Backwater effects in surface water streams and on
adjacent lowland areas caused by mostly complex drainage and flow control
structures are not directly computed with hydrological approaches yet. A
solution to this weakness in hydrological modelling is presented in this
article. The developed method enables transfer of discharges into water
levels and calculation of backwater volume routing along streams and adjacent
lowland areas by balancing water level slopes. The implemented and evaluated
method extends the application of hydrological models for rainfall–runoff
simulations of backwater-affected catchments with the advantages of (1)
modelling complex flow control systems in tidal backwater-affected lowlands,
(2) less effort to parameterise river streams, (3) directly defined input
factors of driving forces (climate change and urbanisation) and (4) runtime
reduction of 1 to 2 orders of magnitude in comparison to coupled
hydrodynamic models. The developed method is implemented in the open-source
rainfall–runoff model Kalypso-NA (4.0). Evaluation results show the
applicability of the model for simulating rainfall–runoff regimes and
backwater effects in an exemplary lowland catchment (Hamburg, Germany) with
a complex flow control system and where the drainage is influenced by a
tidal range of about 4 m. The proposed method is applicable to answer a wide
scope of hydrological and water management questions, e.g. water balances,
flood forecasts and effectiveness of flood mitigation measures. It is
re-usable to other hydrological numerical models, which apply conceptual
hydrological flood-routing approaches (e.g. Muskingum–Cunge or
Kalinin–Miljukov).
Introduction
There is open demand in hydrological modelling of rainfall–runoff regimes in
backwater-affected lowlands. The flow routing in lowland catchments is
characterised by artificially drained catchments using manifold flow control
structures. The occurrence of backwater effects in such complex lowland
river streams and on adjacent lowland areas poses an open research
question in hydrological modelling. Adjacent lowland areas in this article
are distinguished by a low ground level and connection to rivers. The size
of lowlands varies from narrow riparian areas, to wetlands, shallow retention
spaces, floodplains, and vast partly urbanised marshlands or swamplands.
Hydrological models are applied to simulate processes in the compartments of
the (1) surface–atmosphere interaction, (2) the transition between
soil–vegetation–atmosphere, (3) the processes in the vadose zone of the soil
and (4) the flood routing in the receiving surface waters. In lowlands, the
last two issues require more detailed considerations because of mostly high
groundwater levels and the drainage against fast-changing water levels in
tidal streams of complex flow control systems. For simulating the
interaction between groundwater and surface water quite a few approaches are
available (Brauer et al., 2014; Waseem et al., 2020; Sun et al., 2016).
However, modelling backwater effects in tidal streams with fast-changing
water levels in complex flow control systems of lowland catchments directly
with hydrological models has not been implemented in most hydrological approaches
up to now (Waseem et al., 2020).
Simulating backwater effects, velocity fields and the spatial distribution
of water depths for flood inundation maps demands 2D or 3D
hydrodynamic–numerical models with the numerical integration of the partial
differential equations describing the flood-routing processes. To compute
spatial detailed simulation results in river streams and floodplains,
coupled hydrological and hydrodynamic model approaches fit well in terms of meeting the
required modelling objectives. But, hydrodynamic–numerical models require
larger effort to parameterise river streams and simulation times, which are
at least 1 to 2 orders of magnitudes longer in comparison to conceptual
hydrological flood-routing approaches to model river streams. High-resolution data describing the topography of the main channel and the
natural floodplain in the case of bank overflow are necessary. Hence, the
availability of suitable detailed profile data from measurements is
significant for hydrodynamic–numerical modelling. The larger effort in data
resources and runtime for hydrodynamic–numerical model simulations is no
limitation for answering special research questions and creating detailed
inundation maps. However, applying a coupled hydrological–hydrodynamic model
shows disadvantages in the application on mesoscales to regional catchment scales
(>100 km2) and for operational forecast
applications. Therefore, it is proposed in this article that a stand-alone
hydrological approach can be beneficial in flood forecasting models to
enable parsimonious and efficient modelling of flood-routing and backwater
effects in lowlands by a conceptual hydrological method producing less
detailed results.
The demand to solve this weakness in hydrological numerical models is
increasing, since in low-lying tidal catchments, the pressure on current
storm-water flow control systems is rising due to combined impacts of enlarged
urbanisation on the one hand and climate-change-induced sea level rise in
combination with heavy storm events on the other hand (IPCC, 2013, 2014;
UN DESA, 2018). Studies about the combined risk of high tides (storms) and
storm-water events are given by Lian et al. (2013), Nehlsen (2017), Klijn et
al. (2012), Zeeberg (2009), Huong and Pathirana (2013), and Sweet et al. (2017).
These selected examples all show conformity about the tendency that lowlands will be faced by higher pressures to mitigate flooding in the future. A
promising flood mitigation measure against the effects of (high)
precipitation events in low-lying catchments is the controlled temporary
storage of water in retention areas. However, state-of-the-art hydrologic
approaches reveal shortcomings in modelling the flood-routing and retention
volume in backwater-affected lowland catchments.
Objectives
To resolve the previously described shortcomings in hydrological approaches to
model the flood routing in backwater-affected lowland catchments five
objectives are defined. The method shall be (1) applicable to model complex
flow control systems in backwater-affected lowlands, (2) efficient by using
short runtimes for real-time operational model application, (3) open for
further model developments, (4) re-useable for other hydrological model
solutions and (5) parsimonious with regard to the complexity of input
parameters. Reaching a balance between model structure details (namely
complexity) and data availability is an important issue to keep the model as
parsimonious and efficient in runtime as possible, but complex enough to
explain the heterogeneity in the areas and the dynamics in the hydrological
processes. To accomplish the five defined objectives for a re-usable, open,
efficient and parsimonious hydrological method to model backwater effects,
the authors suggest developing a conceptual extension approach for
state-of-the-art flood-routing methods (for instance, Muskingum–Cunge or
Kalinin–Miljukov).
Outline
The literature review in Sect. 2 discusses current
weaknesses in hydrological models to simulate backwater effects and
subsequent flooding of adjacent lowland areas. The theoretical concept in
Sect. 3 and the developed method in Sect. 4 explain the worked-out solution. The
implementation of the methodology is realised in the open-source
hydrological model Kalypso-NA version 4.0 (Sect. 5). The evaluation of the method is done using
observed data for an exemplary lowland catchment study in Hamburg, Germany,
where a complex drainage system and backwater-affected streams have a
significant impact on the flow regime (Sect. 6). A
discussion of results points out the main findings and limitations in Sect. 7. The article closes in Sect. 8 with a summary and an outlook on
follow-up research.
