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Wetropolis extreme rainfall and flood demonstrator: from mathematical design to outreach and research

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Wetropolis is a transportable table-top demonstration model with extreme rainfall and flooding events. It is a conceptual model with random rainfall, river flow, a flood plain, an upland reservoir, a porous moor, representing the upper catchment and visualising groundwater flow, and a city which can flood following extreme rainfall. Its aim is to let the viewer experience extreme rainfall and flood events in a physical model on reduced spatial and temporal scales. In addition, it conveys concepts of flood storage and control, via manual intervention. To guide the building of an operational Wetropolis, we have explored its spatial and temporal dimensions first in a simplified mathematical design. We explain this mathematical model in detail since it was a crucial step in Wetropolis' design and it is of scientific interest from a hydrodynamic modelling perspective. The key novelty is the supply of rainfall every Wetropolis day (unit wd), varied temporally and spatially in terms of both the amount of rain and the rainfall location. The joint probabilities (rain amount times rain location) are determined daily as one 10 of 16 possible outcomes from two asymmetric Galton boards, in which steel balls fall down every wd, with the most extreme rainfall event involving 90 % rainfall on both moor and reservoir. This occurs with a probability of circa 3 % and – by design – can cause severe floods in the city. This randomised rainfall has a Wetropolis' return period of 6:06min, short enough to wait for but sufficiently extreme or long to get slightly irritated as a viewer. While Wetropolis should be experienced live, here we provide a photographic overview. To date, Wetropolis has been showcased to over 200 flood victims at workshops and 15 exhibitions on recent UK floods, as well as to flood practitioners and scientists at various workshops. To enhance Wetropolis' reach, we analyse and report here how both the general public and professionals interacted with Wetropolis. We conclude with a discussion on some ongoing design changes, including how people can experience natural flood management in a revised Wetropolis design, before highlighting how the Wetropolis experience can stimulate new approaches in hydrological modelling, flood mitigation and control in science, education and water management.
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Wetropolis extreme rainfall and flood demonstrator: from
mathematical design to outreach and research
Onno Bokhove1, Tiffany Hicks1, Wout Zweers2, and Tom Kent1
1School of Mathematics, University of Leeds, LS2 9JT, Leeds, UK
2Wowlab, Enschede, The Netherlands
Correspondence: Onno Bokhove (o.bokhove@leeds.ac.uk)
Abstract. Wetropolis is a transportable “table-top” demonstration model with extreme rainfall and flooding events. It is a
conceptual model with random rainfall, river flow, a flood plain, an upland reservoir, a porous moor, representing the upper
catchment and visualising groundwater flow, and a city which can flood following extreme rainfall. Its aim is to let the viewer
experience extreme rainfall and flood events in a physical model on reduced spatial and temporal scales. In addition, it conveys
concepts of flood storage and control, via manual intervention. To guide the building of an operational Wetropolis, we have5
explored its spatial and temporal dimensions first in a simplified mathematical design. We explain this mathematical model in
detail since it was a crucial step in Wetropolis’ design and it is of scientific interest from a hydrodynamic modelling perspective.
The key novelty is the supply of rainfall every Wetropolis day (unit wd), variable temporally and spatially in terms of both the
amount of rain and the rainfall location. The joint probabilities (rain amount times rain location) are determined daily as one
of 16 possible outcomes from two asymmetric Galton boards, in which steel balls fall down every wd, with the most extreme10
rainfall event involving 90% rainfall on both moor and reservoir. This occurs with a probability of circa 3% and – by design
– can cause severe floods in the city. This randomised rainfall has a Wetropolis’ return period of 6:06min, short enough to
wait for but sufficiently “extreme” or long to get slightly irritated as a viewer. While Wetropolis should be experienced live,
here we provide a photographic overview. To date, Wetropolis has been showcased to over 200 flood victims at workshops and
exhibitions on recent UK floods, as well as to flood practitioners and scientists at various workshops. To enhance Wetropolis’15
reach, we analyse and report here how both the general public and professionals interacted with Wetropolis. We conclude
with a discussion on some ongoing design changes, including how people can experience natural flood management in a
revised Wetropolis design, before highlighting how the ‘Wetropolis experience’ can stimulate new approaches in hydrological
modelling, flood mitigation and control in science, education and water management.
20
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1 Introduction
The Boxing Day flood of 2015 caused widespread damage in Yorkshire, UK, due to extreme flooding of the River Aire in
and around Leeds and the River Calder in and around Todmorden, Mytholmroyd and Hebden Bridge. Thankfully no fatalities
occurred but the economic damage was severe and estimated to be around £500M (West Yorkshire, 2016). November 2015
was the third wettest month on record in terms of precipitation and December 2015 was the wettest on record (Environment5
Agency, 2016). As such, the soil was already saturated when 48hrs of extreme rainfall fell in Yorkshire, leading to the Boxing
Day floods of 2015. After this and other recent floods, many flood victims asked why extreme flood events, seemingly occurring
more and more often, were causing such havoc in their communities.
To provide some mathematical background on modelling, mitigation and statistics of extreme flood events, we were asked
to disseminate scientific background on “risk in the age of extremes” at the citizens’ conference “Science of floods” in Hebden10
Bridge (Science of floods, 2016). In the first one-and-half decades of the 2000s, Hebden Bridge was hit by both summer flash-
floods and winter floods leading to concerns amongst flood victims that their lives and properties were insufficiently protected.
It led to further and intense discussions with environmental agencies on the need for more and different types of flood defences.
Important questions in this discussion are the following:
Is it going to rain more in the future in the UK?15
Can we define extreme precipitation and flooding events?
How (well) can we predict heavy precipitation and floods?
Finally, how can we elucidate these questions, their answers and uncertainties, in an interactive table-top demonstration?
We will discuss some answers to these questions in turn.
Is it going to rain more in the future in the UK? Both the IPCC report (IPCC, 2013) and Sanderson’s UKCP09 report of20
the Met Office (Sanderson, 2010) show that there is no increase of significance in average annual rainfall foreseen in climate
projections, not across the globe on average and also not in the UK. However, there are geographical and seasonal variations
foreseen: winter rainfall will generally increase and summer rainfall will thus decrease but with more intense downpours.
Can we define extreme precipitation and flooding events? Extreme events tend to be expressed as the chance that an extreme
event occurs on one day in a year. If that chance is 1%, for example, then we say that this extreme event has a return period of25
1 : 100 years. Flooding events can be classified in terms of such return periods as, e.g., 1 : 10,1 : 20,1 : 50 and 1 : 100 or 1 : 200
year events, with the latter two considered to be extreme. The uncertainty in an event with a 1 : 100 year return period will be
larger than one with a 1 : 20 year return period. An extreme event occurs in the tail of a probability distribution and may have
never been observed. Events with return periods longer than the data record cannot be classified directly from that data and must
be determined using theoretical probability distributions. However, the low number of extreme events in a finite-time data set30
means it is difficult to establish accurately the tail of the distribution. Accordingly, there is a great deal of uncertainty associated
with extreme events, which should ideally be quantified and communicated effectively. By assuming a suitable (parametric)
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probability distribution function (pdf) one can use the data to fit the parameters of the pdf and subsequently generate data
(“sample the pdf”) in the tail for events with return periods beyond the length of the data record. Classical pdfs, such as (half a)
Gaussian distribution or a Gamma distribution, are typically used to model rainfall intensities but are not suitable for extreme
events. Extreme-value theory offers a family of distributions to overcome the limitations of the classical pdfs. Given a sufficient
amount of (rainfall or river level) data over a certain threshold, the Generalised Pareto distribution (GPD) attempts to model5
the probabilities of extreme events beyond the data record. It is also possible to capture both the bulk and the extreme tail of
the distribution in a so-called mixture model by combining a Gamma and a Generalised Pareto distribution (GPD), cf. Wong
et al. (2014).
How (well) can we predict heavy precipitation and floods? The Boxing Day floods of 2015 were caused by large-scale winter
rainfall. The 48 hours of consecutive rainfall in the days leading up to the flood were the wettest on record: in Bradford 69.4mm10
and in Bingley 93.6mm of rainfall were measured over 48 hours. It resulted in the flooding of the River Aire with river-level
records reached in Leeds and elsewhere along the river. In Armley, Leeds, the gauge station measured a maximum river level
of 5.21m while the previous electronic record was 4.03m in the Autumn of 2000 (Environment Agency, 2016). The river level
during the 1866 flood was roughly around 4.5m, cf. Bokhove et al. (2018a) (their Fig. 2). Both rainfall and river levels were
by and large well-predicted by the UK Met Office via Numerical Weather Prediction and by the Environment Agency (–EA,15
cf. a presentation by an EA–Yorkshire leader in Leeds). Predictions are generally quite good for large-scale winter rainfall
and the resultant changes in river levels. Downpours, e.g. in the summer, tend to be more localised and are therefore much
more difficult to predict in terms of location, intensity and duration. The same holds for resulting flash-floods and downpour
induced surface-water flooding. Hence, simply put, fluvial or river flooding in winter tends to be easier to predict than pluvial
or surface-water flooding events in the summer.20
Finally, how can we elucidate these questions, their answers and uncertainties, in an interactive table-top demonstration?
