# The wflow_sediment Model¶

Warning

Experimental version

## Introduction¶

The processes and fate of many particles and pollutants impacting water quality at the catchment level are intricately linked to the processes governing sediment dynamics. Both nutrients such as phosphorus, carbon or other pollutants such as metals are influenced by sediment properties in processes such as mobilization, flocculation or deposition. To better assert and model water quality in inland systems, a better comprehension and modelling of sediment sources and fate in the river is needed at a spatial and time scale relevant to such issues.

The wflow_sediment model was developed to answer such issues. It is a distributed physics-based model, based on the distributed hydrologic wflow_sbm model. It is able to simulate both land and in-stream processes, and relies on available global datasets, parameter estimation and small calibration effort.

In order to model the exports of terrestrial sediment to the coast through the Land Ocean Aquatic Continuum or LOAC (inland waters network such as streams, lakes…), two different modelling parts were considered (see Figure below). The first part is the modelling and estimation of soil loss and sediment yield to the river system by land erosion, called the soil loss model. The second part is the transport and processes of the sediment in the river system, called the river model. The two models together constitute the wflow_sediment model.

Overview of the concepts of the wflow_sediment model.

## Method¶

The wflow_sediment model was developed using the same framework and tool as the wflow_sbm hydrologic model. It uses the results from the hydrology to then deduce soil erosion and delivery to the river system, in what is called the soil loss part of the model. It finally follows the fate and transport of the incoming sediments in the stream network, in what is called the river part of the model. To keep the consistency with wflow_sbm, the model is also developed in Python using PCRaster functions and should use the same datasets.

### Soil Loss part¶

The first process to consider in sediment dynamics is the generation of sediments by land erosion. The main processes behind soil loss are rainfall erosion and overland flow erosion. In order to model such processes at a fine time and space scale, physics-based models such as ANSWERS and EUROSEM were chosen here.

#### Rainfall erosion¶

In wflow_sediment, rainfall erosion can both be modelled using EUROSEM or ANSWERS equation. The main difference between the models is that EUROSEM uses a more physics-based approach using the kinetic energy of the rain drops impacting the soil (Morgan et al, 1998), while ANSWERS is more empirical and uses parameters from the USLE model (Beasley et al, 1991).

In EUROSEM, rainfall erosion is modelled according to rainfall intensity and its kinetic energy while it reaches the soil according to equations developed by Brandt (1990). As the intensity of the rain kinetic energy depends on the length of the fall, rainfall intercepted by vegetation will then be reduced compared to direct throughfall. The kinetic energy of direct throughfall is estimated by (Morgan et al, 1998):

$KE_{direct} = 8.95 + 8.44 * log_{10} R_{i}$

where $$KE_{direct}$$ is kinetic energy of direct throughfall (J m$$^{-2}$$ mm$$^{-1}$$) and $$R_{i}$$ is rainfall intensity (mm h$$^{-1}$$). If the rainfall is intercepted by vegetation and falls as leaf drainage, its kinetic energy is then reduced according to (Brandt, 1990):

$KE_{leaf} = 15.8 * H_{p}^{0.5} - 5.87$

where $$KE_{leaf}$$ is kinetic energy of leaf drainage (J m$$^{-2}$$ mm$$^{-1}$$) and $$H_{p}$$ is the effective canopy height (half of plant height in m). In the global version of wflow_sediment, canopy height can be derived from the global map from Simard & al. (2011) or user input depending on the land use.

Kinetic energies from both direct throughfall and leaf drainage are then multiplied by the respective depths of direct throughfall and leaf drainage (mm) and added to get the total rainfall kinetic energy $$KE$$. The soil detached by rainfall $$D_{R}$$ (g m$$^{-2}$$) is then:

$D_{R} = k * KE * e^{-\varphi h}$

where $$k$$ is an index of the detachability of the soil (g $$J^{-1}$$), $$KE$$ is the total rainfall kinetic energy (J m$$^{-2}$$), $$h$$ is the surface runoff depth on the soil (m) and $$\varphi$$ is an exponent varying between 0.9 and 3.1 used to reduce rainfall impact if the soil is already covered by water. As a simplification, Torri (1987) has shown that a value of 2.0 for $$\varphi$$ is representative enough for a wide range of soil conditions. The detachability of the soil $$k$$ depends on the soil texture (proportion of clay, silt and sand content) and correponding values are defined in EUROSEM user guide (Morgan et al, 1998). As a simplification, in wflow_sediment, the mean value of the detachability shown in the table below are used. Soil texture is derived from the topsoil clay and silt content from SoilGrids (Hengl et al, 2017).

Table: Mean detachability of soil depending on its texture (Morgan et al, 1998).