State of the art in hydrological modelling to compute flood-routing and backwater effects in lowlands
Flood routing describes the processes of translation and retention of a flood
wave moving along a stream in the downstream direction. To simulate the flood
routing in rivers different approaches are applied: (1) pure black box
(namely empirical, lumped), (2) hydrological conceptual or (3)
hydrodynamic–numerical approaches (Maniak, 2016; Hingray et al., 2014). The
applicable flood-routing method needs to be chosen with respect to the
modelling purpose and available data. Computation of water depths and
backwater effects in rivers as well as on forelands by using hydrological
approaches (1 and 2) is rarely done and up to now has mostly been linked with
comparatively high uncertainties. The missing applicability of hydrological
approaches for simulating backwater effects is shown in a recent study
within the North German lowlands (Waseem et al., 2020).
Commonly applied conceptual hydrological approaches are described e.g. in Todini (1991) with the indicated year of publication: storage routing as presented by Puls (in 1928), Muskingum or Muskingum–Cunge routing described by McCarthy (in 1938) and Cunge (in 1969), and Kalinin and
Miljukov routing (in 1958) or linear reservoir and channel cascade routing presented by Maddaus (in 1969). The purpose of hydrological flood-routing approaches is to
compute the discharge hydrographs in the considered stream segments. For
hydrological approaches, conceptual or empirical parameters are calibrated
based on observed events like in the frequently used Muskingum method. A
compromise involves hydrological methods using profile data on streams to model
the flood routing, for example in the Muskingum–Cunge approach
and the approach of Kalinin and Miljukov. These concepts use
profile information in a conceptual way and require far less calculating
effort for mesoscale modelling (>100 km2) than
hydrodynamic–numerical approaches.
Only a few related studies are available with respect to modelling backwater
effects in mesoscale catchments with hydrological approaches, while none of
the reviewed studies enabled the computation of backwater retention in
lowland areas for mitigating backwater-induced flooding. Coupled
hydrological–hydrodynamic computation models like in MIKE SHE coupled with
MIKE 11 (Waseem et al., 2020) or in the German Model NASIM coupled with a
hydrodynamic computation model (Loch and Rothe, 2014; Dorp et al., 2017) are
not part of this comparison because of the previously described disadvantages in
hydrodynamic approaches. A focus is set on direct or stand-alone
hydrological model enhancements.
In Waseem et al. (2020), a review of models is published with regard to
simulating important hydrological processes in coastal lowlands. This review
shows weaknesses in SWIM (soil- and water-integrated model) and
HSPF (hydrological simulation program Fortran). The approaches in the
models SWAT (soil and water assessment tool) and MIKE SHE show good
conformity to simulate processes in lowlands, while both are not applicable
to model backwater effects in the river, on floodplains, or in other adjacent
lowlands as well as backwater effects caused by control structures (sluices,
pumping stations and tide gates). An enhanced approach in SWAT for riparian
wetlands (SWATrw) is presented in Rahman et al. (2016) to compute the
surface water interaction between river streams and explicitly defined
wetlands, while backwater effects in streams are unconsidered. The modified
SWAT-Landscape Unit (SWAT-LU) model enables the computation of horizontal hydraulic
interactions between a river and the aquifer beneath the adjacent floodplain
(Sun et al., 2016). Similarly, in the rainfall–runoff model WALRUS
(Wageningen Lowland Runoff Simulator) a lumped approach is realised to model
the following processes: (1) groundwater–unsaturated zone coupling, (2)
groundwater–surface water feedbacks, and (3) seepage and surface water
supply (Brauer et al., 2014). These are important model features to model
the runoff regime in lowlands, but neither of the approaches enables the
computation of backwater effects (1) along streams, (2) among stream sections and
the land surface, and (3) in river sections influenced by upstream of control
structures.
More nation-specific studies to model backwater effects in streams are
done with the German model ArcEGMO (by the Büro für Angewandte
Hydrologie, Berlin). The hydrological model ArcEGMO takes into account
backwater effects by hindering the downstream flood routing when the water
level at the downstream segment is higher than the upstream one
(Pfützner, 2018). The method presented by the National Hydrological
Forecasting Service (NHFS) in Hungary (Szilagyi and Laurinyecz, 2014)
applies a discrete linear cascade model to account for backwater effects in
flood routing by adjusting a storage coefficient of the cascade. The ArcEGMO
and NHFS methods calculate a retained flood routing, but neither computes
backwater volume being routed into upstream segments by a reverse flow
direction or the backwater-induced flooding of adjacent lowland areas.
In a study by Messal (2000), backwater effects among river streams and the
subsurface flow in riverbanks are modelled exemplarily for the catchment
Stör (1157 km2) in Schleswig-Holstein, Germany. Messal
applies a proportional relationship between upstream and downstream elements
for calibration purposes. The model serves well for the catchment study
Stör, but the parameter values are non-transferable to other catchments
because of a lack of physical descriptions.
Another approach is presented by Riedel (2004) to model the backwater
effects among river streams in German lowlands for the example of two tidal
tributaries of the Weser River. The approach uses the reservoir cascade
theory including the input parameters of the roughness coefficient by
Manning–Strickler and geometric descriptions of the profiles for the flood-routing computation. The river is modelled as a cascade of reservoirs
(namely a NASH cascade), while the water level from the previous time step
of the downstream segments is taken into account to compute the flood
routing. A time step shift in the computational approach is accepted by Riedel (2004) because he reduced the simulation time step size to 1 min. The model computes a reservoir cascade on the basis of a defined
boundary condition at the downstream segment. However, the explicit
simulation of backwater-induced flooding of flood prone areas or adjacent
lowland areas is not included.
These reviewed hydrological methods compute backwater effects in a more or
less conceptual way with the described weaknesses and limitations. None of
these studies analysed the backwater-induced flooding of lowland areas or, in
this specific case, retention areas. Consequently, none of the studies
accomplish simulation of a controlled retention of backwater volume in such
areas, subsequent drainage or the hydrological
processes influenced by backwater-induced flooding. Further on, most studies
do not apply physically based parameters to transfer validated values and
knowledge from one catchment to other studies. A methodology to solve these
shortcomings is proposed in this article.
