M. Esteves, G. Nord · Model description: Soil erosion ¾Soil detachment by rainfall Detachment...

1

DYNAS Workshop, 6th-8th December 2004, INRIA

M. M. EstevesEsteves, G. , G. NordNord

PSEM_2D

A process-based soil erosion model at the plot scale

DYNAS Workshop

Rocquencourt 6th-8th December 2004

2


Introduction

PRIM_2D Plot Runoff and Infiltration Model (1999)PSEM_2D Plot Soil Erosion Model (2003)

These models were designedto dynamically couple hydrological and soil erosion processesto predict the spatial pattern of overland flow hydraulicsto predict the spatial pattern of soil erosionto be used in natural slopes conditionsto consider complex rainfall events

The models work on a rainfall event basisPRIM_2D has been validated (Esteves et al., 2000, J. Hyd.,228)

PSEM_2D model is still under evaluation (Nord and esteves, WRR, submitted)

3


Objectives

The main goals areto improve our understanding of local overland flow hydraulics

to develop a better understanding of soil erosion processes

to bring a better description of the spatial and temporal variability of soil erosion at the plot scale

4


Presentation outline

Description of PRIM_2D and PSEM_2D

Applications of PRIM_2DValidation of the model by comparison with observed data

Effect of the micro-topography

Effect of soil surface features pattern (crusted soils)

Applications of PSEM_2D Evaluation of the model by comparison with experimental data

Some numerical examples to show the capabilities of the model

As a conclusion: Future research

5


Model description Model description

6


The model has three major componentsOverland flow (OF) is generated as infiltration excess

rainfall (hortonian)

OF is routed using the depth averaged two dimensional unsteady flow equations on a finite difference grid

Rainfall and OF hydraulics are used to compute soil erosion

A single representative particle size (D50)

Model description

7


Model description : Infiltration

The infiltration algorithm is based on the Green and Amptequation (1911)

( )f

ff hhZ

ZKIc++

=is

Zθθ

If −

=

( )f

ff hhZ

ZKI cc++

=cf ZZ ≤

In the case of crusted soils the profile is divided in two layers

cf ZZ >( )

f

ff hhZ

ZKI ec++

=

( )c

c

s

ce

KZ

KZZZK

+−

=f

f

Zf

8


Model description : Overland flow

Fully dynamic two dimensional unsteady flow equations (Barré de Saint-Venant)

Continuity equation:

Momentum equations:

),()()( yxIRth

yvh

xuh

−=∂∂

+∂

∂+

∂∂

0=⎥⎦⎤

⎢⎣⎡ −+∂∂

+∂∂

+∂∂

+∂∂

oxfx SSxhg

yuv

xuu

tu

x direction:

y direction:

g gravitational acceleration (m.s-2)h flow depth (m)R rainfall intensity (m s -1)I rate of infiltration (m s -1)Sox ground slope (x direction)Soy ground slope (y direction)Sfx friction slope (x direction)Sfy friction slope (y direction)u flow velocity (x direction) (m s -1)v flow velocity (y direction) (m s -1)

0=⎥⎦

⎤⎢⎣

⎡−+

∂∂

+∂∂

+∂∂

+∂∂

oyfy SSyhg

yvv

xvu

tv

9


Model description : Flow resistance

Friction is approximated using the Darcy-Weisbachequation

The Darcy-Weisbach friction factor is constant

For small depth flows (< 0.1 mm) the velocities are calculated using a kinematic wave approximation

ghvuu

fS fx 8)( 22 +

=

ghvuv

fS fy 8)( 22 +

=

x direction:

y direction:

g gravitational acceleration (m.s-2)h flow depth (m)Sfx friction slope (x direction) Sfy friction slope (y direction)u flow velocity (x direction) (m s -1)v flow velocity (y direction) (m s -1)f Darcy-Weisbach friction factor

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Model description : soil erosion

Transport by runoff

Detachment and re-detachment by raindrop

impactDeposition

Entrainment

Detachment by runoff

Dfd < 0

Dfd > 0

Drd > 0

Tc> qs

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A covering Layer of loose sediment (Hairsine and Rose, 1991) ε is conceptualized as the percentage of a grid cell covered by a deposited layer of depth the median particle diameter D50.