The expression of extreme events in terms of return periods is difficult to grasp and often misunderstood, especially by the
public. The Boxing Day flood of 2015 was classified as an event with a 1 : 200+year return period – including the unclear
meaning of the plus sign in 200+(Environment Agency, 2016)1. That does not mean that it has to take another 200+years,
so until after 2215, before the next Boxing-Day-type flood might occur in Leeds. It does, however, mean that the average time25
between events of similar magnitude will be 200+years, given a sufficiently long record of “stationary” statistical data. To let
people experience such an extreme event in a table-top set-up they can of course not be asked to wait for 200 years on average,
so our design for a flood demonstrator with rainfall must be scaled down both in size and duration. Miniature river flooding
has been demonstrated in small-scale experiments (e.g., as in the Lego model of Pampaloni et al. (2018)) but these all tend
to involve deterministically imposed extreme water input –with water inflow supplied and adjusted deterministically and/or30
manually. The key novelty in our design lies in the way rainfall is supplied randomly to our table-top hydrodynamic set-up for
both river and groundwater flow. We have modified the classical symmetric Galton board, inspired by such a set-up used at
Leeds’ School of Mathematics open days. A typical Galton board has a tilted surface in which a (steel) ball falls down under
gravity and encounters a series of symmetric pins or channel corners, each determining with a pand (1p)chance whether the
1In a recent personal communication with a Yorkshire flood expert, a 1 : 300 year return period was mentioned, with no formal confirmation yet to date.
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ball continues or falls to the left or to the right. The design is usually such that p1/2but small variations can occur in practice
and after a series of nrows of splittings a binomial distribution arises, given a sufficiently large number of trials. Moreover, for
n→ ∞, a Gaussian distribution emerges. The Galton board is often used to visualise and demonstrate statistical distributions in
real time during, e.g., outreach events. To obtain an asymmetric discrete distribution with a discrete tail representing relatively-
extreme events, the standard symmetric Galton board described above was modified as follows. For p= 1/2and n= 1, the5
first and only split leads to a (1,1)/2–distribution. The first split of the second row for n= 2 is now eliminated while the
second split is not, leading to a (3,1)/4–distribution. Continuing to the third row of splittings as usual, for n= 3, we obtain
a(3,4,1)/8–distribution. The last and fourth row for n= 4 yields the final (3,7,5,1)/16–distribution. An image of such a
asymmetric Galton board is given in Fig. 1. Two of these Galton boards will be used to supply rain to our table-top river and
ground-water flow model, one concerning the duration and amount (0.1,0.2,0.4,0.9)r0of rainfall during a Wetropolis day,10
with its unit wd, and another one concerning the location of the rainfall. Rain duration will be either (10,20,40,90)% of the
amount r0of rainfall per wd and rain location will be either rainfall (i) in a reservoir with generally instant run-off into the
river; (ii) in both a reservoir and on a moor; (iii) on the moor with groundwater flow and its nonlinear, delayed release of water
into the river; and, (iv) no rain, in the Wetropolis catchment. Both duration and location are determined by the outcome of one
trial through two Galton boards per wd, together yielding a 4by 4–matrix of joint probabilities, with the no-rain case having a15
rare chance (for the UK) of 1/16th comprised by four of those 16 outcomes. By design, an extreme event occurs when it rains
90% in both locations (i.e. in the reservoir and moor) with a chance of 7/256 0.0273 = 2.73%, which in our construction
will by design lead to flooding of a city further downstream along a (winding) river in the set-up. A Boxing-Day-type event
with two consecutive days of extreme 90%–rainfall then has a chance of 49/(2562)0.000748 = 0.0075%, yielding a return
period of 223min 3 : 43hr. The next and crucial step in the design is to identify and determine the various unknowns in order20
to ascertain whether a feasible design is possible at all.
Given a (winding) river of length Land curvilinear coordinate salong this river, these remaining key unknowns are as
follows:
the influx discharge Q0at the upstream boundary at s= 0,
the locations sres and smwhere the reservoir and moor enter into the river, with a section further downstream along the25
river comprising a city plain that is prone to (extreme) flooding,
the rainfall amount r0(dimensionally a speed, as we will explain in the next section), determining the strength of the
pumps required and whether pumping rates can be realistic at all,
the length of a Wetropolis day, in relation to the extreme rainfall and corresponding extreme flooding event, such that the
viewer experiences some irritation in having to wait for a randomly-induced extreme event but on average will experience30
such an extreme event within a reasonable time, i.e. on average within several minutes.
We chose sres ,sm,Q0a priori and determined wd and r0by simulation of a simplified mathematical model. A plan view of a
sketch of Wetropolis is given in Fig. 2.
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Figure 1. Photographs of asymmetric Galton boards. Test board on the left and a final set-up on the right. At every split the chance of a steel
ball falling to the left or right is 50% for a well-balanced Galton board. When a sufficiently large number of steel balls falls through this
Galton board, the discrete distribution becomes (3,7,5,1)/16. The 4×4possible outcomes in two of such boards, registered in each by four
electronic eyes (located in the black-painted areas along 2×4 = 8 channels marked here by “1,2,4, ... ” and “L,&,M,0”), determine both
the rainfall amount and its location(s) in Wetropolis. The outcome of the random draw, shown by the lit-up lights, will in this instance lead
to 4s of rain in the lake/reservoir. Photos: OB & WZ.
This introduction has given both an anecdotal and scientific background to Wetropolis’ inception. Our intention here is
to document its journey from design to outreach and research; the remainder of the article has the following outline. The
above unknowns were determined by an idealised mathematical model of Wetropolis before any design and construction of
the table-top set-up were undertaken. This mathematical and numerical modelling is explained and used to determine (some
of) the design unknowns in §2. The resulting table-top design of the Wetropolis flood demonstrator is disseminated in §3. Our5
experience in demonstrating Wetropolis to the general public and to flood practitioners is summarised in §4, including the
a-priori surprising outcome that professionals in flood prediction and mitigation have also been inspired by Wetropolis, despite
that our primary aims have been public outreach. A discussion is found in §5, in which we outline some directions for further
scientific research, as gathered from such public-engagement sessions.
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Figure 2. Plan view of the Wetropolis table-top experiment. The main river channel is indicated in white-blue blocks and its one-sided flood
plain extent by a dashed line. A “Leeds-Liverpool” canal with lock-weirs flanks the 1 : 100–sloped river, which has constant upstream inflow
and gets fed by water from a reservoir as well as a porous moor filled with lava grains. In both locations, it can rain intermittently and
randomly. Outflow is at the end of the river channel, after a city plain that can flood and where the canal flows into the river. Water falling in
a full reservoir flows instantly with a manually adjustable fraction of 0< γ 1into the canal and the river, the latter with a fraction (1γ).
The reservoir level can also be adjusted manually, which provides some flood control. This control can be used to demonstrate the role of a
holding reservoir to lessen flooding in cases of extreme rainfall.
2 Mathematical design
The mathematical model of Wetropolis comprises random rainfall and space-time continuous hydraulic modelling of the in-
terconnected river flow, reservoir- and canal-level changes as well as groundwater flow in the moor. Subsequently, we will
establish a numerical discretisation of this space-time continuous model and use numerical simulations to determine the a
priori unknown parameters of rainfall amount r0and wd. Other parameters will be determined heuristically in order to ob-5
tain desirable and practical dimensions of the experimental set-up. We present the model here completely; full details of the
numerical discretisation can be found in Appendix A.
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Table 1. Joint probability matrix of the 16 outcomes of rainfall times 256, with the extreme case of 7/256 shown in bold. The rows
show P(location), and columns P(amount), with both summing to the imposed discrete distribution (3,7,5,1)/16. Since the P(location) and
P(amount) are independent, the joint probability is their product.
r02r04r09r0
reservoir 9 21 15 3
both 21 49 35 7
moor 15 35 25 5
no rain 3 7 5 1
2.1 Statistical modelling of randomised rainfall
As discussed in the introduction, rainfall is modelled stochastically via the outcome of draws from two Galton boards. In
the mathematical model these outcomes are simulated, while in the physical flood demonstrator we have either used two
actual Galton boards with two steel balls or one Galton board with one steel ball running through two consecutive Galton-
board channels. We have discretised rainfall into two categories: location and rain amount, per wd. Rain location has four5
outcomes: rain in reservoir, moor and reservoir, moor, or no rain in the catchment with a discrete distribution of (3,7,5,1)/16.