Theoretical approach to enhance a hydrologic conceptual flood-routing method
to compute backwater effects
To reach the described objectives, a state-of-the art conceptual
hydrological method is extended to be applicable for the computation of
backwater effects in streams and adjacent lowland areas (incl. retention
areas). This section describes the theory of the conventional hydrological
approaches to compute the flood routing (Sect. 3.1), the
concept of modelling control structures in tidal lowlands
(Sect. 3.2), and the approach to compute backwater effects
with a conceptual hydrological approach in streams and adjacent lowland
areas (Sect. 3.3).
Conceptual hydrological flood-routing approach
State-of-the-art hydrological flood-routing theory in free-flow conditions
describes the flood wave propagation in streams which are not affected by
downstream conditions. This means that an afflux in front of obstacles
downstream of the considered stream segment is assumed to have no impact on
the upstream segments. With this assumption, backwater effects are not
considered. Flood-routing processes depend on the characteristics of the
drainage network comprising the geometry of profiles, gradients and roughness
of the streams. Linear or non-linear Muskingum approaches have no physically
based parameterisation and require input parameters which are based on
observed data in upstream and downstream segments of rivers. Therefore,
these hydrological approaches are not suitable for simulation with
changed geometries or changed flow conditions in streams where no observed
data are available. This lack is solved in two approaches, which are based on
physical characteristics such as river geometry, stream length, roughness
coefficient and riverbed slope. On the one hand, the Muskingum–Cunge (often
used in the United States) is applicable, and on the other hand, the Kalinin–Miljukov (KM)
flood-routing approach is applicable. For this work, the approach of
Kalinin–Miljukov is chosen, since this approach is widely applied in Germany
and eastern Europe.
The approach of Kalinin and Miljukov (1957) (KM approach) divides a stream
into a number of characteristic lengths. Each length is considered to be
short enough for assuming a quasi-stationary relationship on the basis of a
hysteresis curve. Different derivations of the KM approach are given in
the literature and discussed, for example, by Koussis (2009). More details
about the applied approach in this work are explained in the Supplement, Sect. S4.
With such conceptual hydrological flood-routing approaches the magnitude and
time of flow along a stream on the basis of stream characteristics are
determined. They describe the (free-flow) propagation of discharge through
streams, whereby translation and retention processes along the stream change
the shape of the hydrograph from an upstream to a downstream point. The explicit direction of
computation from upstream to downstream restrains the modelling of backwater effects. This means that backwater effects caused by an afflux are ignored in these
conceptual hydrological approaches, and an extension is therefore developed
in this article (see Sect. 3.3).
Concept to model control structures in lowland catchments
Backwater effects in river sections are often caused at obstacles like
weirs, (tide) gates, and retention or detention reservoirs, which also function
as control structures in streams. It is required to model these structures
in hydrological models since such control structures are regularly used to
control the flow in catchments. In this article, we focus on control
structures frequently applied in lowland drainage areas. Operation rules of
control structures are mostly pre-defined depending on operative criteria.
The criteria are normally based on thresholds of water level, discharge or
precipitation intensity within hindcasted or forecasted data (see Fig. 1).
Since the data time series influence the status of control structures, they
are defined in this article as drivers. There is a difference between
pre-set and on-the-fly processed driver data. Pre-set data time series are
imported such as observed water level or precipitation data. Additionally,
data series which are computed during runtime (e.g. discharge) can likewise serve
as drivers and are processed on the fly.
(a) Illustration of operative criteria in a control function
depending on driver time series of precipitation, water level and discharge.
(b) Scheme of a control structure with a control function changing the water level W(t), volume V(t) or outflow Q(t) per time step t.
When a threshold of an operative criterion is reached during the runtime of
the model, the status of the system is changed (e.g. opening or closing a
gate). The change in the status based on reached thresholds is described in
control functions, which are checked per time step. In a control structure
the retained water can cause backwater effects in the upstream direction if an
afflux of water occurs. Control structures are one component type within a
hydrological network. Other component types are streams (linear data
structures), areas (spatial data structures) and nodes (point data
structures). An explanation of these components of a hydrological network is
given in the Supplement (Sect. S3).
Concept of the flood-routing enhancement to compute backwater effects
The previously described hydrological conceptual approach (here, of Kalinin and
Miljukov) is enhanced by using the resulting water level, volume and
discharge (WVQ) relation to compute backwater effects per stream element.
The concept enables the computation of a backwater volume routing according to the
water level slope. This is illustrated in a scheme in Fig. 2 for a river
longitudinal segment which is separated in several strands. At the
downstream segment a control structure is located. In stage (1) the free
flood routing in the downstream direction is computed. When the barrier (e.g. a
tide gate) is closed by control functions (stage 2), an afflux of water is
generated (stage 3). The afflux initiates a “backwater volume routing”
(stage 4), meaning that the water volume is routed in the upstream direction to
equalise the surplus water level of the afflux. When the barrier is opened,
the backed-up water volume is routed downstream (stage 5). These five stages
are computed according to the water level slope in each time step. The
methodology to realise the coding of this theoretical concept into a
numerical hydrological model is explained in Sect. 4.
Scheme of five computation steps in the developed concept to
compute backwater effects with a hydrological approach: (1) free-flood-routing computation downstream, (2) control structure simulation, (3) afflux
computation, (4) backwater-volume-routing computation in the upstream direction
including adjacent lowland areas (as well as retention areas) and (5) free-flood-routing computation after opening the barrier.
Methodology to compute backwater effects in rivers and adjacent lowland areas with complex flow control systems
The methodology to calculate backwater effects with a hydrological
conceptual approach consists of three main algorithms: a transfer of
discharges to water levels and volumes per stream segment and time step
(Sect. 4.1), the calculation of (inter)active
control structures (Sect. 4.2), and a backwater
volume routing according to the water level slope along stream segments and
adjacent lowland areas (Sect. 4.3).
Transfer of discharges to water levels and volumes
The flood routing in stream segments of the hydrological network is computed
with conceptual hydrological approaches like Kalinin–Miljukov or
Muskingum–Cunge (see Sect. 3.1). A transfer of
discharges into water levels and volumes is done by calculating the flow
regimes using the approaches of Manning–Strickler or Darcy–Weisbach.