Therefore ε is calculated as:

Model description: Soil erosion

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Sediment mass conservation equation (Bennet,1974)

)(1)()()(fdrd

s

yx DDycq

xcq

thc

+=∂

∂+

∂∂

+∂

∂ρ

h water depth (m)c sediment concentration (m3 m-3)ρs sediment particle density (kg m-3) qx unit runoff discharge (x direction) (m2 s-1 )qy unit runoff discharge (y direction) (m2 s-1 )Drd soil detachment rate by rainfall (kg m-2 s-1 )Dfd soil detachment/deposition rate by runoff (kg m-2 s-1 )

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Soil detachment by rainfall is a function of the rainfall intensity (Li, 1979)

Before sediment movement

(kg m-2 s -1)

α soil detachability coefficient by rainfall (kg m-2 mm-1)p an exponent set to 1.0 according to the results of Sharma et al. [1993]h water depth (m)ld loose sediment depth (m)zm the maximum penetration depth of raindrop splash (m)R rainfall intensity (m s-1)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

mzh1 Damping effect of the water film at

the soil surface 182.069.6 Rzm ×=

where

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Soil detachment by rainfall

Detachment

Re-detachment

After sediment movement

(kg m-2 s -1)

(kg m-2 s -1)

ε function of the area of the covering layer (0-1)α soil detachability coefficient by rainfall (kg m-2 mm-1)αd soil re-detachability coefficient by rainfall (kg m-2 mm-1)p an exponent (1.0) h water depth (m)zm the maximum penetration depth of raindrop splash (m)R rainfall intensity (mm h-1)

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Soil detachment or deposition by runoff : a model proposed by Foster and Meyer [1972]

When • qs<Tc, additional sediment detachment • qs>Tc, excessive sediment deposition

)( scfd qTD −∝ (kg m-2 s-1)

Tc sediment transport capacity of the flow (kg m-1 s-1)qs sediment discharge per unit flow width in theflow direction (kg m-1 s-1)

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When Tc>qs (Dfd>0) net erosion occurs and the detachment and entrainment rates are given by:

Detachment (kg m-2 s -1)

Entrainment (kg m-2 s -1)

τf is the flow shear stress in the flow direction (Pa)τc is the critical shear stress of a spherical sediment particle [Yang, 1996] (Pa)τsoil the critical shear stress of the soil (Pa)Kr is the rill erodibility parameter (s m–1)Tc sediment transport capacity of the flow (kg m-1 s-1)qs sediment discharge per unit flow width in the flow direction (kg m-1 s-1)

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When Tc<qs (Dfd<0) net deposition occurs and the deposition rates is given by [Foster et al., 1995]:

(kg m-2 s -1)

ϕ is a raindrop induced turbulence coefficient assigned to 0.5.Vf is the particle settling velocity (m s–1)q is the water dicharge per unit flow width in the flow direction (m 3 s-1 m-1)Tc sediment transport capacity of the flow (kg m-1 s-1)qs sediment discharge per unit flow width in the flow direction (kg m-1 s-1)

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Flow sediment transport capacity is based on the flow shear stress τf (Foster, 1982)

kcfcT )( ττη −= (kg m-1 s-1)

η coefficient of efficiency of sediment transport (m0.5 s2 kg –0.5 ) τf flow shear stress acting on the soil particles (Pa)τc critical shear stress of sediment (Pa)k an exponent taken as 1.5 (Finkner et al.,1989)

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Model description: Numerical methods

Hydrological model and erosion model are treated independently since it is assumed that the flow dynamics are not affected by the suspended sediment

The Saint Venant equations are solved using the MacCormack scheme

The mass balance equation for sediment is solved using a second-order centered explicit finite difference scheme

For numerical stability of the scheme and computational efficiency the time step is optimised

Topographic elevations are re-estimated at each time step if there is runoff

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Model description

Flow chart of PSEM_2D

Model description: Numerical methods

To avoid directional bias of the Mac Cormack scheme the order is reversed every time step(predictor-forward, corrector-backward then

predictor-backward, corrector-forward).