Independently, rain amount has per location four outcomes (1,2,4,9)r0/wd with again the discrete distribution (3,7,5,1)/16;
hence, there are 4×4 = 16 outcomes determined as a direct product of these two independent distributions, given in Table 1. The
possible rain amounts per wd is therefore (0,1,2,4,8,9,18)r0and the value r0will be determined in the subsequent modelling
such that there is no flooding for (1,2,4)r0/wd rainfall, with potentially limited flooding for (8,9)r0/wd and generally major10
flooding in the city plain for 18r0/wd. The resulting distribution of the rainfall per wd in the ‘Wetropolis catchment’ (to be
read with Table 1) therefore becomes:
0r0:P(no rain)=1/16,(1a)
1r0:P(1r0)P(reservoir) + P(1r0)P(moor)=9/256 + 15/256 = 24/256,(1b)
2r0:P(1r0)P(both) + P(2r0)P(reservoir) + P(2r0)P(moor) = 21/256 + 21/256 + 35/256 = 77/256,(1c)15
4r0:P(2r0)P(both) + P(4r0)P(reservoir) + P(4r0)P(moor) = 49/256 + 15/256 + 25/256 = 89/256,(1d)
8r0:P(4r0)P(both) = 35/256,(1e)
9r0:P(9r0)P(reservoir) + P(9r0)P(moor)=3/256 + 5/256 = 8/256,(1f)
18r0:P(9r0)P(both)=7/256.(1g)
A pdf of this discrete distribution for a computer trial over 500wds is shown in Fig. 3 (blue bars) with the expected values20
overlaid (crosses). Suitable values for r0and wd are established by further modelling, described next.
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Figure 3. The pdf of random rainfall outcomes over 500wds, displayed as a scaled histogram, is compared with the theoretical discrete pdf
(1a), denoted by the black crosses. Extreme cases with 18r0rainfall are observed in moor and reservoir combined, here in this 500wd trial
with an occurrence on the average.
2.2 Mathematical modelling in space-time
The key components of Wetropolis are a river channel with a one-sided flood plain, a groundwater moor, a reservoir, and canals
with three segments separated by lock-weirs. The canal flows into the river in the city plain which lies at the downstream end
of the set-up. We refer to Fig. 2 for a plan view locating these elements in the actual table-top experiments. In the original
design model, the locations of the reservoir and moor have been swapped and we have used a shorter river channel. Simplified5
mathematical sub-models of these different elements are derived next in isolation before being coupled into one complete
mathematical model via suitable boundary and interface conditions.
2.2.1 River dynamics
River flow is often modelled as one-dimensional flow in a channel with a cross-section A=A(s, t)as function of space, with a
horizontal curvilinear coordinate s[0,L]along a winding river channel of length L, and time t. Both A(s,t) = A(h;b(s)) and10
the in-situ water depth h=h(s,t), above a fixed river bottom b=b(s), and the mean flow velocity u(s,t)are all averaged over
the cross section of the river. E.g., for a rectangular river channel we have A=wrhwith fixed width wrbut in general A(h;s)
can depend in a complicated fashion on the river depth hand directly on s– the latter via b=b(s). The governing equations
are the Saint-Venant equations (e.g., Bates et al (2010)) consisting of continuity and momentum equations, augmented with
source terms at the locations smand sres along the river where the water flows from the moor and reservoir enter the river, as15
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well as a parameterization of the channel friction, i.e.
tA+s(Au)Qmδ(ssm) + Qresδ(ssr es)(2a)
tu+u∂su+g∂sh=g(sb+C2
mu|u|/R4/3)(2b)
with the phenomenological Manning friction coefficient Cm[0.01,0.15]m1/6, cf. Munson et al. (2005), the hydraulic radius
R(h) = wet area
wetted perimeter (in m), acceleration due to gravity g, and the discharge rates Qmand Qres of the water flows from the5
moor and reservoir into the river. Partial derivatives have been denoted by s=∂/∂s and t=∂/∂t. These discharge rates are
modelled as point sources using delta functions δ(ssm)and δ(ssres). In reality the same inflow rates will occur along
finite-length, short strips along the river, centred around smand sres . The two unknown fields are Aand uwith h=h(A;s)
an (often) implicit relation at every location s. For the above example with a river channel of rectangular cross-section, we find
R(h) = wrh
2h+wrand h=A/wr. Initial conditions A(s,0) and u(s, 0) have to be imposed at t= 0 as well as boundary conditions10
A(0,t),u(0, t)and A(L, t),u(L,t)at s= 0 and s=L. These latter boundary conditions are used (partially) according to the
way the characteristics at s= 0,L of the hyperbolic equations (2) determine whether the boundary data are flowing into the
domain or not, cf. Toro (2001).
Even though the actual winding river channel with its one-sided flood plain and city plain has a varying cross-section, for the
design calculations we made the simplification to model only a rectangular river channel. In addition, a zeroth-order kinematic15
model approximation to the Saint-Venant equations (2) has been used for the limiting case with positive velocity u > 0and a
constant slope sb > 0of the river channel. To zeroth-order, we assume that the bed slope and friction are locally in balance,
i.e., the underlined terms in (2) are negligible, such that we obtain
u=R(h)2/3psb/Cm.(3)
This is a classical approximation used in hydraulics (Munson et al., 2005): the river flow is thus modelled by a kinematic or20
scalar hyperbolic equation in A, as follows
tA+s(AR2/3psb/Cm)tA+sQf(A) =Qmδ(ssm) + Qresδ(ssres),(4)
with an upwind information speed dQf(A)/dA > 0for flux Q(A) = Au and inflow A(h(0,t);s= 0). Note that the flux Qf=
Qf(A)is an implicit function of Asince h=h(A). For u > 0with A=wrh, it is a kinematic model or nonlinear, scalar
conservation law in the water depth hgiven by25
t(wrh) + swrhR(h)2/3psb/Cmt(wrh) + sQf(h) = Qmδ(ssm) + Qresδ(ssres)(5)
with the flux Qf=Qf(h)rewritten in terms of h, initial condition h(s,0) and upstream influx Q0(0,t) = wrh(0, t)u(0, t)
defining h(0,t)since uis expressed in terms of hthrough (3). The Saint-Venant equations are more advanced than the above
kinematic model and allow both sub- and supercritical flows. An interim and better model arises when we instead of (3) use
the following balance30
u=R(h)2/3ps(b+h)/Cm,(6)
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which, after substitution into the continuity equation, (2a) yields an advection-diffusion equation, cf. Bates et al (2010). Both
these more advanced models are not required for the design estimates but when one wishes to perform accurate predictions of
the hydrodynamics in Wetropolis then such advanced models may be required.
2.2.2 Groundwater flow
Groundwater levels after rainfall are made visible in a transparent and elongated rectangular box filled to a high level with5
porous small lava rocks. The box is open at the top and one side, and has walls at the remaining four sides. Rain falls uniformly
along this box via a copper pipe with a series of holes. The groundwater dynamics are modelled to zeroth-order by assuming
that the surface of the grains is flat, the rainfall uniform per surface area, that there is no surface run-off and that the fallen
rainwater infiltrates sufficiently fast to contribute instantly to the groundwater level hm(y, t)with coordinate yin a different
direction, locally orthogonal to sat the location smwhere the groundwater flows into the river, cf. the delta function in the10
continuity equation (2a) and kinematic equation (5). A depth-averaged groundwater model with level hm(y, t)from Barenblatt
(1996) is used in a cell of width wvand length Ly, e.g. wv= 0.095m and coordinate y[0,Ly= 0.932m]. The nonlinear
diffusion equation for the groundwater level hm(y,t), taken to be uniform in the lateral direction, is
t(wvhm)αg∂ywvhmyhm=wvRm(t)
mporσe
(7)
with moor rainfall Rm(y, t) = Rm(t), porosity mpor [0.1,0.3], the fraction σe[0.5,1] of pores filled with water, α=15
k/(νmporσe)with permeability k= 108m2and viscosity ν= 106m2/s. The boundary conditions are no flux through the
wall at y=Lysuch that yh|y=Ly= 0, while at y= 0 the moor is held at the level h3(t)of canal–3, the upstream branch of
the canal running in parallel to the river, i.e., this a time-dependent Dirichlet condition hm(0, t) = h3(t). The mass flux of moor
water running in the river (at s=sm= 2.038m) is
Qm(t) =(1 γ)Qtm (1 γ)1
2mporσewvαg(yhm)2|y=0,(8)20
where Qtm is the mass flux following from integration of (7) over the domain y[0,Ly]and 0< γ < 1the fraction of moor
water entering into the river. The reason to multiply by mporσeis that the water volume in the matrix of particles in the moor
changes suddenly from a space filled with pores into free space.