According to the Kalinin–Miljukov approach, each stream segment is divided
into a cascade of n reservoirs with a characteristic length Lc and
the coefficient Kc. The WVQ relations for different states (nWVQ) in
the stream segment are defined with an interpolation between supporting
points of water level heights. This results in a division of the bankfull
water level height Hfull (m a.s.l.) into (nWVQ) states with a water
level difference ΔH (m). Three calculation routines are integrated in
the flood-routing method to compute the flow velocity in stream segments. The
appropriate calculation routine is selected according to the stream
segment's profile and data availability. Stream segments with a circular
profile are computed with the Darcy–Weisbach approach and the flood-routing
method of Euler (1983). Stream segments with rectangular or trapezoidal
(angular) profiles are likewise computed with the Darcy–Weisbach or with the
Manning–Strickler approach. The equivalent sand roughness ks (m)
using the Darcy–Weisbach approach and the roughness Kst (m1/3 s-1)
using the Manning–Strickler approach are input parameters. The algorithm of
these three calculation routines is illustrated in the flowchart in Fig. 3.
The Fortran code and equations to compute the following list of flood-routing parameters are explained in the Supplement, Sect. S4: flow velocity v, characteristic
lengths LKM, number of characteristic reservoirs nKM, retention
parameters KKM, water levels W, volumes V and discharges Q, where
KM indicates the parameter calculation according to the Kalinin–Miljukov
approach.
Algorithm to compute the relations between water level, volume and
discharge (WVQ) per stream segment.
Calculating (interactive) control functions of drainage systems
A control structure of a linear stream segment is defined with unsteady
WVQ relations, and the flood routing is modelled with a storage indication
method. In this work the modified Puls method is applied. The outflow of the
control structure can be distributed to four receivers (Fig. 4). Operative
criteria of control structures are defined for three types of driver time
series, which are precipitation intensity, water level stages and discharge
values. Hydrographs of water level stages and discharges are results given
at junction nodes, while precipitation time series are related to
sub-catchments as spatial input data. The status of control structures is
checked per time step during the execution of the numerical model. A
differentiation of control function types is done according to their
operative criteria depending on pre-set (externally pre-processed), on-the-fly
(internally processed) or interactive on-the-fly driver time series. The three
control function types and the dependency on the location of the operative
criteria are listed in Fig. 4. Control function type (1) depends on observed
or externally forecasted driver time series, for instance precipitation
intensity or water level gauge data. These control functions are computed in
the pre-processing phase of the simulation run to set the status of a
control structure. With forecasted data a time duration can be set to change
the status of control functions (closing or opening a gate) with a specific
lead time before the threshold (operative criteria) is reached. In the
control function type (2), criteria depend on the output of computed
parameters of the hydrological network, namely water level or discharge. The
functions are computed during the simulation run on the fly. This
procedure depends on the condition that the driver elements are located
upstream of the control structure and are not influenced by backwater. If the
criteria of a control structure depend on downstream or backwater-affected
conditions in an interactive system, a recursive calculation routine is
started to compute the control function type (3). The recursive calculation
routine is explained in Sect. 4.3.
Scheme of a control structure with discharge distribution
functions to four receivers and the three control function types depending
on operative criteria.
Calculating backwater effects along river streams and adjacent lowland areas
An afflux due to natural or artificial obstructions (for instance, gates or
weirs) leads to a rise of water level in upstream segments. To simulate the
resulting backwater effects, the downstream-directed surplus water volume is
reversed as backwater when the downstream water level is higher than
upstream. This concept is illustrated in the theoretical approach in Sect. 3.3 and comprises the simulation of backwater
effects, which cause the flooding of upstream lowland areas. The developed
algorithm to compute these backwater effects is illustrated in the flowchart
in Fig. 5. The calculation routines are nested in computational loops as
follows: a spatial loop of streams and areas is nested in a time loop. The
time loop is again nested in a backwater system loop.
Algorithm to compute backwater effects in streams and lowland areas
(like retention areas) with the indicated calculation routines (a, b, c, d).
It is realised with a space-before-time algorithm for modelling backwater
effects and control structures per backwater system.
Each backwater system includes several component types of a hydrological
network: linear structures (stream segments), spatial structures
(sub-catchments of lowland areas), junction nodes and a control structure
(tide gate or water level gauge) at the downstream segment. For the control
functions type (1) and type (2) (see Sect. 4.2)
the calculation routines (a) to (c) in Fig. 5 are executed, while at any
element an afflux condition is present (see query “Is backwater system
active?” = yes). Additionally, per backwater system (j) and per time step
(t) a query checks if an interactive backwater system with a control
function type (3) is defined. An interactive system depends on both
downstream and upstream conditions. In the case of an interactive system, the
flag for a “recalculation” loop is activated. The final balanced stage is
reached when in a backwater-affected system the downstream water levels are
not higher than the upstream water levels within a range of a minimum
“tolerated” water level difference. The method demands the definition of a minimum
difference (ΔWmin) according to the application purposes. A
smaller tolerated water level difference increases the accuracy of computed
water level results. At the same time, this increases the number of
backwater computational runs (k=k+1) before reaching a maximum
number (currently: k=10000). This critical state prevents infinite
calculation routines and a warning shows if this limit is reached to check
the input parameters, which include an adjustment of the tolerated water
level difference. In the exemplary evaluation study (see Sect. 6), a water level difference of about ΔWmin=0.01 m gives sufficient results for mesoscale stream segments.
For local-scale stream segments a difference of about ΔWmin=0.001 m gives adequate results (Hellmers, 2020). Backwater
effects are computed in open stream segments and adjacent lowland areas
which are part of the defined backwater system. For intermediate closed
circular profiles having a limited storage capacity, the backwater volume is
routed upstream to the next open stream segment.
In the calculation routine a (Fig. 5), the initialisation of formal parameters of each linear and
spatial data structure for the backwater effect computation is performed.
This includes an initialisation of the water level, volume and discharge per
time step. Discharges are computed with the flood-routing approaches
described in Sect. 3.1. The corresponding water
levels and retained water volumes are derived from the calculated
WVQ relations per stream segment (see Sect. 4.1).
The initialisation of the parameters for the backwater effect computation is
illustrated in Fig. 6. For the computation of backwater effects, the formal
parameters of each linear and spatial data structure are initialised. This
includes an initialisation of the water level, volume and discharge per time
step. Discharges are computed with the flood-routing approaches described in
the Supplement Sect. S3. The corresponding water levels and retained
water volumes are derived from the calculated WVQ relations per stream
segment. When the volume in the control structure is increased (V(t)>V(t-1)), afflux is generated and the flag for afflux conditions is set to “true”. The difference in volume between time steps (ΔV(t-1)h) is revised continuously during the following backwater calculation
routines (b) and (c) (Fig. 5). When the volume in the control system is
decreased (V(t)<V(t-1)) or not changed (V(t)=V(t-1)) the flag for afflux conditions is set to “false” and the volume (ΔV(t-1)) is reduced by the proportion of the changed volume ΔV,
which has already been processed in the time step before. The
upstream-directed backwater routing is computed if the afflux conditions flag
is set to true. The downstream-directed backwater routing is computed if
the afflux conditions flag is set to false.
