There are a number of studies documenting the benefits of beaver dams in terms of habitat improvement, peak flow attenuation and water quality improvements (Puttock et al., 2017, 2018), so it is natural to try and emulate these types of benefits artificially. However, we should also study what happens in nature when things go wrong. Structural failure of natural beaver dams has been reported as occurring frequently by Butler and Malanson (2005), citing numerous cases of dam failure that resulted in outburst floods. These floods have reportedly been “responsible for 13 deaths and numerous injuries, including significant impacts on railway lines”. Engineered NFM measures are likely to be more robust than beaver dams (contingent on maintenance in the longer term), but the relative risks of different configurations, positioning in relation to geometry, slope and proximity to each other, and build design need a mechanism for appraisal. The intention is to help design safer and lower-risk configurations of NFM, which is seen as a potentially low-cost complement to conventional flood risk management strategies.

Failure of beaver dam structures in the US has been reasonably well documented (Hillman, 1998; Butler and Malanson, 2005), and there have been two records (Tom Nisbet, personal communication, 2018) of leaky barrier performance failure at Pickering, UK, to the authors' knowledge, after two large flood events. The first flood event in November 2012 resulted in the washout of one of the larger dams on the main Pickering Beck and a shift to the edge/bank of a second dam below this. These features were relatively tall structures and located within a straightened section of channel alongside a railway line, with limited floodplain storage. The logs from the failed dam were caught within the downstream reach between that and a third dam downstream. The failed and shifted dam plus one other were found to be deflecting flows into the river bank, causing some local scouring, placing a local railway at risk of undercutting so they were removed (2014) and replaced with five new dams on a better reach downstream. The second failure event occurred during the UK Boxing Day floods, 2015, where a total of 11 dams were damaged, all involving a shift/deflection in the dam by edge scour or loss/breakage of top logs, rather than a complete washout. These losses all involved the original, more natural design of cross logs used to construct dams in 2010/11, with no wiring used to secure logs in place. All of these have since been replaced using the now favoured semi-engineered design of horizontal stacked logs secured by wiring (a design also used in Penny Gill – see Fig. 1). Additionally, Addy and Wilkinson (2016) report on complete failure of one structure during a 10 % annual exceedance probability (AEP) event for “engineered log jams” that are albeit designed to trap sediment.

Siting, construction and improvements in engineering design are therefore important, and recent research (Dixon and Sear, 2014) shows logs 2.5 times the channel width provide “near functional immobility” – unlikely to be transported in an extreme event. Such design construction “rules of thumb” can be very useful, but cannot always describe the complexity of the whole-system response, which can be very place-specific, driving the need for a network model that can be rapidly set up to test different situations of the kind described here.

In this paper we explore network issues impacting the three performance issues categorised above, particularly with respect to spatial configuration of leaky barriers in a network that have a probability of failure defined by a fragility curve, an approach commonly used in the systems approach to quantifying flood risk (Hall et al., 2003). The probability of failure is very difficult to define for the range of constructions that are being implemented – and how this varies with age, decay and sedimentation is not known. Thus we attempt to understand what aspects of geometry, slope and proximity are the best trade-offs for a given reasonable assumption about fragility. We later translate this back the real world example on Penny Gill, Cumbria, and the implications for spacing and siting.

We begin by setting up a network model for an arbitrary stream network, breaking the stream up into segments that may each potentially contain leaky barrier designed to attenuate high flows (often referred to generically as runoff attenuation features). Our aim here is to set out a mathematical formulation for the network of features that will enable us to describe and experiment numerically with different configurations of NFM features within a probabilistic analysis. The model is based on a consideration of essential hydraulic principles, with enough simplification to enable solutions to be obtained quickly for idealised cases. Rules for the storage and discharge (flux) in each segment are prescribed based on the slope, stream cross section and roughness. Modifications of these rules to account for the effect of a leaky dam are developed. The model amounts to a series of coupled ordinary differential equations (ODEs) that are solved numerically given prescribed runoff inflow. We then explore solutions for some simple networks forced by idealised flood hydrographs, focussing on the response of the discharge at the downstream end of the network. We then examine the response to failure of the dams including cascade failure.