2.2.3 Reservoir
The reservoir is a rectangular box of dimensions hres ×wres ×Lres with time-dependent water level hres(t), e.g. Lres =25
0.293m and wres = 0.123m. In the physical model, the random rainfall enters either via a pipe or a long pipe with numerous
holes visualising the rainfall and it leaves the reservoir via an overflow pipe, here modelled simply as a straight weir, cf. Munson
et al. (2005). Overflow of the reservoir once it is overfilled is not modelled. The reservoir-level dynamics are governed by
wresLr es
dhres
dt=wresLr esRres(t)Qres with Qr es =Cfg wres max(hres Pwr ,0)3/2,(9)
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in which Lreswr es is the area of the reservoir such that Lreswreshres is its time-dependent volume, Pwr is the overflow height
of the weir, Rres(t)the reservoir rainfall and Qres the flux down into the river. Note that the coefficient Cfis dimensionless.
The weir is located at s=sres = 0.925m, where water flows into the river, cf. the delta function in the continuity equation (2a)
and kinematic equation (5).
2.2.4 Canal sections5
One canal of uniform width wcruns alongside the river, cf. the Leeds-Liverpool canal and the River Aire in central Leeds, in
which the three canal sections are separated by lock-weirs and have mean, time-dependent water depths h1c(t),h2c(t), and
h3c(t). We are thus ignoring currents and height changes along the separate canal sections. Each canal section has a certain
depth and is separated from the river by a berm. Canal–3 is the highest and is blocked off on one end, at s= 0, and has a
weir located at s=L3c= 1.724m. Its level is modelled as the variation of its volume due to partial inflow from the moor and10
outflow of water via a weir in canal–2
wcL3c
dh3c
dt=γQtm Q3cwith Q3c=Cfg wcmax(h3cP3w,0)3/2(10)
with weir height P3w. Canal–2 resides from s[L3c,L2c]with s=L2c= 3.608m and is modelled likewise but with inflow
Q3cfrom canal–3 and outflow Q2cinto canal–1:
wc(L2cL3c)dh2c
dt=Q3c+γmpor wvαg 1
2y(h2)|y=0 Q2cwith Q2c=Cfg wcmax(h2cP2w,0)3/2.(11)15
The section of canal–1 runs from s[L2c,L1c]with L1c= 3.858m, width wcand depth h1c(t). It is modelled in the same
manner with inflow from canal–2 and outflow Q1cinto the river, as follows
wc(L1cL2c)dh1c
dt=Q2cQ1cwith Q1c=Cfg wcmax(h1cP1w,0)3/2.(12)
The weir at s=L1cwhere water flows into the river is assumed to be subcritical, i.e., we assume there is a sufficient drop
from canal–1 to the river level. In terms of height levels, canal–3 has a berm at z= 0.06m and its bottom resides at z= 0.04m;20
canal–2 has a berm at z= 0.04m and its bottom resides at z= 0.02m, and canal–1 has a berm at z= 0.021m and its bottom
resides at z= 0.001m. To wit, the outflow at the two weirs into canal–2 and canal–1 is based on Bernoulli’s relation and flow
criticality, cf. Munson et al. (2005). At s=L1c, e.g., for subcritical flow with flow depth h2cand flow speeds V2c0upstream
as well as critical flow Vcof height hcover the weir, we therefore derive the following
Vc=pghcand gh2c+1
2V2
2c=g(hc+P2w) + 1
2V2
c=3
2ghc+gP2w
25
V2c0s.t. hc= (2/3)(h2cP2w)and therefore:
Q2c=wchcVc=wcgh3/2
c=Cfgwcmax(h2cP2w,0)3/2(13)
with Cf= (2/3)3/2. Similar derivations with suitable adaptations of the quantities involved determine the fluxes Q1c,Q3c,Qres
over the other weirs.
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2.2.5 Fully-coupled system
When all of the above models for the individual components are combined we obtain the entire coupled model, including its
initial and boundary conditions, for the unknowns h(s,t),hm(y, t),hres(t),h1c(t),h2c(t)and h3c(t):
River: t(wrh) + swrhR(h)2/3psb/Cm=Qmδ(ssm) + Qresδ(ssres )
on s[0,L]with Qf|s=0 =wrhR(h)2/3psb/Cm|s=0 =Q0(t), h(s, 0) = h0(s)(14a)5
Moor: t(wvhm)αg∂ywvhmyhm=wvRm(t)
mporσe
on y[0,Ly]with thm|y=Ly= 0, hm(0, t) = h3c(t), hm(y,0) = hm0(y)(14b)
Reservoir: wresLres
dhres
dt=wresLr esRres(t)Qres with hr es(0) = hr0(14c)
Canal–1: wc(L1cL2c)dh1c
dt=Q2cQ1cwith h1c(0) = h10 (14d)
Canal–2: wc(L2cL3c)dh2c
dt=Q3c+γmpor wvαg 1
2y(h2)|y=0 Q2cwith h2c(0) = h20 (14e)10
Canal–3: wcL3c
dh3c
dt=γQtm Q3cwith h3c(0) = h30 ,(14f)
with
Q1c=Cfg wcmax(h1cP1w,0)3/2and Q2c=Cfg wcmax(h2cP2w,0)3/2(14g)
Q3c=Cfg wcmax(h3cP3w,0)3/2and Qm= (1 γ)Qtm (1 γ)1
2mporσeWvαg(yhm)2|y=0 (14h)
Qres =Cfg wres max(hres Pwr,0)3/2and R(h) = wrh/(2h+wr),(14i)15
as well as time-dependent rainfall functions Rres(t), Rm(t)and upstream inflow Q0(t). The remaining parameters are con-
stants, with units and typical values listed in Table 2. The rainfall functions are defined such that, in the absence of other effects,
e.g. unit porosity in the moor, they directly lead to a linear increase of the moor’s ground water level and the reservoir depth.
A space-time numerical discretisation of (14) is given in Appendix A. It involves a second-order finite-difference approxi-
mation of the ground water equation (14b) in y, a first-order finite-volume discretisation of the river equation (14a) in s, and20
straightforward first-order forward-Euler time discretizations of the time derivatives involved.
The rainfall functions are constant during on every wd and generally vary from Wetropolis day-to-day. On a given Wetropolis
day,
Rres(t) = nrnresr0and Rm(t) = nrnmoor r0,(15)
in which nr= 1,2,4,9is drawn daily with probability (3,7,5,1)/16 via one Galton board, while one of the combinations
(nres, nmoor ) = {(1,0),(1,1),(0,1),(0,0)}
is drawn daily with probabilities (3,7,5,1)/16 via the other Galton board, as explained in §2.1. The rainfall speed r0will be25
determined by trial-and-error and has the units of th, i.e. m/s. Hence, the volumetric rate of rainfall per wd on the moor for
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Figure 4. Top panel: the river level h(s,t)of the river (black) and the river velocity Vr(s,t)as function of the along-river coordinate sat
t= 5000s; the bottom panel: topography b=b(s)(in red and fixed), the top of the berm or dike along river/canals in red (fixed); in dashed
blue the bottom of the set of canals; in solid blue the canal levels; in black is the dynamic river level indicated above the bed; all as function
of sat time t= 5000s. When the black line/river level lies above the red lines/berms there is flooding, because at t= 5000s the water level is
seen to be high, cf. Fig.5 bottom-left panel. The black line is seen to have three jumps at s=sres = 0.925m,s=sm= 2.038m and a small
one at s=Lc1= 3.858m where water comes in from the reservoir, moor and canal respectively. At s= 0 there is constant water influx.
Flooding is just observed: in this simplified design model there is no actual water going out the river.
unit nr= 1 can then be calculated, yielding
Vrate =Lywvr0wd.(16)
2.3 Numerical results
Given the choice of parameter values with (or near) the values given in Table 2, the goal is to determine a suitable rainfall
speed r0and length wd via trial-and-error through numerical simulation. As initial conditions we take h(s,0) = 0.0135m,5
hm(y, 0) = 0, zero canal and reservoir levels h10 =h20 =h30 =hres0= 0, and an upstream influx of river water corresponding
to the mass flux Q0=Qf(h(0,0)) associated with h(0,0). In reality, rainfall will be varied daily by changing the action of
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Table 2. Parameters: units and values used. Note that α=k/(mpor νσe).γ[0,1].
Parameter Units Value Parameter Units Value
gm/s29.81 P1wm0.01
Lm4.211 P2wm0.0125
Cf-(2/3)3/2P3wm0.0125
Cmm1/60.02 L1cm3.858
db/ds- 0.01 L2cm3.608
wrm0.05 L3cm1.724
wvm0.095 sres m0.932
Lym0.925 wres m0.123
mpor - 0.3 Lres m0.293
σe- 0.8 Pwr m0.1
k m2108smm2.038
νm2/s 106γ-0.2
wcm0.02
two pumps, which require a fraction of a second to change gear. For someone viewing Wetropolis, a length of day between 5s
and 20s seems reasonable so we have chosen wd = 10s as a first guess. In the design phase, values of r0have been chosen and
tuned in simulations of 100 to 500wd’s, i.e. 1000s to 5000s, which can be simulated in about 10% of real time.