In the calculation routine b (Fig. 5), the backwater effect computational loop in the upstream
direction is activated, while afflux conditions are present in the backwater
system. The calculation is done per stream segment in a computational loop
starting at the downstream element (i=n). If the difference in water
levels between the current and the upstream segment is larger than the defined
tolerated water level difference ΔWmin, an algorithm to compute
the backwater effect is activated. The backwater quantity derived from an
afflux at the downstream segment is routed to the upstream segments. Along
the streams, spatial structures (like lowland catchments) are linked where
the water is retained or causes backwater flooding. This developed concept is
illustrated in the scheme in Fig. 7, where the backwater effect computation
between stream segments with linked spatial structures (retention areas) is
shown. The formal parameters of the WVQ relations of the current (i) and the
upstream (i-1) segment are processed. The computation is done in three
sub-calculation routines (namely A, B and C) to compute the water level and
volume stages.
Explanation of the sub-calculation routine (A): in the case of adjacent lowland areas (linked spatial data structures), a
portion of water flows from the stream segment (i) into the respective linked
areas (i) if the water level exceeds the riverbank. The inflow continues
until the water level in the stream Wi(t) is in balance with the water
level in the linked spatial data structures Wi,areas(t). The result is
a decreased difference in volume ΔVi(t) to be routed to the
upstream segment (i-1) per time step.
Explanation of the sub-calculation routines (B) and (C): the computed backwater effect in the
calculation routine (B) describes how the water volume ΔVi(t)
is added to the upstream linear data structure V′i-1(t)=Vi-1(t)+ΔV(t), whereupon the water level is derived from the
WVQ relations. If the upstream segment is linked with another spatial data
structure as illustrated in Fig. 7 (case C), the balancing of water level
and volume is done according to the procedure in (A). As long as a
backwater effect is present in any river segment or adjacent lowland area,
the calculation is repeated (until k=10000). The algorithm to compute
upstream-directed backwater effects on the water levels and volumes is
illustrated in Fig. 8. If the following queries are true, the upstream
backwater effect computation is executed. These queries are called at the
beginning of the calculation routine (see “Are afflux conditions present?”)
with the following equation:
is Wi(t)-Wi-1(t)>ΔWmin? and is Vi(t)>Vi,free(t)?,
where the water level Wi(t) (m a.s.l.) and volume Vi(t) (m3)
are defined by the WVQ relation per stream segment with the index (i). ΔWmin(t) is the tolerable backwater-affected water level rise given for
the stream segments (m) in the backwater system. Vi,free(t) is the
water volume in the segment without backwater effects, which is computed with
the flood-routing method.
While afflux conditions are present, the water level in the current stream
segment (i) is reduced by the minimum water level difference ΔWmin(t). The adjusted storage volume of the stream segment V′i(t)
is defined accordingly by the WVQ relation. The adjustment of the stream
segment (i) is done with Eqs. (2) to (4).
2Wi′(t)=Wi(t)-ΔWmin3Vi′(t)=f(Wi′(t))→Derivation of the WVQ relations4ΔVi(t)=Vi(t)-Vi′(t)
Here, i′ indicates the adjusted stage in the stream segment (i). This
results in a difference of volume ΔVi(t), which is routed to a
linked spatial data structure (for example, retention areas). This
calculation routine is indicated with (A). Otherwise, the backwater is
directly routed to the upstream linear data structure (i-1). These
calculation routines are indicated as (B) and (C) in Fig 8.
In the calculation routine c (Fig. 5), the backwater volume is routed downstream if the afflux
conditions at the downstream segment of the backwater system are no longer present, for instance by opening a gate or starting additional pumping. The
water level and storage volume in the stream segments are reduced per time
step until free-flow conditions are reached. In the developed calculation
routine the drainage process of the backed-up water volume is calculated.
The stream segments are computed in the order from upstream (i=1) to
downstream (i=n). The algorithm for the computation of the subsequently
drained backwater in the downstream direction is applied stepwise with the current
(i) and the downstream (i+1) data structures using the sub-calculation
routines (C) to (A) in reversed order (see Figs. 7 and 8).
In calculation routine d (Fig. 5) interactive systems are computed. When a control structure
depends on criteria of a downstream backwater-affected system, an interactive
computational loop is activated. In this case a recalculation loop is
started and revises control structure settings if the results of the
interactive backwater system are available. Then the recalculation loop
restarts the computation of the calculation routines (a) to (c) (Fig. 5).
The results of this developed algorithm to compute backwater effects are the
time series of water levels (m a.s.l.), discharges (m3 s-1) and volumes
(m3) for stream segments and linked spatial data structures (e.g.
lowland catchments). Additionally, the activated control functions per
control structure are given as time series for verification purposes.
Algorithm to initialise WVQ relations in streams, control
structures and areas per backwater system. Details are illustrated in Fig. 5 in the algorithm (a).
Scheme of the sub-calculation routines (A), (B) and (C) to compute
backwater effects in stream segments and adjacent lowland areas (spatial data
structures). The sub-calculation routines are part of the main calculation routine (b and c, Fig. 5).
Computation of the upstream-directed backwater effect over
stream segments and adjacent lowland areas (e.g. retention areas).
Details are illustrated in Fig. 5 in the algorithm (b).
Implementation of the hydrological method for calculating backwater effects in Kalypso-NA (4.0)
Implementing the developed method into target software is done for
evaluation and application purposes. The implementation is realised in the
open-source model Kalypso-NA (4.0), which has constantly been under development
and applied for more than 20 years in research and practice. The numerical
model features are semi-distributed, deterministic, multi-layered and
combined conceptually–physically based. The model shows strengths in short
computation times, which is in the range of max 3 min on typical desktop
computers (with e.g. i7-5600U CPU processor) for large catchments (ca. 200 km2) using a time step size of 15 min for a 14 d
simulation. It is applicable for real-time operational simulations in flood
forecasting. In combination with the Kalypso Project providing a user
interface, the model Kalypso-NA is applicable for calculating the
rainfall–runoff regime in catchments by users who are not familiar with
input scripts. Open access for developments and user application is
supported by an online-accessible commitment management via the Source Forge
platform and a wiki as an online manual. More information about the software
product Kalypso and the model Kalypso-NA is provided in the Supplement Sect. S1. Such an open-source module provides accessibility to the implemented methods and
therewith supports re-use in other hydrological models. The
purpose is to support good scientific practice towards open and reproducible
science.