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Boundary conditionsIn the plot version the boundaries are 3 non porous walls and an open boundary (outlet)

Dummy cells are added to model wall boundary

At the outlet no condition is required because the flow is supercritical

Initial conditionAt the beginning of the simulation

h(x,y,0) = 0 u(x,y,0) = 0 v(x,y,0) = 0

c(x,y,0) = 0

Model description: Initial and boundary conditions

u = 0 u = 0

v = 0

inwardboundaries

upslope

downslope

y

dummy cellsu=0 v=0h=h_inwardc=c_inward

We consider that rainsplash transportation outside the plot isbalanced by sediment coming from the area surrounding the plot.

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Model description : Calibrated parameters

Transport by runoff

Detachment and re-detachment by raindrop

impactDeposition

Entrainment

Detachment by runoff

Dfd < 0

Dfd > 0

Drd > 0

Tc> qs

τsoil

Kr

α

αd

η

hf,f

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Model description: Data

The model needs information on

Slopes and elevations (Digital Elevation Model)

Map of soil surface features distribution

Infiltration parameters (hf, initial WC,Kc,Ks)

Map of DW friction factor

Soil erosion parameters (η,Kr, τsoil, D50)

Map of α and αd, (αd=10 α)

Rainfall (time, intensities)

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Model description: Parameter identfication

The parameter identification is carried out in three stages

We started with parameters estimation based on physical characteristics and published data

Some of soil erosion parameters are defined using data available in the literature (δs=0.047, τsoil is estimated using the WEPP soil database)

Calibration is undertaken for hf (crusted soils) and/or f on one rainfall event

Calibration is undertaken for η,Kr, τsoil using the ranges of values found in the literatureWEPP: Water Erosion Prediction Project (US Dept. Agr.)

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Applications of Applications of PRIM_2DPRIM_2D

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Examples of application PRIM_2D

Two runoff plots located on the same hillslopeHomogeneous soil surface feature (ERO)

one type of crust: erosion

Heterogeneous surface feature (JAC)

erosion crust and sandy aeolian micro mounds

• Grid resolution 0.25 by 0.25 m• Both plots have the same subsoil• Initial soil water content were obtained from

neutron probe measurements• Verification runs

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Examples of application PRIM_2D

Homogeneous soil surface feature (ERO)

Heterogeneous surface feature (JAC)

Runoff plots in Niger(West Africa)

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DYNAS Workshop, 6th-8th December 2004, INRIA0 2 40

2

4

6

8

10

12

14

ERO

0 2 40

2

4

6

8

10

12

14

16

18

20

0.95

Sandy mounds

Erosion crusts

JAC EROSoil properties Soil texture Loamy sand Loamy sand θs sat. W C (-) 0.296 0.296

hf (m) 1.3795 1.3795 Ks (m/s) 2.15 E-05 2.15 E-05Surface properties

ErosionZc (m) 0.005 0.005

hf (m) 1.3795 1.3795 Ks (m/s) 1.70 E-08 1.70 E-08 f 0.25 0.25

Sandy moundsZc (m) 0.05

hf (m) 0.18 Ks (m/s) 1.90 E-06 f 0.70

JAC EROLength (m) 20.0 14.25Width (m) 5.0 5.0Max slope (x) 0.19 0.12Max slope (y) 0.17 0.21

Examples of application PRIM_2DJAC

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PRIM_2D Validation

04 september 94

0

20

40

60

80

100

120

140

160

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Time (s)

Disch

arge

and

rainf

all i

nten

sity

(mm/h

)

Rainfall Observed Calculated

An exemple of validation run

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PRIM_2D Validation

0

10

20

30

40

0 5 10 15 20 25 30 35 40

Observed (mm)

Calc

ulat

ed (m

m)

calibrationvalidation1:1

Runoff depth

05

101520

25303540

0 5 10 15 20 25 30 35 40

Observed (mm)

Calc

ulat

ed (m

m)


Infiltration depth

0

1000

2000

3000

4000

0 1000 2000 3000 4000

Observed (s)

Calc

ulat

ed (s

)


Time to peak

Calc

ulat

ed (m

m/h

)

0

25

50

75

100

125

150

175

200

0 25 50 75 100 125 150 175 200

Observed (mm/h)


Maximum discharge

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PRIM_2D Validation

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700

Observed (s)

Calc

ulat

ed (s

)


