b}}_{\mathit{j}}$ |
---|---|---|---|---|

$\mathbf{m}$ | ${\mathbf{m}}^{3-{\mathit{b}}_{\mathit{j}}}/\mathbf{s}$ | $\mathbf{m}$ | - | |

1 | 0.685 | 30.69 | 0.156 | 1.115 |

2 | 1.917 | 27.884 | 0.028 | 1.462 |

3 | 4.17 | 30.127 | 0.153 | 1.502 |

**Table 2.**Hypothetical flood-alleviation scheme FASII${}^{+}$: a summary of the various protection measures, their function (qualitative) and capacity (quantitative), and in which scenarios (S0–S4) they feature. A detailed description of each component is given in Section 4.1, Section 4.2 and Section 4.3.

Measure | Function | Capacity | S0 | S1 | S2 | S3 | S4 |
---|---|---|---|---|---|---|---|

Flood walls | Increase local flow capacity (effectively increasing ${h}_{T}$) | Various (depending on scenario), see Table 3 | X | X | X | X | X |

NFM: Natural flood management | Store water in the catchment headwaters | Leaky dams & increase tree coverage by 8% | X | X | X | X | X |

GRR: Giving-room-to- the-river | Increase local flow capacity (by river-bed widening & removing constrictions) | $0.65\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ | X | X | X | X | |

Beavers | Store water in the catchment headwaters | 85 beaver colonies storing $1100\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ each | X | X | X | X | |

Calverley site | Flood-plain storage | Total: $1\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$. Available: $0.7\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$. | X | X | |||

Rodley site | Flood-plain storage | Total: $2.2\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$. Available: $1.0\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$. | X | X | |||

Cononley Washlands & Holden Park sites | Flood-plain storage | $2\times 2.1\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ | X |

**Table 3.**The height of the defence walls proposed in FASII for protection against the 1:200-year flood and height reduction in percentage, along two different river stretches for the cases with only walls, walls with also the Rodley flood storage, or also the Apperley Railway Bridge/Calverley storage, or both. The wall-height reduction is given as a percentage for each location and can be compared against the extra flood-storage reduction percentage with respect to the flood excess volume (FEV) of ${V}_{e}=9.34\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ chosen. Percentages relative to ${V}_{eGRR}=8.69\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mm}}^{3}$ are calculated by us, with an adjustment by a factor ${V}_{eGRR}/{V}_{e}=0.93$ added in parentheses giving the percentage relative to ${V}_{e}$, and wall heights are from a table in [21] based on Leeds City Council’s (LCC’s) calculations/simulations. The underlined percentages are used in FASII${}^{+}$.

Location | Walls Only | Walls Plus Rodley | Walls Plus Calverley | Walls Plus Rodley & Calverley |
---|---|---|---|---|

Whitehall Rd to Wellington Br | 1.74 m | 1.53 m | 1.60 m | 1.50 m |

- | −12% (−11%) | −8% (−7%) | −14% (−13%) | |

Viaduct Rd to Cardigan Fields | 2.27 m | 2.00 m | 2.09 m | 1.96 m |

- | −12% (−11%) | −8% (−7%) | −14% (−13%) | |

Reduction of FEV by upstream retention | $0\%$ | $\frac{2.2}{9.34}=\underline{24\%}$ | $\frac{1}{9.34}=\underline{11\%}$ | $\frac{3.2}{9.34}=\underline{34\%}$ |

($\frac{2.2}{8.69}=25\%)$ | $(\frac{1}{8.69}=12\%)$ | $(\frac{3.2}{8.69}=37\%)$ |

**Table 4.**Table summarising the figures for the five scenarios S0, S1–S4 of FASII${}^{+}$; total costs for FPS (minimum case) are minus the FASI improvements needed, which were costed at $\pounds 14\phantom{\rule{3.33333pt}{0ex}}\mathrm{M}$ for scenario S1. Tentative wall heights are for the stretch including Viaduct Road, cf. Table 3.

GRR % | $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}$ | $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}/\%$ | FPS % | min $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}$ | Case $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}/\%$ | HW % | $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}$ | $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}/\%$ | m | FPS % | Max $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}$ | Case $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}/\%$ | Total $\mathit{\pounds}\phantom{\rule{3.33333pt}{0ex}}\mathbf{M}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

S0 | 0 | - | - | 0 | - | - | 100 | 88 | $0.88$ | $2.44$ | 0 | - | - | 88 |

S1 | 7 | 10 | $1.43$ | 7 | 10 | $1.43$ | 86 | $75.6$ | $0.88$ | $2.10$ | 12 | 10 | $0.83$ | $95.6$ |

S2 | 7 | 10 | $1.43$ | 11 | 14 | $1.27$ | 82 | $72.2$ | $0.88$ | $2.00$ | 24 | 14 | $0.58$ | $96.2$ |

S3 | 7 | 10 | $1.43$ | 13 | 24 | $1.85$ | 80 | $70.4$ | $0.88$ | $1.95$ | 34 | 24 | $0.71$ | $104.4$ |

S4 | 7 | 10 | $1.43$ | 45 | 35 | $0.78$ | 55 | $48.4$ | $0.88$ | $1.34$ | 45 | 35 | $0.78$ | $93.4$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bokhove, O.; Kelmanson, M.A.; Kent, T.; Piton, G.; Tacnet, J.-M.
A Cost-Effectiveness Protocol for Flood-Mitigation Plans Based on Leeds’ Boxing Day 2015 Floods. *Water* **2020**, *12*, 652.
https://doi.org/10.3390/w12030652

**AMA Style**

Bokhove O, Kelmanson MA, Kent T, Piton G, Tacnet J-M.
A Cost-Effectiveness Protocol for Flood-Mitigation Plans Based on Leeds’ Boxing Day 2015 Floods. *Water*. 2020; 12(3):652.
https://doi.org/10.3390/w12030652

**Chicago/Turabian Style**

Bokhove, Onno, Mark A. Kelmanson, Thomas Kent, Guillaume Piton, and Jean-Marc Tacnet.
2020. "A Cost-Effectiveness Protocol for Flood-Mitigation Plans Based on Leeds’ Boxing Day 2015 Floods" *Water* 12, no. 3: 652.
https://doi.org/10.3390/w12030652