We construct a network model in which segments of a channel (“reaches”) are described in a lumped fashion (Fig. 2). The primary variables are the average cross-sectional area Ai and discharge Qi, which flows into the next channel segment downstream. The channel segments correspond to nodes of a graph, and the edges that transfer water downstream can be thought of as potential dams (i.e. the positions at which dams might be added). The connections between the channel segments are described using an adjacency matrix (aij). The ith row of this matrix is all zeros except for in the jth column, where j indexes the node immediately downstream of the ith node. Idealised network structures, with uniform widths and slopes are used, although positions and connections between the channel segments, as well as their lengths and slopes, might be determined from studying a real drainage network, for example based on a two-dimensional digital elevation model (DEM) such as that used by Metcalfe et al. (2017), or the Penny Gill example discussed further below.

Taking li to represent the length of the channel segments, volume conservation requires 1lidAidt=∑j=1NajiQj-Qi+qi, for each node (i=1 … N). The sum represents the fluxes from the immediately upstream nodes, and qi represents the inflow to each segment from rain/runoff from the surrounding land. It may be more convenient to think of (1) in terms of the water volumes Vi=Aili stored in each channel segment. We assume that the lateral inflows, qi(t), are prescribed, although in a more complete treatment they might be taken from a two-dimensional model (using the shallow water equations for example), or they might be derived from rainfall data using a filter to represent the time delay due to subsurface and/or overland flow.

Given the known slope of each channel segment Si (which may be related to the bed angle θ by Si=tan⁡(θ)), we could relate the discharge and cross-sectional area. However, it turns out to be more convenient to express the discharge in terms of the water depth hi behind the potential dam in each reach. In the case that there is no dam, or when the depth is below the bottom of the dam, this is simply the average water depth and we can relate this to the cross-sectional area and flow.

The relationship depends on the assumed shape of the channel and on a parameterisation of turbulent flow. If we assume for simplicity that the channel has a rectangular cross section with fixed width wi and use Manning's law, we have 2Ai=wihi,Qi=wi5/3hi5/3Si1/2wi+2hi2/3n, where n is the Manning roughness coefficient and Si is the slope. Since we can then relate Qi directly to Ai (by eliminating hi), we can interpret Eq. (1) as a set of coupled ordinary differential equations for the Ai, forced by the inputs qi. These can be solved numerically using a variety of methods. More generally, when we include dams, we write 3Ai=Ãhi;⋅,Qi=Q̃hi;⋅, where Ã(hi;⋅) and Q̃(hi;⋅) are known functions, and the extra parameters (⋅) will describe the dam as well as the cross section and slope (see below). We also define h̃(A;⋅) to be the inverse of Ã (which is a monotonically increasing function of h and therefore has a well-defined inverse). Thus, we will still have a direct relationship between Qi and Ai.

We take h to represent the height of the water behind the dam. The dam has its bottom at height b above the stream bed and its top at height H. The description of the flow past the dam can then be divided into three modes represented in Fig. 3, h<b (corresponding to the water level being below the bottom and the dam doing nothing), b≤h<H (when the dam is operating normally) and h≥H (when the dam is overspilling).

For the first mode we use the same Manning relationship as given above to relate Q to h. For the second two modes we adopt relationships from hydraulic theory for the flow beneath sluice gates and over weirs (e.g. Munson et al., 2013). When the water depth is part of the way up the face of the dam, the flow underneath is given by Bernoulli's equation to be 4wbh2gb+h. An empirical correction factor to account for losses is often included in this formula, but we neglect it for simplicity. The flow through the (leaky) dam is assumed to similarly vary with the water depth (due to the hydrostatic pressure), and we write this as 5kw(h-b)2gh, where k should be interpreted as the dam permeability. When the water depth is above the level of the dam, the overflow is described as for flow over a weir, giving 622g3w(h-H)3/2. The leaky flow through the dam is then 7kw(H-b)2gh. In summary, therefore, 8Q̃(h;w,l,b,H,k,S)=w5/3h5/3S1/2(w+2h)2/3n0≤h<bw2gbh(h+b)1/2+k(h-b)h1/2b≤h<Hw2gbh(h+b)1/2+k(H-b)h1/2+23(h-H)3/2H≤h. When there is a dam, Ai no longer represents the uniform cross section of the stream, but rather its average over the length. It is most straightforward to calculate the volume AiIi in terms of the water depth h for each of the three cases mentioned above. This again depends upon the precise geometry; for the rectangular channel we have 9Ã(h;w,l,b,S)=wh0≤h<bwb+w(h-b)22Slb≤h. The terms in the second expression here represent the volume of water in the stream up to the depth of the bottom of the dam, plus the volume of water stored in the triangular wedge that forms behind the dam. These relationships are shown in Fig. 4.