Time to begin runoff

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Plot scale results

ERO 25 august 94 20:31:00

020406080

100120140160

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

Time (s)

Run

off a

nd R

ainf

all

inte

nsity

(mm

/h)

Rainfall Observed Computed

JAC 25 august 94 20:31:00

020406080

100120140160

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

Time (s)

Run

off a

nd R

ainf

all

inte

nsity

(mm

/h)

Rainfall Observed Computed

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Plot scale results

Plot Rain(mm)

Ov. flow(mm)

Peak disch.(mm/h)

Infiltration(mm)

ERO obs. 23.9 14.3 91.6 9.6

ERO cal. 23.9 14.5 95.2 9.4

Rel. error - -1.9 % + 3.9 % + 1.3 %

JAC Obs. 23.9 11.9 68.7 12.0

JAC cal. 23.9 11.3 69.2 12.6

Rel. error - - 5.0 % + 0.7 % + 5.0 %

Efficiency ERO : 0.879 Efficiency JAC : 0.913

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DYNAS Workshop, 6th-8th December 2004, INRIA0 1 2 3 4 50

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4

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14

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2

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2

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0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Velocities (m/s)

0.5

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

JACJAC

ERO

EROWater

depth (m)

Time 789 s(max discharge)

Distributed results

35


2

4

6

8

10

12

14

16

18

20

0.043

0.048

0.053

0.058

0.063

0.068

0.073

0.078

0.083

0.088

0.093

0.098

0 1 2 3 4 50

2

4

6

8

10

12

14

0.043

0.0435

0.044

0.0445

0.045

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 50

2

4

6

8

10

12

14

JACJAC

ERO

EROInfiltration depth (m)

Shear velocities(m/s)

Time 789 s

Distributed results

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DYNAS Workshop, 6th-8th December 2004, INRIA0 2 4

0

2

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0

50

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0 500 1000 1500 2000 2500

mm

/h

A small pond

B rill

D top

C rill

0

0.1

0.2

0.3

0 500 1000 1500 2000 2500

(m/s

)

0

0.0025

0.005

0.0075

0.01

0 500 1000 1500 2000 2500

(m)

0

0.02

0.04

0.06

0 500 1000 1500 2000 2500

(m/s

)

0

500

1000

1500

0 500 1000 1500 2000 2500Time (s)

Velocities

Water depth

Shear velocities

ReynoldsRainfall

Point results

37


Effect of the microtopography

The microtopography is represented by

the topographic map of the plot (JAC)

a plane surface with the same mean slope

All other parameters are the same

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25 august 94 20:31:00

0102030405060708090

100

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400Time (s)

Run

off a

nd R

ainf

all

inte

nsity

(mm

/h) Observed

Computed Plan. Computed Topo.

Simulation Rain(mm)

Ov. flow(mm)

Peak disch.(mm/h)

Infiltration(mm)

Topography 23.9 11.3 69.2 12.6

Plane 23.9 11.1 73.0 12.8

Diff. - - 1.8 % + 5.5 % + 1.6 %

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2

4

6

8

10

12

14

16

18

20

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Vel. (m/s)

0.5

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

0 2 40

2

4

6

8

10

12

14

16

18

20

0 2 40

2

4

6

8

10

12

14

16

18

20

JAC PLAN PLANJAC

Water depth (m)

Distributed resultsTime 789 s


40


Effect of the surface features distribution

The soil surface features are represented by

the soil surface feature map (JAC)

the dominant surface feature (erosion crust)

All the other parameters are the same

41



25 august 94 20:31:00

0102030405060708090

100

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400Time (s)

Run

off a

nd R

ainf

all

inte

nsity

(mm

/h)

Observed Computed 1 SF Computed 2 SF

Simulation Rain(mm)

Ov. flow(mm)

Peak disch.(mm/h)

Infiltration(mm)

2 Surf. feat. 23.9 11.3 69.2 12.6

1 Surf. Feat. 23.9 14.4 84.6 9.5

Diff. - + 27.4 % + 22.3 % - 24.6 %

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JAC 2 SF JAC 1 SF

0 2 40

2

4

6

8

10

12

14

16

18

20

0 2 40

2

4

6

8

10

12

14

16

18

20

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

Water depth (m)