To monitor whether r0has the (approximately) desired value during a simulation, major flooding is defined to occur when
the river level significantly, i.e. by 0.01m or more, exceeds the canal–1 berm along the strip of river bordering the city plain.5
This is monitored visually in daily snapshots, one of which is given in the lower panel of Fig. 4. It contains a compound of
levels to enable this flood monitoring, which requires some explanation. While the canal water enters the river in the city,
for simplicity flood waters of the river are not modelled numerically to enter the canal or city, which suffices for our design
purposes. The information displayed in the lower panel of Fig. 4 is as follows. The vertical axis has units in mso the range
across the length L= 4.21m of the set-up is about 0.06m with the horizontal s–axis lying along the river and canal. The10
zero-level of the canals and rivers in the vertical is put at the river exit (s,z)=(L, 0) with vertical coordinate z. In reality, the
length of river and canal differ slightly owing to the curved channels; for design and our numerical model, this difference is
considered negligible and it is sufficient to assume they are of the same length. The lower, solid and thick red line displays
the fixed sloped bottom of the river (with 1% downhill gradient, i.e., sb=0.01). The thinner solid-black line displays the
river level with the upstream input depth of h(0, t) = h0(t). This is uniform except at the reservoir influx (sres = 0.932m),15
the moor influx (sm= 2.038m) and the canal–1 influx (s= 3.858m) which cause sudden increases in the river levels. These
larger and smaller jumps are indeed visible and identifiable in the lower panel of Fig. 4. The flux into the river from canal–1 is
comparatively small, so the rise in the river level here is much smaller than the time-varying influx of water from the reservoir
and moor. The bottom of the three canal sections is displayed by the (stepped) thick-blue dashed line with the upper canal–3
level at z=b3= 0.04m, the middle canal–2 level at z=b2= 0.02m and the lower canal–1 level at z=b1= 0.0m. The three20
berm (or dike) heights are 0.02m higher at {d3,d2, d1}={0.0,0.04,0.02}m. Canal berm or dike levels are displayed with a
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Figure 5. A four-panel figure in which: (i) the first top-left panel contains the three canal levels and the level of the reservoir as function of
time tall initialised at zero in this run (reservoir: red; canal–1: black, canal–2: blue; canal–3: cyan); (ii) the second top-right panel displays
the river level at s= 0 in blue and the river level at one point in the city in black as function of time; (iii) the bottom left panel shows moor
groundwater level hm(y,t)as function of space yin a snapshot at t= 5000s; and, (iv) panel four, bottom right, shows rainfall per wd = 10s
scaled with the magic factor r0versus time.
(stepped) thick solid-red line, while the three varying canal levels are displayed as the (stepped) solid-blue line. Steps in the
berms occur where the weirs are placed and the jumps in the varying canal levels are determined by the hydraulic weir relations
at these weirs. Some river flooding can occur when the river level, the (stepped) solid-black line, exceeds the canal–2 berm
downstream of the second weir, as is visible in the lower panel of Fig. 4. Major flooding is defined when the river levels exceeds
the canal–1 berm in the city section, i.e. at s=L1c= 3.858m the water depth h(L1c,t)significantly exceeds 0.02m, visible as5
snapshot in the lower panel of Fig. 4, where the solid-black line of the river level is seen to exceed the solid-red canal–1 berm
level downstream of the last weir around s= 3.7m. Via visual optimisation, i.e., monitoring when major flooding occurred in
the city for the extreme or rare events of 90% rainfall in both the reservoir and moor, a suitable value of the rainfall speed is
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found to lie around the value
r0= 2.05104m/s.(17)
The corresponding water volumes for the various Galton-board outputs required in the moor per wd are then
(1,2,4,8,9,18)Vrate = (0.18,0.36,0.72,1.44,1.62,3.24)l/wd = (0.018,0.036,0.072,0.144,0.162,0.324)l/wd.(18)
Consequently, the pump supplying the rainfall on the moor should have a maximum discharge of about 324ml/s, which is5
a manageable amount from a design perspective. Such a discharge is feasible by using inexpensive off-the-shelf aquarium
pumps, both for the supply of the upstream river influx Q0and the varying rainfall amounts in reservoir and moor.
An example of a simulation over 500wds is summarised in Fig. 5. Since reservoir and canals are empty at t= 0, we observe
that it takes about 25wd’s before they are filled. During this time major flooding is lessened, or prevented completely, because
the reservoir and canal in essence act like flood-attenuation storage sites, supplying passive flood control. Extreme rainfall in10
this start-up period tends to be buffered such that city flooding is prevented. Reservoir and canal levels are displayed in the
top-left panel of Fig. 5 versus time. The (constant) upstream river level h(0, t)and city river level h(L1c,t)are displayed as
function of time tin the top-right panel of Fig. 5, in which extreme events with nr= 18 are clearly identifiable as flood peaks
at time twhen h(L1c,t)>0.02m. A snapshot of the groundwater level hm(y, t)in the moor is displayed in the bottom-left
panel of Fig. 5; it shows the no-flux upstream boundary condition at y=Lyand the gradual decrease of the groundwater level15
towards its outflow location at y= 0. The rain units for the moor (being 1,2,4or 9), reservoir (being 1,2,4or 9) and their
summation nr, with the discrete values of 1,2,4,8,9or 18, are displayed in the bottom-right panel of Fig. 5. The peaks of
extreme rainfall are, of course, seen to match the peaks in extreme flooding in the panels on the right except, possibly, during
the first circa 25wd’s when the canals and reservoir tend to act as flood-attenuation buffers.
3 Table-top design20
After the design calculations commenced on May 29th 2016 and were completed at June 8th 20162, the Wetropolis flood
demonstrator was constructed and finalised between June 4th and August 31st by OB and WZ 3. The final design was limited
by the demand to transport it in the back of a car. We note that the reservoir and moor have been swapped in the actual set-up,
relative to the mathematical design, and that the river-channel length has been increased to 5.2m.
The physical Wetropolis model consists of several elements which we describe next:25
The topographic landscape with a winding river channel, one-sided flood plain, canals and the city-plain has been routed
out of two standard polystyrene foam plates each of dimension 5×60 ×120cm3(plus a small extra foam plate) with an
overlay to fit two plates together. A smaller third piece was added to extend the river length after the city which enhanced
2The first design and complete model calculations were presented during a seminar at Imperial College London on June 1st 2016. A week later an error in
the use of the Manning coefficient Cmwas fixed, leading to an increase of the river channel length Lby a factor of four. Hence, the winding channel.
3See public postings in that period around 08-06-2016 and 31-08-2016 on https://www.facebook.com/resurging.flows
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Figure 6. Overview of the Wetropolis flood demonstrator with its winding river channel of circa 5.2m and the slanted flood plains on one
side of the river, a reservoir, the porous moor, the (constant) upstream inflow of water, the canal with weirs (the three small blue foam-wedges
seen in the photograph), the higher city plain, and the outflow in the water tank/bucket with its three pumps. Two of these pumps switch on
randomly for (1,2,4) or 9s of each 10s or wd. Photo compilation: Luke Barber.
flooding in the city plain. An overview is given in Fig. 2 and a photograph is found in Fig. 6. Drawings have been made in
the CAD programs Rhino/Grasshopper and used to steer the router. Routing precision is circa 0.8mm. After the routing,
the river channel and its flood plain have been roughened by varnishing fine sand to the base.
A framework of wooden support slabs has been made that fits on four A-frames. This framework is put together with a
bolt-nut system such that it can be disassembled for transport. Wooden wedges are used to squeeze and level the foam5
pieces within the slab-framework. The three foam pieces are squeezed together to limit leakage. Aluminum one-side-
sticky tape is used locally to seal two sections of the river channel together and thus bridge two adjacent channels.
Rather than sealing off all leakage, which in practice becomes impossible, an “aquifer” system of two interconnected
gutters underneath the seams of the foam pieces leads leaked water back to the holding reservoir with the three aquarium
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pumps. Hence, the water budget is closed in the absence of evaporation, the latter which is negligible on the time scale
of operation, of typically one to a few hours.
Three aquarium pumps with a maximum pumping capacity of 0.375ml/s are placed in a holding reservoir, a rectan-
gular bucket with dimensions 0.3×0.40 ×0.22m3, which bucket is hung underneath the wooden framework in a
rectangular area adjacent to the upstream inflow point of river water. Plastic tubing with inner and outer diameters of5
circa (1.8,2.2)mm leads the water from this reservoir to the upstream point and, depending on whether it rains or not,
to the reservoir and the moor. Rainfall on the moor is spread out and visualised using a copper pipe with numerous
downward-facing holes over the lava grit.
The moor unit is made of acrylic and on the open side-face of the box a gauze prevents the lava grit from avalanching
into the river. The acrylic reservoir is open from the top and water can enter through a hole near the top edge. Outflow10
of water in the river is regulated via an internal pipe which outflow level can be manually adjusted. Hence, active flood
control can be demonstrated by manually adjusting this outflow level. Outflow into the canal can be arranged separately
via an adjustable valve. Note that this is slightly different from the set-up in the mathematical and numerical design
model, where the outflow of moor water was partitioned between the river and canal.