We expect that the flow through the dam will be small compared to that under and over it (the fact that it is allowed to be leaky makes it easier to construct, but the leakiness between logs is not fundamental to its operation in that there is leaking from underneath the barriers). Thus, for the results presented here, we assume this can be ignored and set the dam permeability coefficient k to zero for simplicity.

We choose scales, denoted by square brackets, such that 10[Q]=[w][h]5/3[S]1/2n,[A]=[w][h],[t]=[l][w][h][Q]. For given typical values of [Q], [w], [S] and [l], these determine the scales [h], [A] and [t]. Using typical values for small headwater drainage channels in the UK or other humid temperate environments, [w]=2 m, [Q]=1 m3 s-1, [S]=0.01, [n]=0.01 s-1 m-1/3 and [l]=100 m, we find [h]=0.17 m, [A]=0.33 m2 and [t]=33 m2.

In non-dimensional form, and assuming negligible dam permeability, these are 11Q̃(h;w,l,b,H,k,S)=w5/3h5/3S1/2(w+2αh)2/30≤h<bwγbh(h+b)1/2b≤h<Hwγbh(h+b)1/2+23(h-H)3/2H≤h,12Ã(h;w,l,b,S)=wh0≤h<bwb+βw(h-b)22Slb≤h, where the dimensionless parameters are 13α=[h][w]β=[h][S][l]γ=2g[w][h]3/2[Q]. These represent the ratio of depth to width (this is of little importance), the ratio of depth to elevation change across the segments and the strength of gravity compared to friction – Manning's coefficient is implicit from the relationship for [Q] in Eq. (10). For the values given above we find α∼0.08, β∼0.17, γ∼0.6.

The parameter β helps link this mathematical analysis back to the real world, in that it is related to “rule of thumb” estimates of backwater length used by hydraulic engineers to understand influence upstream as 0.7×h/S (for example see Environment Agency, 2010). This estimate, like β, tells us that for a significant backwater (and therefore storage) we need to have a small slope and larger depth. In other words, small β indicates that the capacity to hold back a significant volume of water behind the dams is very limited. However, β can also be large due to large h – which can lead to increased probability of failure if h>H, so in Sect. 4 the model is applied to understand different configurations.

A small value of β is also the first indication of why a large number of dams may be required to have even a noticeable effect on the discharge downstream. The small value here is an artefact of the assumed uniform width of the channel, but it is also consistent with the work of Metcalfe et al. (2017), where 57 leaky barriers were required in the 29 km2 Brompton catchment, along a 4.7 km area of the main stem and Ing Beck, before significant attenuation was achieved. It is likely that the locations for the dams may in reality be chosen to dam “reservoirs” that are wider than the average stream width, where the stream bed is particularly flat or where there is capacity for significant overflow onto the floodplain, or additional off-line storage (e.g. Quinn et al., 2013; Nicholson et al., 2019). We suggest that the contribution to the cross-sectional area due to the volume in the reservoir in Eq. (9) is therefore underestimated and should be increased. Thus, we modify the cross-sectional area to 14Ã(h)=wb+λw(h-b)22Sl,H≤h, where λ≥1 is this enhancement factor that accounts for a larger volume being stored behind the dam. In practice, this would have to be estimated for each dam location but provides a useful mechanism for exploring different NFM designs, for instance it can be used to conceptualise reaches where leaky barriers are being used to enhance floodplain reconnection, thus accessing additional storage with greater potential for peak flow attenuation.

One potential concern with the above formulation is the discontinuity in the discharge–depth relation when the water depth reaches the bottom of the dam (Fig. 4). This occurs in the model because the physics used to relate the depth to discharge is different in the two cases of free-stream flow (when we use Manning's law to describe turbulent drag) and flow under the dam (when we use an essentially inviscid formula for flow beneath a sluice gate). Mathematically, provided the discontinuity in flux involves a reduction as h increases, there should be no problem. As the water depth reaches the bottom of the dam, the flow past it suddenly decreases and the water quickly fills up behind the dam until the depth has increased to allow sufficient flow to balance the inflow from upstream. There can be problems, however, if the discharge suddenly increases when h increases past b which, from Eq. (8), occurs if the slope is sufficiently small, or n sufficiently large. If that occurs, we continue to use the frictional formula in the first case of Eq. (8) until the second formula gives a lower value for the flux.