Texture (USDA system)

Mean detachability $$k$$ (g/J)

Clay

2.0

Clay Loam

1.7

Silt

1.2

Silt Loam

1.5

Loam

2.0

Sandy Loam

2.6

Loamy Sand

3.0

Fine Sand

3.5

Sand

1.9

Rainfall erosion is handled differently in ANSWERS. There, the impacts of vegetation and soil properties are handled through the USLE coefficients in the equation (Beasley et al, 1991):

$D_{R} = 0.108 * C_{USLE} * K_{USLE} * A_{i} * R_{i}^{2}$

where $$D_{R}$$ is the soil detachment by rainfall (here in kg min$$^{-1}$$), $$C_{USLE}$$ is the soil cover-management factor from the USLE equation, $$K_{USLE}$$ is the soil erodibility factor from the USLE equation, $$A_{i}$$ is the area of the cell (m $$^{2}$$) and $$R_{i}$$ is the rainfall intensity (here in mm min$$^{-1}$$). In wflow_sediment, there are several methods available to estimate the $$C$$ and $$K$$ factors from the USLE. They can come from user input maps, for example maps resulting from Panagos & al.’s recent studies for Europe (Panagos et al, 2015) (Ballabio et al, 2016). To get an estimate of the $$C$$ factor globally, the other method is to estimate $$C$$ values for the different land use type in GlobCover. These values, summed up in the table below, come from a literature study including Panagos & al.’s review (2015), Gericke & al. (2015), Mansoor & al. (2013), Chadli & al. (2016), de Vente & al. (2009), Borrelli & al. (2014), Yang & al. (2003) and Bosco & al. (2015).

The other methods to estimate the USLE $$K$$ factor are to use either topsoil composition or topsoil geometric mean diameter. $$K$$ estimation from topsoil composition is estimated with the equation developed in the EPIC model (Williams et al, 1983):

$K_{USLE} = \left\{ 0.2 + 0.3exp\left[-0.0256SAN\frac{(1-SIL)}{100}\right] \right\} \left(\frac{SIL}{CLA+SIL}\right)^{0.3}$
$* \left(1-\frac{0.25OC}{OC+exp(3.72-2.95OC)}\right) * \left(1-\frac{0.75SN}{SN+exp(-5.51+22.9SN)}\right)$

where $$CLA$$, $$SIL$$, $$SAN$$ are respectively the clay, silt and sand fractions of the topsoil (%) from SoilGrids, $$OC$$ is the topsoil organic carbon content (%) from SoilGrids and $$SN$$ is $$1-SAN/100$$. The $$K$$ factor can also be estimated from the soil mean geometric diameter using the formulation from the RUSLE guide by Renard & al. (1997):

$K_{USLE} = 0.0034 + 0.0405exp\left[-\dfrac{1}{2}\left(\dfrac{log_{10}(D_{g})+1.659}{0.7101}\right)^{2}\right]$

where $$D_{g}$$ is the soil geometric mean diameter (mm) estimated from topsoil clay, silt, sand fraction.

Table: Estimation of USLE C factor per Globcover land use type

GlobCover Value

Globcover label

$$C_{USLE}$$

11

Post-flooding or irrigated croplands (or aquatic)

0.2

14

Rainfed croplands

0.35

20

Mosaic cropland (50-70%) / vegetation (grassland/shrubland/forest) (20-50%)

0.27

30

Mosaic vegetation (grassland/shrubland/forest) (50-70%) / cropland (20-50%)

0.25

40

Closed to open (>15%) broadleaved evergreen or semi-deciduous forest (>5m)

0.0065

50

Closed (>40%) broadleaved deciduous forest (>5m)

0.001

60

Open (15-40%) broadleaved deciduous forest/woodland (>5m)

0.01

70

Closed (>40%) needleleaved evergreen forest (>5m)

0.001

90

Open (15-40%) needleleaved deciduous or evergreen forest (>5m)

0.01

100

Closed to open (>15%) mixed broadleaved and needleleaved forest (>5m)

0.02

110

Mosaic forest or shrubland (50-70%) / grassland (20-50%)

0.015

120

Mosaic grassland (50-70%) / forest or shrubland (20-50%)

0.03

130

Closed to open (>15%) (broadleaved or needleleaved, evergreen or deciduous) shrubland (<5m)

0.035

140

Closed to open (>15%) herbaceous vegetation (grassland, savannas or lichens/mosses)

0.05

150

Sparse (<15%) vegetation

0.35

160

Closed to open (>15%) broadleaved forest regularly flooded (semi-permanently or temporarily) - Fresh or brackish water

0.001

170

Closed (>40%) broadleaved forest or shrubland permanently flooded - Saline or brackish water

0.0005

180

Closed to open (>15%) grassland or woody vegetation on regularly flooded or waterlogged soil - Fresh, brackish or saline water

0.04

190

Artificial surfaces and associated areas (Urban areas >50%)

0.0

200

Bare areas

0.0

210

Water bodies

0.0

220

Permanent snow and ice

0.0

230

No data (burnt areas, clouds,…)

0.0

#### Overland flow erosion¶

Overland flow (or surface runoff) erosion is induced by the strength of the shear stress of the surface water on the soil. As in rainfall erosion, the effect of the flow shear stress can be reduced by the soil vegetation or by the soil properties. In wflow_sediment, soil detachment by overland flow is modelled as in ANSWERS with (Beasley et al, 1991):

$D_{F} = 0.90 * C_{USLE} * K_{USLE} * A_{i} * S * q$

where $$D_{F}$$ is soil detachment by flow (kg min$$^{-1}$$), $$C_{USLE}$$ and $$K_{USLE}$$ are the USLE cover and soil erodibility factors, $$A_{i}$$ is the cell area (m$$^{2}$$), $$S$$ is the slope gradient and $$q$$ is the overland flow rate per unit width (m$$^{2}$$ min$$^{-1}$$). The USLE $$C$$ and $$K$$ factors can be estimated with the same methods as for rainfall erosion and here the slope gradient is obtained from the sinus rather than the tangent of the slope angle.