The set-up for the Galton boards including the Galton boards themselves, the accompanying Arduino control units,15
power sockets and plugs to operate the three aquarium pumps. The two draws from the discrete probability distribution
are either computer generated or determined from the random paths of (a) steel ball(s) through the asymmetric Galton
boards. In the latter case, the steel ball triggers a signal by interrupting optical sensors in one of the four channels on each
Galton board, cf. Fig. 1. The signal subsequently steers either the reservoir pump, moor pump, both or none as arranged
via the Arduino technology.20
Further specifications, instructional photographs, a link and design drawings of various components have been provided in
Appendix B.
4 Wetropolis illustrated and demonstrated
Illustrative images of Wetropolis in action are shown in the photographs of Figs. 7 and 8. It includes close-ups of excessively
flooded river-bends and city plain, the reservoir and its outlets as well as the moor under heavy rainfall in Fig. 7. During25
extremely heavy rainfall (90% per wd) after a relatively wet period, the moor becomes supersaturated and the groundwater
level can rise through the lava grit and trigger fast surface run-off. In other situations the groundwater level is below the surface
of the lava grit. Under varying rainfall the rising and falling groundwater level can be observed through the transparent acrylic
walls. This visualisation was inspired by cartoons in hydrological textbooks, in which we often find similar cross-sections of
the earth and its groundwater levels.30
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a) b)
c) d)
Figure 7. Action shots of Wetropolis: a) overview of overflowing reservoir on the left, the lave-grit filled moor under heavy rainfall in the
middle and a flooded city in the background on the right during tests with massive flooding and 100% rainfall over several days; b) zoom-in
of the final river bend and its one-sided flood plain and the canal before the city as well as a flooded city plain in the background on the right
during massive flooding; c) zoom-in of the reservoir with water streaming through the manually adjustable outflow pipe into the river and
the separate valve-adjustable underflow into the canal on the right; and, d) zoom-in of the holding reservoir with the three aquarium pumps
and tubing leading to the constant upstream inflow at the start of the river at s= 0 on the right and two other tubes leading to the reservoir
and moor.
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Figure 8. Photograph of the entire set-up at the Churchtown Flood Action Group workshop on 28-01-2017, with the winding river channel
in the foreground, the city plain with a few smurfs, the groundwater moor, the reservoir on the left behind the moor, and in the background
on the right, the control table with the Arduino units and the two Galton boards as well as an informative poster on Wetropolis.
To date Wetropolis Flood Demonstrator has been showcased to flood victims at public events, attendees of science fairs, and
to scientists and flood professionals at bespoke workshops on natural hazards. Wetropolis has been shown: 4
first at the general assembly of the EPSRC UK network Maths Foresees in Edinburgh in September 2016 to an audience
of scientists with expertise in environmental fluid dynamics and representatives from stakeholders such as the Met Office,
JBA Trust and the Environment Agency; Wetropolis has been created as an outreach project within this Maths Foresees5
network;
at the Churchtown flood action group conference in January 2017 in Lancashire to circa 140 flood victims and several
experts on flooding, see Fig. 8;
as part of two public exhibitions on the Boxing Day 2015 floods in Leeds’ Armley Industrial Museum in December 2017
and March 2017;10
4For movie footage see the posts dated 31-08-2016 [with an extremely rare Boxing-Day-2015-type flood after two consecutive days of extreme rainfall],
06-09-2016, 16-01-2017, 08-12-2016, and 07-04-2017 on https://www.facebook.com/resurging.flows, or at https://youtu.be/1FIHFOn6IPQ and the Boxing-
Day-equivalent flood at https://youtu.be/N4Sp5gHXcz4
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at a bespoke Canal & River Trust workshop in Liverpool in March 2017 by students of Leeds Centre for Doctoral
Training in Fluid Dynamics;
at the “Be Curious” science festival in March 2017 at the University of Leeds;
at the Maths Foresees “Environmental Modelling in Industry” study group at the Turing Gateway to Mathematics,
Cambridge, in April 2017, which particular event focussed on solving mathematical challenges related to flooding,5
see https://gateway.newton.ac.uk/event/tgmw41; and,
at the Yorkshire iCASP confluence (integrated catchment program) in June 2018 for a range of scientists, flood profes-
sionals, stakeholders and politicians, see the right panel of Fig, 1.
The strength of Wetropolis is that it is a physical visualisation of the probability of extreme rainfall and flooding events
including actual and visual river hydraulics, groundwater level changes and interactive flow control. We recall that the reservoir10
has valves such that the audience can store and release water interactively to control and possibly prevent flooding in the city.
Wetropolis is, however, a conceptual model of flooding rather than a literal scale model of a specific catchment. It has, however,
been inspired by the Boxing Day Floods of 2015 of the River Aire, in and upstream of Leeds, UK. This conceptuality is both
a strength and weakness because one needs to explain the translation of a 1 : 200 year return period for a realistic extreme
flooding and rainfall event such as the Boxing Day 2015 flood of the River Aire into one in Wetropolis with its one in 6:06min15
return period, and one also needs to explain that the moor and reservoir are conceptual valleys where all the rain falls, since rain
cannot fall everywhere in the Wetropolis catchment, in contrast to rainfall in real catchments. This scaling and translation step
is part of the conceptualisation, which the audience, whether public or scientific, needs to grasp. The visualisations of flooding
in the city and the ground water level also involve learning steps. Hitherto, this conceptualisation step was either explained
by the Wetropolis wardens in attendance at a demonstration, via our bespoke poster, or both. Alternatively, we aim to arrange20
bespoke audiovisual material.
Due to this learning curve, the most receptive public audiences have been flood victims or people with friends or family
who went through the unpleasant and potentially devasting experience of being flooded. We have perceived such audiences
to be the most receptive, inquisitive and interactive because they have an intrinsic interest in flooding phenomena and wish
particular questions to be addressed in order to gain more understanding as to what causes flood hazards, how these hazards25
can be predicted and how such floods can possibly be tackled through flood mitigation and/or management. Therefore, they
are exactly interested to obtain answers to the questions which were raised in the introduction. In particular, Wetropolis aids
in raising awareness of the probabilistic character and randomness of rainfall and flooding events, also in connection with the
difficulties in predicting some of these extreme events. Combining showcasing Wetropolis with a general public lecture on
the science of flooding has proven to be particularly successful, cf. (Science of floods, 2016; Potter, 2016), owing to such a30
presentation whetting the appetite to view a scale model with rainfall and river flooding. While Wetropolis was designed as a
public outreach project, the reception from flood practitioners and scientists working in environmental fluid dynamics has been
surprisingly positive; we will discuss this reception later.
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5 Summary and discussion
In summary, we have told the story of how the Wetropolis demonstrator of extreme rainfall and floods was constructed after
an efficient mathematical and numerical design enabled us to estimate the characteristic components of the envisioned set-up.
This efficient mathematical model was first presented as a coupled system of ordinary and partial differential equations which
we subsequently solved numerically to define a near-optimal design. While that mathematical model is close to a prediction5
model for river and groundwater flows in Wetropolis, due to its relatively minimalist nature and purpose to facilitate the design,
it is likely not quite sophisticated enough to make bonafide predictions. In a final modelling step, we determined the reasonable
rainfall and flow rates through numerical simulation, on which rates we based the actual design and construction of Wetropolis.
Several improvements and extensions of Wetropolis are under exploration, including the following:
the suggestion to accentuate an actual flooding event in the city more prominently e.g. by measuring the actual flood10
waters of each flood event via a separate drain into a measuring cup or by triggering flashlights in the city to go on by
closing an electric circuit by the flood water and to go off when the circuit is broken; and,
to visualise key principles of Natural Flood Management (NFM) or more broadly Nature Based Solutions (NBS) (e.g.,
Hankin et al. (2017); Lane (2017); Potter (2016); Cabaneros et al. (2018); Bokhove et al. (2018b)) by visualising the
effects of different riverbed roughness to slow down the flow in a cut-out river-bed segment via removable river-channel15
inserts and by including a porous upland catchment with various small-scale river channels and flow-attenuation features
to enhance water storage, the features of which can be manipulated by the audience.
5.1 Games
One of the shortcomings of the current Wetropolis set-up is that it lacks to date bespoke educational material and games. The
game suggestions for the current Wetropolis set-up, which have arisen during various workshops, are as follows:20
The notion of (theoretical and sampled) pdfs can be developed, based on recording histograms of actual Galton board out-
comes and comparison of these outcomes against the theoretical outcomes. Games can be created to determine whether
the outcomes have been tampered with by human intervention, e.g., a game in which one team is allowed to trigger
extreme flooding through tricking the electronic eyes by a finger or by purposely misaligning the Galton boards without
telling the other team whether or not such unnatural interventions took place. While the audience generally likes massive25
flooding to occur more often in Wetropolis, by secretly triggering daily 90% rainfall in both moor and reservoir, record-
ing the Galton board outcomes would immediately reveal that such tampering is unrealistic in that it makes no rain and
low to intermediate rainfall into rare rather than common events.