The algorithms in the source code of Kalypso-NA are extended for the
integration of the developed methods for backwater effect computation in
rivers and adjacent lowland areas. The hydrological numerical model
comprises algorithms in the form of time loops executed within a spatial
tree structure (time-before-space algorithm) and spatial calculation
routines executed within a time loop (space-before-time algorithm). Both
approaches are integrated in the source code of Kalypso-NA (4.0) as
illustrated in Fig. 9. A time loop nested in a spatial loop accomplishes the
simulation of data structures (such as sub-catchments, stream segments,
junction nodes or retention areas) in the downstream direction on the basis of
the overall results of the upstream data structures. This means that the
data structures are computed for the whole simulation period consecutively
in the order given by the hydrological network from upstream to downstream.
More information about the hydrological network is given in the Supplement Sect. S3. The first
implementation (Part A) provides actual time-dependent results of data
structures to set control functions or drainage criteria in the hydrological
network. This method is applied in the extended algorithm to model processes
in sub-catchments like the soil water balance and the downstream-directed
flood routing. This algorithm is explained in more detail
in Hellmers and Fröhle (2017).
Structure of the implemented primary and secondary algorithm in
the source code of Kalypso-NA (4.0). The enhancement of the primary
algorithm is published in Hellmers and Fröhle (2017). The new
(secondary) algorithm is explained in Sect. 4.3.
Additionally, an algorithm is implemented when spatial calculation routines
are nested in time loops. This secondary algorithm provides the overall
results of a backwater-affected system per time step before calculating the
next time step. The time loop is additionally nested in a backwater system
loop. In that calculation routine the backwater effects in streams and
adjacent lowland areas as well as the evaporation from submerged water
surfaces are computed. This implementation is labelled as space-before-time
algorithm and is illustrated in Fig. 5. The implemented hydrological model
approach is applicable to other catchment studies, while using
physically based input parameters. The input and output parameters are listed
in the Supplement Sects. S2 and S5.
Exemplary model application and evaluation
The objective of the model evaluation is to determine the reliability of the
numerical model results to be in a sufficient range of accuracy for the
designated field of application (Law, 2008; Oberkampf and Roy, 2010;
Refsgaard and Henriksen, 2004; Sargent, 2014). An evaluation of the extended
model Kalypso-NA (4.0) is performed by comparing the results of the
numerical model with observed data from gauging stations in the mesoscale
catchment Dove–Elbe. This exemplary catchment comprises a tide gate and several sluices, weirs and low-lying catchments drained by pumping
stations. The drainage through the tide gate depends on low tide conditions.
At high tide, the gate is closed, causing backwater effects in the streams.
Description of the backwater-affected lowland catchment Dove–Elbe
The mesoscale catchment area Vier- und Marschlande has a size of 175 km2 and is located in the south-east of Hamburg, Germany (see
Fig. 10). The downstream river segment Dove–Elbe is a stream of 18 km in
length and a tributary of the tidally influenced Elbe River. Further tributary
streams which drain into this main river segment are the Gose Elbe,
Schleusengraben, Brookwetterung and a downstream segment of the Bille. These
streams are part of the analysed mesoscale catchment. The soil is mainly
peat and clay with a varying spatial distribution and thickness. Another
regional-scale catchment (namely of the river Bille) with a size of about
337 km2 drains into the study area Vier- und Marschlande.
Thus, an overall catchment area of about 512 km2 is drained
through the tide gate Tatenberger Deichsiel.
Map of the application study area Vier- und Marschlande (175 km2): sub-catchments, gauging stations (1 to 7), studied
backwater-affected streams of the Dove–Elbe, three retention areas in the
main stream and control structures (A to G).
The downstream-situated water level in front of the tide gate is affected by
a mean tidal range of about 3.7 m (Nehlsen, 2017). The mean low water (MLW)
is at about -1.5 m a.s.l., and the mean high water (MHW) is at about 2.2 m a.s.l. The tide gate closes when a water level of about 0.9 m a.s.l. is
exceeded in the Elbe River. During the closure period of the tide gate,
water is retained in the stream segments of the Vier- und Marschlande
catchment, leading to an afflux of water, which causes backwater effects. The
numerical model includes 75 sub-catchments, 75 junction nodes, 75 mesoscale
stream segments, 7 gauging stations and 7 control structures. These control
structures comprise gates, weirs, pumping stations and a tide gate (see Fig. 10). The control functions comprise the opening and closure of gates
and sluices or starting of pumps according to defined criteria. The backwater-affected river segments in the Dove–Elbe with a length of about 12.5 km are
characterised by wide profiles (width >100 m) and wide flood-prone areas (width >200 m) on the mesoscale.
For the computation of the flood routing, the Kalinin–Miljukov method for
mainly irregular profiles with five reservoir parameterisations is applied.
An explanation is given in the Supplement Sect. S4.3. Additionally, a scenario simulation is
performed within the research project StucK (Long-term
drainage management of tide-influenced coastal urban areas with
consideration of climate change; https://www.stuck-hh.de, last access: 1 May 2021) with three retention areas (300 000 m2), which are indicated in Fig. 10. The application and
evaluation results of the research project StucK for the Dove–Elbe streams
as part of the Vier- und Marschlande catchment are summarised in the
following section.
Application and evaluation results
An evaluation of the developed method to compute backwater effects with
Kalypso-NA (4.0) is done by comparing numerical model results with data from
gauge measurements along the river stream segments of the Dove–Elbe. The
analysis of two flood events is presented. Measurements of five gauging
stations in the Dove–Elbe stream segments are available for a flood event in
February 2011, and the measurements of the downstream gauging station are
available for a flood event in February 2002. The locations of gauging
stations and control structures are indicated in Fig. 10.
The results at the downstream gauging station (Allermöher Deich) are
illustrated in Fig. 11 for the opening and closing function of the tide gate
(in red) according to water levels at the downstream gauging station
Schöpfstelle in the Elbe River (in dotted violet) for the event in
2002. The tide gate closes when a water level of 0.9 m a.s.l. is exceeded at
the downstream gauging station Schöpfstelle. In the illustrated
example of February 2002, the tide gate remained closed two times during
storm tides, meaning that the Elbe River water level during low tide periods did
not fall below the required minimal water level of 0.9 m a.s.l. The long
closure times generated a large afflux up to a water level of 1.7 m a.s.l. and
consequently large backwater effects in the Dove–Elbe streams. The simulated
and observed peak water levels show an average difference of about 0.02 m.