#### Delivery to the river system¶

Once the amount of soil detached by both rainfall and overland flow has been estimated, it has then to be routed and delivered to the river network. Inland routing in sediment models is usually done by comparing the amount of detached sediment with the transport capacity of the flow, which is the maximum amount of sediment than the flow can carry downslope. There are several existing formulas available in the literature. For a wide range of slope and for overland flow, the Govers equation (1990) seems the most appropriate choice (Hessel et al, 2007). However, as the wflow_sediment model was developed to be linked to water quality issues, the Yalin transport equation was chosen as it can handle particle differentiation (Govers equation can still be used if wflow_sediment is used to only model inland processes with no particle differentiation). For land cells, wflow_sediment assumes that erosion can mobilize 5 classes of sediment:

• Clay (mean diameter of 2 $$\mu$$ m)

• Silt (mean diameter of 10 $$\mu$$ m)

• Sand (mean diameter of 200 $$\mu$$ m)

• Small aggregates (mean diameter of 30 $$\mu$$ m)

• Large aggregates (mean diameter of 500 $$\mu$$ m).

To deduce the amount of small and large aggregates from topsoil clay, silt and sand contents, the following equations from the SWAT model are used (Neitsch et al, 2011):

$PSA = SAN *(1-CLA)^{2.4}$
$PSI = 0.13SIL$
$PCL = 0.20CLA$
$SAG = 2.0CLA for CLA < 0.25$
$SAG = 0.28(CLA-0.25)+0.5 for 0.25 \leq CLA \leq 0.5$
$SAG = 0.57 for CLA > 0.5$
$LAG = 1 - PSA - PSI - PCL - SAG$

where $$CLA$$, $$SIL$$ and $$SAN$$ are the primary clay, silt, sand fractions of the topsoil and $$PCL$$, $$PSI$$, $$PSA$$, $$SAG$$ and $$LAG$$ are the clay, silt, sand, small and large aggregates fractions of the detached sediment respectively. The transport capacity of the flow using Yalin’s equation with particle differentiation, developed by Foster (1982), is:

$TC_{i} = (P_{e})_{i} * (S_{g})_{i} *\rho_{w} * g * d_{i} * V_{*}$

where $$TC_{i}$$ is the transport capacity of the flow for the particle class i, $$(P_{e})_{i}$$ is the effective number of particles of class i, $$(S_{g})_{i}$$ is the specific gravity for the particle class i (kg m$$^{-3}$$), $$\rho_{w}$$ is the mass density of the fluid (kg m$$^{-3}$$), $$g$$ is the acceleration due to gravity (m s$$^{-2}$$), $$d_{i}$$ is the diameter of the particle of class i (m) and $$V_{*}=(g* R * S)^{0.5}$$ is the shear velocity of the flow (m s$$^{-1}$$) with $$S$$ the slope gradient and $$R$$ the hydraulic radius of the flow (m). The detached sediment are then routed downslope until the river network using the accucapacityflux, accupacitystate functions from the PCRaster Python framework depending on the transport capacity from Yalin.

Finally, the different processes happening for a land cell in the soil loss part of wflow_sediment are summarized in the figure below:

Overview of the different processes for a land cell in wflow_sediment (adapted from Kinnell et al, 2010).

### River part¶

As wflow_sediment was developed for applications across Europe, it must be able to simulate sediment dynamics both for small and large catchments. Which is why, for large catchments, it needed to model more precisely processes happening in the stream network. Thus once sediments coming from land erosion reach a river cell in the model, processes and equations change. There are not so many models available to model in-stream sediment dynamics with only hydrology. In the end, the more physics-based approach of the SWAT model was chosen as it requires little or no calibration and it can separate both the suspended from the bed load (Neitsch et al, 2011). As in SWAT, in the river part of wflow_sediment, 5 particles class are modelled: clay, silt, sand, small and large aggregates and gravel. Small and large aggregates are assumed to only come from land erosion, gravel only from river erosion, while clay, silt and sand can both come from either land or river erosion. In the river, suspended sediment load is assumed to be the sum of clay and silt and the bed sediment load is assumed to be composed of sand, gravel, small and large aggregates.

#### Sediment inputs in a river cell¶

The first part of the river model assesses how much detached sediment are in the river cell at the beginning of the timestep t. Sources of detached sediment are sediments coming from land erosion, estimated with the soil loss part of wflow_sediment model, the sediment coming from upstream river cells and the detached sediment that were left in the cell at the end of the previous timestep (t-1):

$(sed_{in})_{t} = (sed_{land})_{t} + upstream\left[(sed_{out})_{t-1}\right] + (sed_{riv})_{t-1}$

Sediment coming from upstream river cells is estimated using the PCRaster upstream function and the local drainage direction map to spot the upstream river cells.