The system of equations Eq. (1), coupled with the expressions for Q̃(h) and Ã(h) in Eqs. (8) and (9), is a system of non-linear equations for the temporal evolution of the water depths hi. The equations are forced by the source terms qi, and each equation is coupled with the equations corresponding to the upstream edges of the network. The whole system is solved numerically using an ordinary differential equation solver in MATLAB, using code that we have published in a repository as cited at the end of this paper. Whilst there are a number of hydraulic modelling packages solving similar equations with a diverse range of hydraulic units, these do not permit rapid assessment of collapse and cascade collapse of barriers having leakiness factors and a channel storage multiplier, making it easier to test arbitrary networks of configurations.

In this section we consider a simple example of the model, using the one-dimensional network shown in Fig. 2b. We suppose that each of the channel segments is the same (i.e. equal widths, lengths and slopes), and the discharge in the final segment is of most interest for the community requiring protection. For these calculations (and all others shown in this report) we use the original flux and area formulas (Eqs. 8 and 9), with the enhancement factor described in Eq. (12).

The model is forced with a “storm” input in the form of a hydrograph based on a simplified Gaussian functional form, as an approximation to a typical design storm estimated using the unit hydrograph approach (used in the application to the real case in Sect. 4). Here an extreme flow of 10 m3 s-1 is used to fully stress-test the system, also permitting flows and depths with magnitudes capable of failing leaky barriers. 15q(t)=q0+qmaxe-t2/σ2, where q0 is a baseline inflow (groundwater flow into the channel, say), qmax is the peak flood inflow at time t=0 and the flood is spread over a time period σ. In Fig. 5 we compare the resulting modelled discharge in each channel segment with a large λ factor, between the case of no dams and the case of having a dam on each segment (we use only five segments for ease of illustration; using more segments allows for greater potential of reducing the peak discharge).

Figure 6 shows an example when the input has a double peak. In this case, as might be expected, the dams are less effective at reducing the height of the second peak, because they are already holding back a lot of water and have less capacity to store and delay water for the second storm. This indicates that testing of the performance of NFM, or any risk reduction measures, should potentially consider testing resilience against real storm series or double peaks and not simply single-peaked storm events, as are commonly assumed in practice when considering flood storage design analysis.

Here we consider a simple two-dimensional network as shown in Fig. 2c, which reflects a more likely pattern given the dendritic nature of channel formation in headwater catchments. There are more interesting questions to consider about the positioning of dams in this case. For example, if one has funding to build a certain number of dams, which of the channel segments are the best ones on which to put them? Putting them on the central trunk is likely to ensure that they are used (performance issue 1), but also means that they may more easily overspill and lose their effectiveness. They may also be more susceptible to cascade failure (performance issue 3 – discussed in the next section).

In Fig. 7 we show two examples of the response to a flood input of the form given by Eq. (15). In the first case, four dams are placed on the main trunk (nodes 1–4), whereas in the second case four dams are placed on the upper branches (nodes 5, 6, 9, 10). The discharge from the final segment (node 4) is plotted, along with its maximum value. Both dam placements have the effect of slightly delaying and reducing the peak discharge, with the second design being marginally more effective. This is because the dams near the bottom of the central trunk are overspilling and losing their effectiveness, whereas the dams on the side branches are all having a significant effect.

However, for different sized floods or realistic spatial patterns of extreme rainfall (see Hankin et al., 2017a), the optimal arrangement can vary. Unfortunately, there does not appear to be a clear rule for the most effective dam placement, even in this simple example, where the resilience of distributed NFM in terms of temporary storage and tree-planting was tested against different storm extremes having spatially realistic patterns (Lamb et al., 2010). In this network study, the on-average performance of one particular system of NFM was tested, allowing for utilisation and the risk-reducing or risk-increasing impacts of changes to tributary synchronisation (performance issue 2), using average annual losses as the integrated measure of risk reduction. However, the high-resolution model, with 180 million cells, took over 26 h to run so only 30 extreme events were simulated with and without NFM measures, and alternative spatial strategies were not tested to understand which were more advantageous. Simplified network analyses such as those presented here could be used to rapidly explore such spatial strategies, without resorting to highly complex and relatively slow models.

One of the potential risks of installing many dams in a catchment is the possibility that they all collapse in sequence, creating a flood surge that is larger than would have occurred if no dams had been installed at all. Provided each dam stores only a small reservoir of water, the collapse of one dam on its own should not be catastrophic. But if the collapse of one dam causes others further downstream to collapse too, there is the obvious danger of the surge escalating. This risk may be an important factor in deciding the best placement of dams (perhaps outweighing the efficiency of peak-flow reduction under “normal” operating conditions).