Building a game on flood prevention in the city by controlling the valves on the reservoir, say over 30 to 100wds,
with the winning team having the least or zero amount of flooding in the city. This can also include a discussion on the30
possibility that the winning team can win by chance over a limited set of trials rather than by virtue of optimal flood
control.
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The audience can play with the set-up. The set-up can namely be modified by interchanging the two-out-of-three loca-
tions where rainfall is random, changing the locations of the reservoir and moor, for example by bringing one unit in
close proximity to the city, including an investigation as to what consequences these changes entail in observed spatio-
temporal patterns.
Each of these suggestions requires further development.5
5.2 New approaches in science and water management
Flood practitioners from various stakeholders have been quite positive about Wetropolis’ novel way to visualise the probability
of extreme rainfall and flooding events via the asymmetric Galton boards and how the outcomes from these boards directly
lead to observable rainfall and river dynamics in the set-up. Stakeholders such as JBA Trust and the Environment Agency see
Wetropolis as a potentially useful tool to trigger discussions about innovations in flood mitigation and water management, as10
part of workshops and brainstorm sessions. To date, Wetropolis has triggered two innovations: one on the use of the revisited
concept of flood-excess volume in devising a novel and graphical cost-effectiveness analysis to flood mitigation, in particular
meant for decision makers, and one on education in water engineering and management.
Flood-excess volume (FEV) concerns the volume of flood waters that caused flood damage. It is the flood volume of the river
flow beyond a certain, chosen and relevant threshold water level. This FEV, expressed in cubic metres (m3), or expressed more15
visually and comprehensively as a square lake of 2m in depth with a certain side length, is a useful measure to devise flood-
mitigation strategies. It allows us to quantify what fraction of the FEV is mitigated by a certain strategy. Our cost-effectiveness
analysis results in a series of square-lake graphs, one for each flood-mitigation scenario envisioned, which express both the
flood volume mitigated by a particular flood-mitigation measure, its cost, its cost per percent mitigated, and the overall costs.
When an accumulation of flood-mitigation measures captures the entire FEV, the FEV is essentially reduced to zero. Building20
higher flood-defence walls in a city at or just above the maximum river level to be mitigated does, for example, reduce the FEV
to zero in one fell swoop. However, building high walls around a river in a city is only one type of flood-mitigation scenario, one
that may be undesirable, so in general flood-mitigation scenarios, expressed visually as square-lake graphs, will consist of an
accumulation of measures such as river-bed widening, i.e. giving-room-to-the-river (GRR), active flood-storage plains, higher
flood walls and NBS. Each measure cuts a certain fraction as a rectangular strip off the square flood lake, with accompanying25
costs displayed. The graphical cost-effectiveness analysis has been developed by us, in a series of papers Bokhove et al. (2018a,
b, c), for several extreme river floods in the UK and France in order to facilitate and improve evidence-based decision-making
by city councils and concerned citizens’ groups.
The R&D consortium “Wetropolis, tangible models for education and water management” is a regional EU EDRF-project5
in the Dutch counties of Salland and Twente that has begun to develop strategy tools to support the dialogue on climate adap-30
tation to flooding and drought. It consists of four parallel activities with an overarching theme:
– Development of educational content and tools to raise awareness in primary schools and extend the Dutch GRR-programme
5The Dutch Wetropolis’ project is led by Dr. Henk de Poot from Nobis, Enschede, The Netherlands, and involves a consortium of Dutch SMEs, schools
and universities –https://www.wetropolis.nl
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in secondary school levels, cf. §5.1; and,
– Build-up of ‘citizen-sensing’6experiments to measure climate-related indicators such as groundwater, drought, and temper-
ature in urban areas.
– Development of outreach models on the water cycle, drought, and heat stress phenomena in the local environment suitable
for hands-on exploration in public settings such as museum exhibits. The overarching theme of the project is to stimulate a5
climate-resilience dialogue on municipal levels.
A particular question pertaining to a revised Wetropolis is how a combined flood-and-drought demonstrator can be created
with a visual appeal similar to, or beyond, the one presented here, given the expectation that extremely dry European summers,
such as those of 1976 and 2018, are likely to occur more often in the future.
Appendix A: Numerical discretisation of entire system10
Allowing for irregular time steps tn=tn+1 tnwith t0= 0, the entire system (14) has the following space-time discretisa-
tion, using regular finite differences for the groundwater equation, a first-order finite-volume method with upwinding for the
river equation, and a first-order forward-Euler time discretisation for all differential equations involved, as follows
River: (hn+1/2
khn
k)
tn
+(Qn
k+1/2Qn
k1/2)
xk
=Qm
wr
δkm +Qres
wr
δkr for k= 1,...,Nx
with Qn
k+1/2=hn
kR(hn
k)2/3sb
Cm
, Qn
1/2=Qn
0=Q0(tn), h0
k=h0k(A1a)15
Moor: (hn+1
jhn
j)
tnαg
y2hn
j+1/2(hn
j+1 hn
j)hn
j1/2(hn
jhn
j1=Rn
m
mporσe
for j= 1,...,Ny1
on yj=jywith (hn
Nyhn
(Ny1))=0, hn
0=hn
3c, h0
j=h0j(A1b)
Reservoir: wresLres
(hn+1
res hn
res)
tn
=wresLr esRn
res Qn
res (A1c)
Canal–1: wc(L1cL2c)(hn+1
1chn
1c)
tn
=Qn
2cQn
1c(A1d)
Canal–2: wc(L2cL3c)(hn+1
2chn
2c)
tn
=Qn
3c+γmpor wvαg 1
2(hn
1)2hn
3cQn
2c(A1e)20
Canal–3: wcL3c
(hn+1
3chn
3c)
t=γQn
tm Qn
3c(A1f)
with
Qn
1c=Cfg wcmax(hn
1cP1w,0)3/2and Qn
2c=Cfg wcmax(hn
2cP2w,0)3/2(A1g)
Qn
3c=Cfg wcmax(hn
3cP3w,0)3/2and Qn
m= (1 γ)Qtm (1 γ)
2mporσewvαg (hn
1)2(hn
3c)2
y(A1h)
Qn
res =Cfg wres max(hn
res Pwr ,0)3/2and R(h) = wrh/(2h+wr),(A1i)25
6A relatively recent form of community-based participatory environmental monitoring with success in the Netherlands, Spain, and Kosovo, cf. Woods et
al. (2016).
24
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94.4
50.0
109.0 99.0
463.1
R65.0
R85.0
R100.0
R150.0
R200.0
R235.0
R215.0
R150.0 R50.0
R85.0
R65.0
R100.0
R250.0
R300.0
R200.0
R150.0
130.0
130.0
100.0
R250.0
R150.0
20.0
50.0100.0
26.9
A
B C
E
F G
J
K
M
N
O
P
Q
Mp C
Mp E
Mp F
Mp N
D-Lock3
H-Lock2
L
Figure A1. Drawings of the basic topographic landscape of Wetropolis with letter indications on the left matching coordinates on the right.
in which Nyis the number of regular grid points in the moor across Ly,y=Ly/Nyand hn
j=hm(jy, tn)and hj+1/2=
(hj+1 +hj)/2;Nxfor k= 1,...,Nxthe number of finite-volume cells xk=sk+1/2sk1/2for the river with cell faces
sk±1/2and cell average
xkhn
k=
sk+1/2
Z
sk1/2
h(s,tn) ds, (A2)
and in which the Kronecker delta stymbol δkm = 1 for the cell kin which smresides and is zero elsewere and likewise δkr = 15
in the cell in which sres resides, etc. A stable, explicit time step is determined using suitable CFL conditions based on the
information speed and the nonlinear diffusion.
25
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Appendix B: Wetropolis’ design details
A GitHub site contains information on the materials used as well as building instructions7. Some design tools and materials
are briefly outlined below:
Matlab programs of the numerical model are available on GitHub, concerning three versions. The third version “table-
topt3v2019.m” was used here, which equals “tabletopt2v2016.m”, except for some relabelling of figure axes.5
Blue polystyrene plates were used “Isolatieplaat polystyreen XPS” of dimensions 120 ×60 ×5cm3. Yacht varnish was
mixed with fine calcinated sand and shells sieved with a sieve (with holes of 0.9m and wire thickness of 0.1m). The
following CAD programs were used: Solidworks for the designs, saved as a Step file for import in Rhino (V5); plugin
in Rhino for routing: Rhinocam 5, which generates routing/freesfiles (NC files); foams were routed on a BZT 1400 PF
router/frees with Winpc-nc driver.10
Aquarium pumps are used: Syncra 1.5, 234240V,50Hz,23W,Qmax = 1350l/h,Hmax = 1.8m.
Design drawings for the topographic foam plates are given in Fig. A1 and photos of the wooden support system in
Fig. B1.