#### River transport and erosion¶

Once the amount of sediment inputs at the beginning of the timestep is known, the model then estimates transport, and river erosion if there is a deficit of sediments. Transport in the river system is estimated via a transport capacity formula. There are several transport capacity formulas available in wflow_sediment, some requiring calibration and some not. Choosing a transport capacity equation depends on the river characteristics (some equation are more suited for narrow or wider rivers), and on the reliability of the required river parameters (such as slope, width or mean particle diameter of the river channel). Available transport capacity equations are:

• Simplified Bagnold: originally more valid for intermediate to large rivers, this simplified version of the Bagnold equation relates sediment transport to flow velocity with two simple calibration parameters (Neitsch et al, 2011):

$C_{max} = c_{sp} * \left( \dfrac{prf * Q}{h * W} \right) ^{sp_{exp}}$

where $$C_{max}$$ is the sediment concentration (ton m$$^{-3}$$ or kg/L), $$Q$$ is the surface runoff in the river cell (m:math:^{3}/s), $$h$$ is the river water level (m), $$W$$ is the river width (m) and $$c_{sp}$$, $$prf$$ and $$sp_{exp}$$ are calibration parameters. The $$prf$$ coefficient is usually used to deduce the peak velocity of the flow, but for simplification in wflow_sediment, the equation was simplified to only get two parameters to calibrate: $$sp_{exp}$$ and $$c_{Bagnold} = c_{sp} * prf^{sp_{exp}}$$. The coefficient $$sp_{exp}$$ usually varies between 1 and 2 while $$prf$$ and $$c_{sp}$$ have a wider range of variation. The table below summarizes ranges and values of the three Bagnold coefficients used by other studies:

Range of the simplified Bagnold coefficients (and calibrated value)

Study

River

$$prf$$ range

$$c_{sp}$$ range

$$sp_{exp}$$ range

Vigiak 2015

Danube

0.5-2 (/)

0.0001-0.01 (0.003-0.006)

1-2 (1.4)

Vigiak 2017

Danube

/

0.0001-0.01 (0.0015)

1-2 (1.4)

Abbaspour 2007

Thur (CH)

0.2-0.25 (/)

0.001-0.002 (/)

1.35-1.47 (/)

Oeurng 2011

Save (FR)

0-2 (0.58)

0.0001-0.01 (0.01)

1-2 (2)

• Engelund and Hansen: not present in SWAT but used in many models such as Delft3D-WAQ, Engelund and Hansen calculates the total sediment load as (Engelund and Hansen, 1967):

$C_{w} = 0.05 \left( \dfrac{\rho_{s}}{\rho_{s} - \rho} \right) \left( \dfrac{u * S}{\sqrt{\left( \dfrac{\rho_{s}}{\rho_{s} - \rho} \right) * g * D_{50}}} \right) \theta^{1/2}$

where $$C_{w}$$ is the sediment concentration by weight, $$\rho$$ and $$\rho_{s}$$ are the fluid and sediment density (here equal to 1000 and 2650 g m$$^{-3}$$), $$u$$ is the water mean velocity (m/s), $$S$$ is the river slope, $$g$$ is the acceleration due to gravity, $$D_{50}$$ is the river mean diameter (m) and $$\theta$$ is the Shields parameter.

• Kodatie: Kodatie (1999) developped the power relationships from Posada (1995) using field data and linear optimization so that they would be applicable for a wider range of riverbed sediment size. The resulting equation, for a rectangular channel, is (Neitsch et al, 2011):

$C_{max} = \left( \dfrac{a* u^{b}* h^{c} * S^{d}}{V_{in}} \right) * W$

where $$V_{in}$$ in the volume of water entering the river cell during the timestep (m:math:^{3}) and $$a$$, $$b$$, $$c$$ and $$d$$ are coefficients depending on the riverbed sediment size. Values of these coefficients are summarized in the table below:

Range of the simplified Bagnold coefficients (and calibrated value)

River sediment diameter

a

b

c

d

$$D_{50} \leq$$ 0.05mm

281.4

2.622

0.182

0

0.05 $$< D_{50} \leq$$ 0.25mm

2 829.6

3.646

0.406

0.412

0.25 $$< D_{50} \leq$$ 2mm

2 123.4

3.300

0.468

0.613

$$D_{50} >$$ 2mm

431 884.8

1.000

1.000

2.000

• Yang: Yang (1996) developed a set of two equations giving transport of sediments for sand-bed or gravel-bed rivers. The sand equation ($$D_{50} < 2mm$$) is:

$\begin{split}log\left(C_{ppm}\right) = 5.435 - 0.286log\frac{\omega_{s,50}D_{50}}{\nu}-0.457log\frac{u_{*}}{\omega_{s,50}} \\ +\left(1.799-0.409log\frac{\omega_{s,50}D_{50}}{\nu}-0.314log\frac{u_{*}}{\omega_{s,50}}\right)log\left(\frac{uS}{\omega_{s,50}}-\frac{u_{cr}S}{\omega_{s,50}}\right)\end{split}$

And the gravel equation ($$2 \leq D_{50} < 10 mm$$) is:

$\begin{split}log\left(C_{ppm}\right) = 6.681 - 0.633log\frac{\omega_{s,50}D_{50}}{\nu}-4.816log\frac{u_{*}}{\omega_{s,50}} \\ +\left(2.784-0.305log\frac{\omega_{s,50}D_{50}}{\nu}-0.282log\frac{u_{*}}{\omega_{s,50}}\right)log\left(\frac{uS}{\omega_{s,50}}-\frac{u_{cr}S}{\omega_{s,50}}\right)\end{split}$

where $$C_{ppm}$$ is sediment concentration in parts per million by weight, $$\omega_{s,50}$$ is the settling velocity of a particle with the median riverbed diameter estimated with Stokes (m/s), $$\nu$$ is the kinematic viscosity of the fluid (m$$^{2}$$/s), $$u_{*}$$ is the shear velocity ($$\sqrt{gR_{H}S}$$ in m/s with $$R_{H}$$ the hydraulic radius of the river) and $$u_{cr}$$ is the critical velocity (m/s, equation can be found in Hessel, 2007).