The differences in peak water levels are in the range of 0.01 to 0.10 m.
This corresponds to a variation of 1 % to 10 % in the streams with a
backwater-affected water level variation larger than 1 m. The root mean
square error (RMSE) (<0.12 m) and coefficient of determination
(R2) (>0.9) of the flood event analysis confirm
the good result evaluation. The RMSE and R2 show a very good
fit for the rising limb of the flood event. Because of an exceptional manual
pre-opening of the tide gate by the authority, ca. 1.5 h before reaching
the water level of 0.9 m a.s.l. in the Elbe, the simulated control function
and observed status of the control structure are not comparable for the
falling limb (details are illustrated in Supplement Sect. S6). During the rainfall storm event in
February 2011, the water level increased due to backwater effects caused by
high flood discharge from upstream catchments. Here, a difference of less
than 0.01 m is shown between observed and simulated peak water levels. The
scatter plot, the R2 and the RMSE for the flood event
analysis on 7 to 8 February 2011 show a good concordance. An
interactive backwater system is present for the downstream Dove–Elbe river
section, which is influenced by the control structures Reitschleuse
(blue, Fig. 11) and Dove–Elbe Schleuse (green, Fig. 11). Both control
structures depend on thresholds of the downstream water levels in the
Dove–Elbe stream segments (black, Fig. 11). In this case, the method to
model interactive control systems is applied. The evaluation results show a
good performance of the model: the closing and opening times of the sluices
according to the thresholds are met.
Closure and opening state of the control structures as well as
simulated and observed water levels at the downstream gauge Allermöher
Deich for the event in February 2002 and February 2011. The tide gate remained
closed two times during the storm events in February 2002, meaning that the Elbe
River water level during low tide period did not fall below the required
minimal water level of 0.9 m a.s.l. The simulated and observed water levels
depict a difference of 0.02 to 0.01 m in a stream with a water table
fluctuation of about 1 m. The RMSE for the flood event analysis shows a
deviation of up to 0.12 m.
Details and further results of the events in February 2002 and February 2011
for the control structures (Tatenberger Schleuse, Reitschleuse and
Dove–Elbe Schleuse) are given in the Supplement Sect. S6. The average difference in observed
and simulated water level peaks is about ΔW=0.04 m. This
corresponds to a difference of about 5 % in relation to the 1 m large
fluctuation range of the water table in the stream segments of the Dove–Elbe
catchment. In additional to the good fit in peak values, the hydrographs in
the Supplement of this article show that the temporal sequence (1) of
opening and closing the control systems and (2) of the rising and
falling limb in the hydrographs in the river segments is well
simulated. The results show a good reliability of the computed flood-routing
and backwater effects in streams. It is stated that with these findings the
reliability of the numerical model results is in a sufficient range of
accuracy for the designated field of application.
In additional to the presented evaluation studies, a flood peak reduction
measure is analysed in the research project StucK. By excavating three
retention areas with a total size of 330 000 m2 from +2 to +1 m a.s.l., an additional retention volume of 330 000 m3 is created when the water level exceeds the riverbanks
at +1 m a.s.l. The location of retention areas is indicated in Fig. 10.
With the additional retention volume, the peak water level can be reduced by
0.08 m. For the event in 2011 the result is shown in the Supplement Sect. S6. More results of the
model application for the research project StucK are published in Fröhle and Hellmers (2020).
Discussion of model results and limitations
In this section, achieved objectives, limitations and evaluation results of
the developed conceptual method are presented. The method facilitates
modelling backwater effects in lowlands caused by flow control structures using
a stand-alone hydrological model.
The first reached objective is the possibility to model the effects of tidal
ranges on the setting of flow control structures and the resulting backwater
effects on the flow regime in lowland streams. This objective is reached
using a conceptual hydrological method. The developed, implemented and
evaluated method transfers discharges into water levels, which means that
backwater volume routing is calculated by taking into account the water
level slope along streams and adjacent lowland areas. It applies a
pre-defined water level tolerance to calculate the backwater volume routing.
The use of physically based input parameters (e.g. profile geometries) extends
the application of this hydrological model to other catchment studies. The
input parameters comprise, for example, data on the stream profiles, gradients
and roughness along the flow path (river streams).
A second reached objective is the parsimony of the numerical model. In
comparison to coupled hydrodynamic models, the input parameters are
parsimonious, meaning less complex and easier to derive. A third reached
objective of the developed method is the direct computation of hydrological
processes in backwater-affected areas and streams. For example, the
infiltration, groundwater recharge and evaporation of water from submerged
areas are simulated. To simulate prospective changes in urbanisation or
effects by climate change on precipitation, patterns can be directly defined
in the hydrological numerical model. The implementation of the method is
realised in the open-source rainfall–runoff model Kalypso-NA (version 4.0).
The conceptual method is re-useable to extend other hydrological models
which are based e.g. on the often-applied flood-routing methods of
Kalinin–Miljukov and Muskingum–Cunge.
There are limitations of the conceptual method in the modelling of spatial and
temporal details like the variability in the velocity fields and (tidal)
flow regime. In the conceptual method, each stream section is computed as
a “reservoir” according to the linear reservoir theory, meaning that the
backwater profile is assumed to be flat within each river section. The
exactness of the water level heights depends on the defined water level
tolerance and the scale of the stream segments in the model. This means, in
contrast to hydrodynamic–numerical approaches, that the developed hydrological
model does not compute velocity fields within streams and water levels but
represents average values per stream segment. This hydrological flood-routing
method enables modelling regional-scale backwater-affected catchments
(>100 km2) with the requirement to keep the
computing times small and with a parsimonious parameterisation. The presented
method does not mean to replace coupled hydrodynamic–numerical models to
answer specific research questions, e.g. for which two- or three-dimensional
velocity fields and a spatial distribution of water levels within river
streams or on submerged areas need to be computed.