Acknowledgements. The Wetropolis Flood Demonstrator originally started as an outreach project in the EPSRC UK Living with Envi-
ronmental Change (LWEC) Network Maths Foresees, under grant EP/M008525/1 for TK, OB, TH and the materials. The project is also15
partially stimulated by funding for OB and TK from EP/P002331/1 and the EU EDRF “Wetropolis decision-making, participation, education
and outreach with physical and augmented hydrological models (AR)” project for WZ and the Stichting Free Flow Foundation.
7https://github.com/obokhove/wetropolis20162020
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a) b)
c) d)
Figure B1. a,b) The making of the wooden support frame with its bolt-nut system. c) Overview with moor and the first reservoir; notice the
aluminum tape sealing two foam plates. d) Detail of canal and sluice gate as well as a water-level measurement device involving Arduino
technology.
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Conference Paper
https://cmswebonline.com/esrel2021-epro/pdf/134.pdf Flood risk protection measures are designed to reduce intensity, frequency and extent of feared events. For any category of measures ranging from classical civil engineered measures to Nature-Based Solutions (NBS), being able to assess their physical and technical capacity remains a starting key issue and requirement. It is essential both to design effective solutions and also to analyze their reliability during their lifetime. For hydraulic applications, the analysis of this capacity consists in checking that proposed solutions are able to evacuate flood water discharge or to store water volume. The protocol described in this paper provides an easy-to-understand framework to assess and represent the effect of measures on the considered flood event and to compare it with their relative costs. It can therefore be considered as a basis to help decision-making within the risk management process and also as a contribution to the analysis of the safety and reliability of planned measures. The protocol enables rapid a priori, as well as thorough a posteriori, comparisons to be made of the efficacy of various flood-mitigation options and scenarios. We have considered a concept called "dynamic flood-excess volume" (dFEV or FEV) and revisited it in a three-panel graph comprised of the (measured) in-situ river-level as function of time, the rating curve and the hydrograph, including critical flooding thresholds and error estimates. FEV is the amount of water in a river system that cannot be contained by existing flood defences. The new tool deliberately eschews equations and scientific jargon and instead uses a graphical display with FEV displayed as a (dynamic) hypothetical square lake two metres deep. This square-lake graphic is overlaid with the various mitigation measures necessary to capture the floodwaters and how much each option will cost. The tool is designed to help both the public and policymakers grasp the headline options and trade-offs inherent in flood-mitigation schemes. It has already led to better understanding and decision-making regarding flood defences in the UK, Slovenia and France, particularly where a number of alternatives are being considered. Three realistic cases-from the UK, Slovenia and France-will be reviewed, including insights on dealing with uncertainty and on the communication of multiple benefits of Nature-Based Solutions, followed by a Socratic-method dialogue. (... the latter dialogue was set up for our secret admirers.) https://cmswebonline.com/esrel2021-epro/html/134.xml https://cmswebonline.com/esrel2021-epro/pdf/134.pdf 10.3850/978-981-18-2016-8_134-cd
Book
Full-text available
We produced this book as a part of the Making Sense project, which draws on nine citizen sensing campaigns in Holland, Kosovo and Spain in 2016 and 2017. In them, we have developed a form of citizen participation in environmental monitoring and action which is bottom-up, participatory and empowering to the community: this is called citizen sensing. If you are interested in best practices and tools for community engagement and co-creation, this book if for you.
Article
Full-text available
Precipitation is highly variable in space and time; hence, rain gauge time series generally exhibit additional random small-scale variability compared to area averages. Therefore, differences between daily precipitation statistics simulated by climate models and gauge observations are generally not only caused by model biases, but also by the corresponding scale gap. Classical bias correction methods, in general, cannot bridge this gap; they do not account for small-scale random variability and may produce artifacts. Here, stochastic model output statistics is proposed as a bias correction framework to explicitly account for random small-scale variability. Daily precipitation simulated by a regional climate model (RCM) is employed to predict the probability distribution of local precipitation. The pairwise correspondence between predictor and predictand required for calibration is ensured by driving the RCM with perfect boundary conditions. Wet day probabilities are described by a logistic regression, and precipitation intensities are described by a mixture model consisting of a gamma distribution for moderate precipitation and a generalized Pareto distribution for extremes. The dependence of the model parameters on simulated precipitation is modeled by a vector generalized linear model. The proposed model effectively corrects systematic biases and correctly represents local-scale random variability for most gauges. Additionally, a simplified model is considered that disregards the separate tail model. This computationally efficient model proves to be a feasible alternative for precipitation up to moderately extreme intensities. The approach sets a new framework for bias correction that combines the advantages of weather generators and RCMs.
Conference Paper
the abstract is available at https://meetingorganizer.copernicus.org/EGU2019/EGU2019-9112.pdf
Article
Please see https://rdcu.be/bAJQk Natural flood management (NFM) has been advocated as a sustainable alternative to traditional flood management. NFM is based upon the well‐established principle that instead of locally defending floodplains from inundation, it is possible to manipulate river flow at the catchment‐scale (catchment‐based flood management, CBFM) to reduce flood inundation downstream. NFM is a subset of CBFM because the focus is on more ‘natural’ approaches to doing this, even if the associated measures may not be strictly natural. The options for doing this are classified and explained in terms of: (1) reducing the rate of rapid runoff generation on hillslopes; (2) storage of water during high river flows; and (3) slowing flow by reducing the ease of connection between runoff sources and zones of potential flood inundation. NFM is argued to have potential at certain sizes of river catchment but it is argued that there are fundamental arguments and scientific uncertainties in concluding that its potential will also apply at larger spatial scales. WIREs Water 2017, 4:e1211. doi: 10.1002/wat2.1211 This article is categorized under: • Engineering Water > Sustainable Engineering of Water
Article
The paper examines the concept of self-similarity and demonstrates how certain problems are studied with the idea of establishing self-similarity of the solution. Dimensional analysis is performed to derive some basic relations concerning similarity criteria, and these criteria are applied in some sample problems to see how similarity of the solution or of some of the variables can be established. The example of a heat source in a medium with temperature-dependent thermal conductivity is considered. Another problem illustrated is that of strong explosion. A classification of self-similar solutions is given, and the concept of incomplete self-similarity is defined. The intermediate-asymptotic nature of all self-similar solutions is demonstrated. Incomplete self-similarity in the problem of isotropic homogeneous turbulence is studied.
Article
This paper describes the development of a new set of equations derived from 1D shallow water theory for use in 2D storage cell inundation models where flows in the x and y Cartesian directions are decoupled. The new equation set is designed to be solved explicitly at very low computational cost, and is here tested against a suite of four test cases of increasing complexity. In each case the predicted water depths compare favourably to analytical solutions or to simulation results from the diffusive storage cell code of Hunter et al. (2005). For the most complex test involving the fine spatial resolution simulation of flow in a topographically complex urban area the Root Mean Squared Difference between the new formulation and the model of Hunter et al. is ~1cm. However, unlike diffusive storage cell codes where the stable time step scales with (1/Δx)2, the new equation set developed here represents shallow water wave propagation and so the stability is controlled by the Courant-Freidrichs-Lewy condition such that the stable time step instead scales with 1/Δx. This allows use of a stable time step that is 1-3 orders of magnitude greater for typical cell sizes than that possible with diffusive storage cell models and results in commensurate reductions in model run times. For the tests reported in this paper the maximum speed up achieved over a diffusive storage cell model was 1120×, although the actual value seen will depend on model resolution and water surface gradient. Solutions using the new equation set are shown to be grid-independent for the conditions considered and to have an intuitively correct sensitivity to friction, however small instabilities and increased errors on predicted depth were noted when Manning's n=0.01. The new equations are likely to find widespread application in many types of flood inundation modelling and should provide a useful additional tool, alongside more established model formulations, for a variety of flood risk management studies.
On using flood-excess volume in flood mitigation, exemplified for the River Aire Boxing Day 5 Flood of
  • O Bokhove
  • M A Kelmanson
Bokhove, O., Kelmanson, M.A., and Kent, T.: On using flood-excess volume in flood mitigation, exemplified for the River Aire Boxing Day 5 Flood of 2015. Archived at https://eartharxiv.org, 2018a.
On using flood-excess volume to assess natural flood management
  • O Bokhove
  • M A Kelmanson
Bokhove, O., Kelmanson, M.A., and Kent, T.: On using flood-excess volume to assess natural flood management, exemplified for extreme 2007 and 2015 floods in Yorkshire. Archived at https://eartharxiv.org, 2018b.
Using flood-excess volume to show that upscaling beaver dams for protection against extreme floods proves unrealistic
  • O Bokhove
  • M A Kelmanson
Bokhove, O., Kelmanson, M.A., and Kent, T.: Using flood-excess volume to show that upscaling beaver dams for protection against extreme floods proves unrealistic. Archived at https://eartharxiv.org, 2018c.