• Molinas and Wu: The Molinas and Wu (2001) transport equation was developed for large sand-bed rivers based on the universal stream power $$\psi$$. The corresponding equation is (Neitsch et al, 2011):

$C_{w} = \dfrac{1430*(0.86+\sqrt{\psi})* \psi^{1.5}}{0.016+\psi}* 10^{-6}$

where $$\psi$$ is the universal stream power given by:

$\psi = \dfrac{\psi^{3}}{\left(\dfrac{\rho_{s}}{\rho}-1\right)* g* h* \omega_{s,50} * \left[ log_{10}\left(\dfrac{h}{D_{50}}\right)\right]^{2}}$

Once the maximum concentration $$C_{max}$$ is established with one of the above transport formula, the model then determines if there is erosion of the river bed and bank. In order to do that, the difference $$sed_{ex}$$ between the maximum amount of sediment estimated with transport ($$sed_{max} = C_{max}* V_{in}$$) and the sediment inputs to the river cell ($$sed_{in}$$ calculated above) is calculated. If too much sediment is coming in and $$sed_{ex}$$ is negative, then there is no river bed and bank erosion. And if the river has not reach its maximum transport capacity, then erosion of the river happens.

First, the sediments stored in the cell from deposition in previous timesteps $$sed_{stor}$$ are eroded from clay to gravel. If this amount is not enough to cover $$sed_{ex}$$, then erosion of the local river bed and bank material starts.

Instead of just setting river erosion amount to just cover the remaining difference $$sed_{exeff}$$ between $$sed_{ex}$$ and $$sed_{stor}$$, actual erosion potential is adjusted using river characteristics and is separated between the bed and bank of the river using the physics-based approach of Knight (1984).

The bed and bank of the river are supposed to only be able to erode a maximum amount of their material $$E_{R,bed}$$ for the bed and $$E_{R,bank}$$ for the river bank. For a rectangular channel, assuming it is meandering and thus only one bank is prone to erosion, they are calculated from the equations(Neitsch et al, 2011):

$E_{R,bed} = k_{d,bed} * \left( \tau_{e,bed} - \tau_{cr,bed} \right)* 10^{-6} * L * W * \rho_{b, bed} * \Delta t$
$E_{R,bank} = k_{d,bank} * \left( \tau_{e,bank} - \tau_{cr,bank} \right)* 10^{-6} * L * h * \rho_{b, bank} * \Delta t$

where $$E_{R}$$ is the potential bed/bank erosion rates (tons), $$k_{d}$$ is the erodibility of the bed/bank material (cm$$^{3}$$ N$$^{-1}$$ s$$^{-1}$$), $$\tau_{e}$$ is the effective shear stress from the flow on the bed/bank (N/m$$^{2}$$), $$\tau_{cr}$$ is the critical shear stress for erosion to happen (N/m$$^{2}$$), $$L$$, $$W$$ and $$h$$ are the channel length, width and water height (m), $$\rho_{b}$$ is the bulk density of the bed/bank of the river (g/cm$$^{3}$$) and $$\Delta t$$ is the model timestep (s).

In wflow_sediment, the erodibility of the bed/bank are approximated using the formula from Hanson and Simon (2001):

$k_{d}=0.2* \tau_{cr}^{-0.5}$

Normally erodibilities are evaluated using jet test in the field and there are several reviews and some adjustments possible to this equation (Simon et al, 2011). However, to avoid too heavy calibration and for the scale considered, this equation is supposed to be efficient enough. The critical shear stress $$\tau_{cr}$$ is evaluated differently for the bed and bank. For the bed, the most common formula from Shields initiation of movement is used. For the bank, a more recent approach from Julian and Torres (2006) is used :

$\tau_{cr,bank} = (0.1+0.1779* SC+0.0028* SC^{2}-2.34* 10^{-5} * SC^{3})*C_{ch}$

where $$SC$$ is the percent clay and silt content of the river bank and $$C_{ch}$$ is a coefficient taking into account the positive impact of vegetation on erosion reduction. This coefficient is then dependent on the land use and classical values are shown in the table below. These values where then adapted for use with the GlobCover land use map. Percent of clay and silt (along with sand and gravel) for the channel is estimated from the river median particle diameter assuming the same values as SWAT shown in the table below. Median particle diameter is here estimated depending on the Strahler river order. The higher the order, the smaller the diameter is. As the median diameter is only used in wflow_sediment for the estimation of the river bed/bank sediment composition, this supposition should be enough. Actual refined data or calibration may however be needed if the median diameter is also required for the transport formula. In a similar way, the bulk densities of river bed and bank are also just assumed to be of respectively 1.5 and 1.4 g/cm$$^{3}$$.