The main method suggested for analysing this risk is to run an ensemble of simulations of flood events, assigning a failure depth to each dam using the probability distribution suggested by a fragility curve. Ideally, this ensemble should include a range of storm conditions too. Such analysis could in principle use dynamical weather models or rainfall records to construct statistical models and then sample from the modelled joint (spatial) distributions of rainfall forcing. Both approaches have been considered in the context of reviewing flood resilience in the UK (HM Government, 2016) and for flood risk analysis over large and complex infrastructure networks (Lamb et al., 2019). This type of spatially structured risk analysis can be expensive, and so it may be desirable to establish some rules of thumb about which dam placements are more, or less, at risk of cascade failure.

Knowledge gained from this type of analysis might be used to plan for the size and strength of dams that should be built at different locations. For example, it could be that certain locations are particularly prone to collapse (downstream of merging tributaries for example), and building one stronger “buffering” dam could significantly reduce the risk of a cascade.

As an example of cascade failure in the network model, we return to the one-dimensional example shown in Fig. 2b. We impose a regular storm inflow to the upstream node of the form given in Eq. (15) and examine an ensemble of 50 possible system states (describing different combinations of survival or failure of the individual dams). Each of the dams is assigned a critical water depth hci such that when hi>hci the dam collapses; the critical depth is sampled from a normal distribution with a mean of 3.5 m and standard deviation of 0.5 m (the top of the dam is at 1.5 m, so dam collapse usually occurs when the dam is already submerged). The results are shown in Figs. 8 and 9. Figure 8 shows the peak discharge Qmax at the downstream node for each simulation.

Figure 8 helps illustrate whole-system resilience across a wide range of events and is coloured by the number of dams that are predicted to collapse within each ensemble member; small blue dots indicate that no dams failed, and since the forcing is identical in each case, the peak discharge in this case is always the same. It is lower than what the peak would have been in the absence of any dams, so the dams are proving effective in these cases. Larger dots correspond to more dams having failed. In most of the ensemble members only one dam collapses, and the peak discharge recorded downstream is strongly dependent on which one fails (the red dots in Fig. 8). A larger peak occurs if the collapsed dam is further downstream, since if an upstream dam fails (and importantly does not precipitate a cascade of downstream failure) the sudden release of water from that dam is buffered by the dams further downstream.

If, on the other hand, a single dam failure leads to further collapse of two or more dams, the peak discharge can be much larger. In one example, all five dams collapse in quick succession, and the time series of this example (the black dot in Fig. 8) is shown in Fig. 9, where it is compared to an example with no failure. We have found that the pattern of failure in this one-dimensional model, including the likelihood for cascading failure, depends heavily on the assumed dam sizes, critical water depth distribution and magnitude of rainfall events.

As a second more instructive example of cascade failure, we revisit the herringbone network in Fig. 2c. We consider the two possible placements of four dams that were discussed earlier: either along the main trunk (nodes 1–4) or on the upstream side branches (nodes 5, 6, 9, 10). In Fig. 7 we found that there was relatively little difference in the peak discharge measured at the downstream node with these different placements. However, in Fig. 10 we see that the first case is much more at risk from cascade failure, and the whole system of dams is not resilient in this spatial configuration. This figure shows the peak downstream discharge in an ensemble of simulations, with the failure depths for each dam being different each time. The failure depths are sampled from the same distribution in each case.

In almost every run with dams on the trunk, we see cascade failure occurring so that three or four of the dams collapse; this leads to extremely high (though short-lived) peak discharge. In contrast, when the dams are placed on the side branches, there is no possibility of cascade failure (the individual branches do not communicate with each other), and it is unlikely that more than one dam collapses. Thus, having used the model to consider the resilience of the whole system of barriers for two cases, it can be seen that although each of these dam placements is similarly effective at reducing the peak discharge, there may be strong reason for preferring the second design that places them on the upstream tributaries because this configuration is a more resilient system (even though the resilience of the individual dams is the same in both designs). The large surges predicted in the simple network model when multiple dams fail have some support in the literature. For example, Hillman (1998) describes a June 1994 outburst flood in central Alberta, Canada, releasing 7500 m2 of water and a flood wave 3.5 times the maximum discharge recorded for that creek over 23 years. Although not reported as a cascade failure, large trees and debris from older beaver dams were carried further downstream, and five hydrometric stations downstream were destroyed.