The evaluation results in Sect. 6 show the
applicability of the model for simulating rainfall–runoff regimes and
backwater effects in an exemplary lowland catchment (175 km2, Hamburg, Germany). This catchment is characterised by a complex flow
control system, wherein the drainage is influenced by a tidal range of about 4 m. The flood event analyses confirm good evaluation results: the comparison
of observed with simulated results shows a low RMSE (<0.12 m) and
a high R2 (>0.90). For these application
studies, a standard desktop computer with an i7-5600U CPU processor and 2.6 GHz
is applied. The computation time is in the range of a maximum of 3 min even
for large catchments (here: 175 km2) using a time step size
of 15 min for a 14 d simulation period. With these short computation
times, the presented method shows good potential to be used in flood
forecast simulation models for which results in the form of time series (e.g. water
levels and discharges) per river section and flood-prone area are
sufficient.
Summary and outlook
Numerical models are required in forecast simulations and to assess the
consequences of future impacts like changes in magnitude and
probability of storm-water events, changes in urbanisation, and changes in predicted mean
sea level rise on the runoff regime in catchments. Especially in coastal
lowlands, the pressure on storm-water drainage and flow control systems
is rising due to a combination of all three impacts. The literature review
shows weaknesses in modelling water depths and backwater effects in streams
and lowland areas using stand-alone hydrological numerical models. A method
to resolve these weaknesses is presented in this article. The developed
numerical method is:
applicable to model complex drainage and flow control systems in backwater-affected lowlands,
efficient by using short runtimes for real-time operational model
application,
open for further model developments,
re-useable for other hydrological model solutions and
parsimonious with respect to the complexity of input parameters.
The evaluation results in the application study of the complex and tidally
influenced lowland catchment Vier- und Marschlande illustrate good
conformance in the simulated backwater effects on the flow regime.
In addition to the findings in this article, the published outcomes in
Hellmers (2020) and Fröhle and Hellmers (2020) show that the reliability of
the numerical model results is in a sufficient range of accuracy for the
designated field of application to answer a wide range of hydrological and
water management questions. The numerical model is suitable for operational
flood forecasting, real-time control, risk analyses, scenario analyses and
time series gap filling in micro-scale to regional-scale catchments. The presented
method is re-useable for other hydrological numerical models which apply
conceptual hydrological flood-routing approaches (e.g. Muskingum–Cunge or
Kalinin–Miljukov).
Outlook
The presented method in the model Kalypso-NA (4.0) to compute backwater-affected flood routing will be adapted to model hydrological processes in
local-scale drainage measures (e.g. SUDS, GI and BMP as parts of nature-based
solutions). Preliminary research study results of local-scale drainage
measures are published in Hellmers and Fröhle (2017) and in Hellmers (2020). The integration of Kalypso-NA in flood forecasting systems (e.g.
Delft-FEWS) is in progress.
Data availability
Applied data on Hamburg for hydrological modelling are provided by the geoportal of Hamburg https://geoportal-hamburg.de (LGV, 2022a). In the following list, the references to data on Hamburg are provided. WMS protocols can be directly added and visualised in GIS software without a download of data.
Land use data and information are given at https://metaver.de/trefferanzeige?docuuid=DC71F8A1-7A8C-488C-AC99-23776FA7775E#detail_links and as WMS in LGV (2022b).
Groundwater data are provided as a zip file for direct download at https://daten-hamburg.de/geographie_geologie_geobasisdaten/hydrogeologie/Hydrogeologie_Grundwassergleichen_UBKS_2008_HH_2016-05-18.zip and as WMS in LGV (2022c).
Data from the digital elevation model (DEM) using a resolution of 1 m × 1 m are given as a direct download at https://daten-hamburg.de/geographie_geologie_geobasisdaten/Digitales_Hoehenmodell/DGM1/dgm1_2x2km_XYZ_hh_2021_04_01.zip or WMS in LGV (2022d).
Data for hydrogeological profile types are provided at https://daten-hamburg.de/geographie_geologie_geobasisdaten/hydrogeologie/Hydrogeologie_Profiltypen_HH_2016-05-18.zip and as WMS in LGV (2022e).
Data from the rainfall station “Wettermast” are available at https://wettermast.uni-hamburg.de/frame.php?doc=Downloads.htm (UH, 2022).
Data from gauging stations in Hamburg are illustrated for a period of 1 year by the Agency of Roads, Bridges and Water at https://www.wabiha.de/karte.html (WABIHA, 2022); longer time series can be requested by BUKEA.
Code availability
Name of the modified computation model: Kalypso-NA (version 4.0)
Developer of the modified part: (IWB) Institute of River and Coastal Engineering (TUHH-Hamburg University
of Technology)
Home page: https://www.tuhh.de/wb/forschung/software-entwicklung/kalypso/kalypso-na.html (TUHH, 2020)
First time available: BCENA renamed to Kalypso-NA (around 2000)
License: GNU Lesser General Public License (LGPL) as published by the Free Software Foundation, version 2.1
Hardware required: PC
Program language: Fortran
Program size: 5.8 MB
Availability and cost: compiled code is freely available at http://kalypso.wb.tu-harburg.de/downloads/KalypsoNA/ (TUHH, 2021), and a user manual is available at
https://kalypso.bjoernsen.de/manual/index.php/Contents/hydrology (last access: 1 May 2021)
(German/English version). Long-term access is given by means of an open
research platform (TORE) to a frozen version of the source code of the
presented method in this article: 10.15480/882.3522 (Hellmers, 2021). Excerpts of main code sections, flow diagrams and equations are additionally published
in the Supplement of this article.
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-15-1061-2022-supplement.
Author contributions
The lead author of this article, SH, formulated the research topic. She
placed the topic in the current state of research and defined the purpose of
the work. The presented approaches, methods, implementations and evaluation
results have been worked out by SH and were discussed with PF. The
conceptualisation of the paper was a joint effort from SH and PF, as were
the discussion and refinement of the presented methods.
Competing interests
The contact author has declared that neither they nor their co-author have any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The model development and evaluation study is part of the research project
StucK (Long-term drainage management of tide-influenced coastal urban areas
with consideration of climate change; 2015–2019; https://www.stuck-hh.de, last access: 1 May 2021). The joint
project in the funding measure “Regional Water Resources Management for
Sustainable Protection of Waters in Germany” (ReWaM) is sponsored by the
German Federal Ministry of Education and Research (BMBF). Publishing fees are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation – project number 491268466) and the Hamburg University of Technology (TUHH) in the funding programme “Open Access Publishing”. The authors gratefully acknowledge this
support.
Financial support
This research has been supported by the German Federal Ministry of Education and Research (BMBF) (grant no. 033W031).
Review statement
This paper was edited by Jeffrey Neal and reviewed by two anonymous referees.
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