Classical values of the channel cover vegetation coefficent (Julian and Torres, 2006)

Bank vegetation

$$C_{ch}$$

None

1.00

Grassy

1.97

Sparse trees

5.40

Dense trees

19.20

Table : Composition of the river bed/bank depending on the median diameter (Neitsch et al, 2011)

Sediment Fraction

$$d_{50}$$ ($$\mu$$m)

$$\leq$$ 5

5 to 50

50 to 2000

$$>$$2000

Sand

0.15

0.15

0.65

0.15

Silt

0.15

0.65

0.15

0.15

Clay

0.65

0.15

0.15

0.05

Gravel

0.05

0.05

0.05

0.65

Then, the repartition of the flow shear stress is refined into the effective shear stress and the bed and bank of the river using the equations developed by Knight (1984) for a rectangular channel:

$\tau_{e,bed} = \rho g R_{H} S * \left(1 - \dfrac{SF_{bank}}{100}\right) * \left(1+\dfrac{2h}{W}\right)$
$\tau_{e,bank} = \rho g R_{H} S * \left( SF_{bank}\right) * \left(1+\dfrac{W}{2h}\right)$

where $$\rho g$$ is the fluid specific weight (9800 N/m$$^{3}$$ for water), $$R_{H}$$ is the hydraulic radius of the channel (m), $$h$$ and $$W$$ are the water level and river width (m). $$SF_{bank}$$ is the proportion of shear stress acting on the bank (%) and is estimated from (Knight, 1984):

$SF_{bank} = exp \left( -3.230* log_{10}\left(\dfrac{W}{h}+3\right)+6.146 \right)$

Finally the relative erosion potential of the bank and bed of the river is calculated by:

$RTE_{bed} = \dfrac{E_{R,bed}}{E_{R,bed}+E_{R,bank}}$
$RTE_{bank} = 1 - RTE_{bed}$

And the final actual eroded amount for the bed and bank is the maximum between $$RTE * sed_{exeff}$$ and the erosion potential $$E_{R}$$. Total eroded amount of sediment $$sed_{erod}$$ is then the sum of the eroded sediment coming from the storage of previously deposited sediment and the river bed/bank erosion.

#### River deposition¶

As sediments have a higher density than water, moving sediments in water can be deposited in the river bed. The deposition process depends on the mass of the sediment, but also on flow characteristics such as velocity. In wflow_sediment, as in SWAT, deposition is modelled with Einstein’s equation (Neitsch et al, 2011):

$P_{dep}=\left(1-\dfrac{1}{e^{x}}\right)* 100$

where $$P_{dep}$$ is the percentage of sediments that is deposited on the river bed and x is a parameter calculated with:

$x = \dfrac{1.055* L * \omega_{s}}{u* h}$

where $$L$$ and $$h$$ are channel length and water height (m), $$\omega_{s}$$ is the particle settling velocity calculated with Stokes formula (m/s) and $$u$$ is the mean flow velocity (m/s). The calculated percentage is then subtracted from the amount of sediment input and eroded river sediment for each particle size class ($$sed_{dep} = P_{dep}/100 * (sed_{in} + sed_{erod})$$). Resulting deposited sediment are then stored in the river bed and can be re-mobilized in future time steps by erosion.

#### Mass balance and sediment concentration¶

Finally after estimating inputs, deposition and erosion with the transport capacity of the flow, the amount of sediment actually leaving the river cell to go downstream is estimated using:

$sed_{out} = (sed_{in} + sed_{erod} - sed_{dep}) * \dfrac{V_{out}}{V}$

where $$sed_{out}$$ is the amount of sediment leaving the river cell (tons), $$sed_{in}$$ is the amount of sediment coming into the river cell (storage from previous timestep, land erosion and sediment flux from upstream river cells in tons), $$sed_{erod}$$ is the amount of sediment coming from river erosion (tons), $$sed_{dep}$$ is the amount of deposited sediments (tons), $$V_{out}$$ is the volume of water leaving the river cell (surface runoff $$Q$$ times timestep $$\Delta t$$ in m$$^{3}$$) and $$V$$ is the total volume of water in the river cell ($$V_{out}$$ plus storage $$h* W * L$$ in m$$^{3}$$).

A mass balance is then used to calculate the amount of sediment remaining in the cell at the end of the timestep $$(sed_{riv})_{t}$$:

$\begin{split}(sed_{riv})_{t} = (sed_{riv})_{t-1} + (sed_{land})_{t} + upstream\left[(sed_{out})_{t-1}\right] \\ + (sed_{erod})_{t} - (sed_{dep})_{t} - (sed_{out})_{t}\end{split}$

Finally, the different processes happening for a land cell in the river part of wflow_sediment are summarized in the figure below:

Overview of the different processes for a river cell in wflow_sediment.

#### Lake modelling¶

Apart from land and river, the hydrologic wflow_sbm model also handles lakes and reservoirs modelling. In wflow_sbm, lakes and large reservoirs are modelled using a 1D bucket model at the cell corresponding to the outlet. For the other cells belonging to the lake/reservoir which are not the outlet, processes such as precipitation and evaporation are filtered out and shifted to the outlet cell. wflow_sediment then handles the lakes in the same way. If a cell belongs to a lake/reservoir and is not the outlet then the model assumes that no erosion/deposition of sediments is happening and the sediments are only all transported to the lake/reservoir outlet. Once the sediments reach the outlet, then sediments are deposited in the lake/reservoir according to Camp’s model (1945) (Verstraeten et al, 2000):

$TE = \dfrac{\omega_{s}}{u_{cr,res}} = \dfrac{A_{res}}{Q_{out,res}} * \omega_{s}$

where $$TE$$ is the trapping efficiency of the lake/reservoir (or the fraction of particles trapped), $$\omega_{s}$$ is the particle velocity from Stokes (m/s), $$u_{cr,res}$$ is the reservoir’s critical settling velocity (m/s) which is equal to the reservoir’s outflow $$Q_{out,res}$$ (m$$^{3}$$/s) divided by the reservoir’s surface area $$A_{res}$$ (m$$^{2}$$).

## Configuration¶

The wflow_sediment model was developed as part of the wflow hydrologic platform and is therefore a nother wflow module, developed in Python, and using the same framework than wflow_sbm. First, the model case is set up and run normally with wflow_sbm. Then wflow_sediment is run using the outputs of the hydrologic model. As settings for wflow_sbm are explained in the corresponding part of this documentation, only specific details regarding the run of wflow_sediment are developed here.

### Running wflow_sbm¶

To model sediment dynamics, the first step is to build a wflow_sbm model and to run it for the catchment considered. Apart from the usual settings for the wflow_sbm model, additional ones for a run with wflow_sediment are to save the following variables in the outputmaps section of the wflow_sbm.ini file:

• Precipitation “self.Precipitation” (can also be taken directly from the wflow_sbm forcings)

• Surface runoff from the kinematic wave “self.SurfaceRunoff”

• Water level in the kinematic wave “self.WaterLevel”

• Rainfall interception by the vegetation “self.Interception”.

wflow_sediment also needs some static output maps which are saved by default by wflow_sbm. These maps are the map of the actual width and length of the flow volume (Bw and DCL.map). After the set up, wflow_sbm is run normally either via a batch file or via the command line.

### Running wflow_sediment¶

As wflow_sediment is built in the same way as wflow_sbm, its settings and use are very similar. First, some additional data must be downloaded. Then the corresponding ini file that summarizes all inputs and outputs of the model run is completed and the model can finally be run.

Apart from many data, such as landuse, catchment map, ldd map etc, that are already needed for the wflow_sbm run, wflow_sediment requires some extra additional data which are:

• Map with topsoil percent of clay: this can be download, as for wflow_sbm other soil data, from the SoilGrids database (Hengl et al, 2017). Values then needs to be resampled and adjusted to the model grid size (for the global version of wflow by averaging). This data is mandatory for the sediment model to run.

• Map with topsoil percent of silt: this can also be downloaded from SoilGrids and processed in the same way as the topsoil clay map. This data is mandatory for the sediment model to run.

• Map with topsoil percent of organic carbon: this data can be downloaded from SoilGrids. Units should be in percent (SoilGrids gives it in per-mille) and adjusted to the model grid cells. This data is only needed if the user wishes to calculate the USLE K parameter of soil erosion using the EPIC formula.

• Map of vegetation height: this is available globally using the map published by Simard & al (2011). Other sources can however be used. Units should be in meters. Vegetation height is only needed if the EUROSEM model is used to calculate rainfall erosion.

#### Setting the ini file¶

As for wflow_sbm, the setting up of the wflow_sediment model is also done via an ini file and its different sections. A complete example is given in the wflow examples folder. The main sections and options needed are:

• inputmapstacks: Links to the dynamic outputs of the wflow_sbm run either stored as maps in the outmaps folder of the sbm run or in the netcdf file. Dynamic data needed are Precipitation, SurfaceRunoff, WaterLevel and Interception.

[inputmapstacks]
# Outputs from wflow_sbm
Precipitation           = /inmaps/P
Interception            = /inmaps/int
SurfaceRunoff = /inmaps/run
WaterLevel = /inmaps/levKin

• framework: As for wflow_sbm, specifies if the inputs or outputs of the model are in netcdf format or PCRaster maps. If the results of wflow_sbm are saved in a netcdf file, link to this file is precised in the netcdfinput argument.

• run: Info on the run parameters of the model. The start time, end time and timesteps of the model are written in this section. The reinit argument precise if the model should start from cold states (all the states maps of the model are set to zero if reinit = 1) or from the states maps given in the instate folder of the model (reinit = 0).

• modelparameters: Other parameters used by the model. This section should include the same inputs as the wflow_sbm.ini file for reservoir modelling and Leaf Area Index data.

• model: Parameters and settings for the sediment model. It contains both links to the staticmaps of the model (DEM, LDD etc.) and settings to decide which equations to use. These switches are used to choose if both the soil loss and river part of the model should be run (runrivermodel = 1 or 0 for just the soil loss part) and how the model should compute the USLE K factor (uslekmethod = 1 for a staticmap, 2 for geometric mean equation and 3 for EPIC equation), the USLE C factor (uslecmethod = 1 for a staticmap, 2 for a table based on land use), rainfall erosion (rainerodmethod = 1 for EUROSEM and 2 for ANSWERS), inland transport (landtransportmethod = 1 for Yalin with particle differentiation, 2 for Govers total transport and 3 for Yalin total transport) and the river transport (rivtransportmethod = 1 for Engelund and Hansen, 2 for simplified Bagnold, 3 for Kodatie, 4 for Yang and 5 for Molinas and Wu).

# Model parameters and settings
[model]
modeltype= sediment
configfile = wflow_sediment.ini
intbl = intbl
# Run only the soil erosion model (0) or also the river transport model (1)
runrivermodel = 1
#Use original (0) or minimum dem (1) for river slope computation
slopecorr = 1
#USLE K computation method
#1=map ; 2=geometric mean ; 3=EPIC
uslekmethod = 2
#USLE C computation method
#1=map ; 2=TBL based on land use
uslecmethod=2
#Rainfall erosion
rainerodmethod = 1
#Inland sediment transport formula
#1=Yalin (particle differentiation) ; 2=Govers (total) ; 3=Yalin (total)
#If the river transport model is run, will be set to 1
landtransportmethod = 1
#River sediment transport formula for erosion threshold
#1=Engelund and Hansen ; 2=Bagnold ; 3=Kodatie ; 4=Yang ; 5=Molinas and Wu
rivtransportmethod = 2
#sCatch = 0
#Model maps from wflow_sbm
wflow_dem = staticmaps/wflow_dem.map
wflow_landuse = staticmaps/wflow_landuse.map
wflow_soil = staticmaps/wflow_soil.map
wflow_subcatch = staticmaps/wflow_subcatch.map
wflow_Hype = staticmaps/SUBID-HYPE-Rhine.map
wflow_ldd = staticmaps/wflow_ldd.map
wflow_river = staticmaps/wflow_river.map
wflow_riverwidth = staticmaps/wflow_riverwidth.map
wflow_dcl = staticmaps/DCL.map
wflow_streamorder = staticmaps/wflow_streamorder.map
wflow_clay = staticmaps/percent_clay.map
wflow_silt = staticmaps/percent_silt.map
wflow_oc = staticmaps/percent_oc.map
wflow_canopyheight = staticmaps/canopy_height.map

• layout: Specifies if the cell size is given in lat-lon (sizeinmetres = 0) or in meters (1). Should be set as in wflow_sbm.

• outputmaps: As in wflow_sbm, this section is used to choose which dynamic data to save from the wflow_sediment run. These are:

##### Output grids #####
[outputmaps]
#Gross precipitation [mm] (input)
self.Precipitation=P
#Surface runoff in the kinematic wave [m^3/s]
self.SurfaceRunoff=run
#Water level in the kinematic wave [m] (above the bottom)
self.WaterLevel=levKin
#Overland flow [m3/s]
self.OvRun = ovRun
#Soil loss by surface runoff erosion [ton/timestep/cell]
self.SedOv=sedov
#Soil loss by splash erosion [ton/timestep/cell]
self.SedSpl=sedspl
#Total soil loss [ton/timestep/cell]
self.SoilLoss=soilloss
#Total sediment input in the river per subcatchment [kg/ha/timestep]
self.HYPEOvSedCatch = sedcatch
#Sediment from land erosion entering the river [ton]
self.InLandSed = landsed
#Total river inputs [ton]
#River erosion [ton]
self.RivErodSed = erodsed
#Deposition in rivers [ton]
#Sediment stored on the river bed [ton]
self.RivStoreSed = rivstore
#Sediment output [ton]
#Final sediment load in river cells [ton]
#Total sediment concentration [mg/L]
self.SedConc = sedconc
#Suspended sediment concentration [mg/L]
self.SSConc = ssconc

• summary: Used to save summary maps of wflow_sediment outputs such as yearly average or yearly sum etc. It works in the same way than for wflow_sbm (see wflow documentation for more details).

• outputcsv and outputtss: Used to save the evolution of wflow_sediment outputs for specific points or areas of interest in csv or tss format. It works in the same way than for wflow_sbm (see wflow documentation for more details).

#### Running the model¶

Once all the settings are ready, the wflow_sediment model is run similarly to wflow_sbm via the command line or a batch file. The minimum command line requires:

• The link to the wflow_sediment script.

• -C option stating the name of the wflow case directory.

• -R option stating the name of the directory of wflow_sediment outputs.

• -c option stating the link to the wflow_sediment ini file.

As in wflow_sbm, the outputs of the wflow_sediment model can both be dynamic netcdf/pcraster maps data, static data, or dynamic data for points/areas of interest. The main outputs variables are soil loss by rainfall and overland flow erosion (“self.SedSpl” + “self.SedOv” = “self.soilloss” in ton/timestep/cell), all the arguments from the sediment mass balance in the river and the total, suspended and bed sediment concentration (“self.SedConc”, “self.SSConc” and “self.BedConc” in mg/L). The other outputs are some default summary maps, saved in the outsum folder, and are the USLE C and K factors, the topsoil percent sand (deduced from clay and silt content) and the median river sediment diameter $$d_{50}$$ in mm. The last outputs are the final states of the model, stored in the outstate folder. In the river model, wflow_sediment need to use the sediment load in the river at the beginning of the timestep, the sediment load coming from upstream river cells at the beginning of the timestep and the amount of deposited sediment that are stored in the river bed and are available for erosion. The three variables are stored in state map files for all particle size class (clay, silt, sand, small and large aggregates, gravel) and total sediment resulting in 21 state maps. If the model is run for the first time and the states are not available, then, as in wflow_sbm, a cold run with initial zero maps for the states can be done first for a year (at least 6 months depending on the size of the model).

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