Time 589 s

For this storm the time to ponding is

390 s for erosion crust

625 s for sandy mounds

43


Key results

Even in low relief plots, OF is not a sheet of flowing water, uniform in depth and velocity across the slope. OF concentrates downslope into deeper flow pathways

Small surface feature may play a major role in the OF production from a plot

A good reproduction of discharges at the outlet of a plot does not imply that OF hydraulics is correctly simulated

Infiltration is not homogeneous all over the plot which is partly due to the effect of micro-topography

Large variations in the OF hydraulics are due to the variable rainfall rates and to the characteristics of the uphill areas

44


PSEM_2D EvaluationPSEM_2D Evaluation

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Psem_2D Evaluation : Experimental data

Kilinc and Richardson (1973) experimental data

A 1.52 m wide × 4.58 m long flume with an adjustable slope and a rainfall simulator. Each run was one hour long

The flume was filled with compacted sandy soil composed of 90 % sand and 10 % silt and clay.

The soil had a non-uniform size distribution with a median diameter D50 of 3.5 × 10-4 m.

The soil surface was levelled and smoothed before each run.

46



Kilinc and Richardson (1973) experimental dataThe major controlled variables were rainfall intensity and soil surface slope.

Infiltration and erodibility of surface were supposed constant.

Six slopes (5.7, 10, 15, 20, 30, and 40 %) were tested

Four rainfall intensities (32, 57, 93, and 117 mm h-1).

Calibration was carried out using a run with 20 % slope and 93 mm h-1 rainfall intensity.

47



Data available

• Flow discharge at the outlet of the flume

• Mean sediment concentration in the flow at the outlet

• Mean infiltration rate

• No data were collected on microtopography and Overland flow hydraulics (water depth, velocity)

48


Psem_2D Evaluation : Results

Rain intensity, 93 mm h-1. Slopes, 15, 20, and 30 %

0

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 50 60

Time (min)

Sed

imen

tdis

char

ge(k

g/m

/s)

Observed, 30 % slope

PSEM_2D, 30 % slope

Govindaraju and Kavvas[1991], 30 % slope


PSEM_2D, 20 % slope(CALIBRATED)



PSEM_2D, 15 % slope


49



Rain intensity, 117 mm h-1. Slopes, 15, 20, and 30 %

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 60

Time (min)

Sed

imen

t dis

char

ge (k

g/m

/s)


PSEM_2D, 30 % slope



PSEM_2D, 20 % slope



PSEM_2D, 15 % slope


50



0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35Time (min)

Sed

imen

t con

cent

ratio

n (g

/l)

Observed, 50 mm/h Observed, 100 mm/hPSEM_2D, 50 mm/h (calibrated) PSEM_2D, 100 mm/hGovindaraju and Kavvas [1991], 50 mm/h Govindaraju and Kavvas [1991], 100 mm/h

Singer and Walker [1983] experimentSlope 9%

D50 of the soil: 2. 10-5 m

51


Psem_2D Evaluation : Sensitivity analysis

The range of variation of the parameters calibrated with the data of Singer and Walker [1983]

9 % slope and 50 mm h-1 rainfall intensity.

52


Psem_2D Evaluation : Sensitivity analysis

-200

-100

0

100

200

300

400

500

-500 0 500 1000 1500 2000 2500

parameter variation (in %)

C v

aria

tion

(in %

)

δs

Kr

f

D50

τsoil

α

η

ld_initia l = 0.01 m

Variations in percentage of the mass sediment concentration versus variations in percentage of each tested parameter, all the

other parameters keeping the calibrated value

53


PSEM_2D PSEM_2D ApplicationsApplications

54


Psem_2D application

Plot 5 by 15 m a grid of 0.2 by 0.2 m

Parameter values of Singer and Walker experiment

Average slopes are 0.02 and 0.06 in the x and y directions.

55


Psem_2D application: Effect of initial condition

Effect of the formation of a deposited layer before the rainfall

0

20

40

60

80

100

120

140

160

180

2000 20 40 60 80 100 120 140

time (min)

rain

fall

inte

nsi

ty(m

m/h

)

0

20

40

60

80

100

120

140

160

180

200

wat

erdi

sch

arge

(mm

/h)

and

sedi

men

tco

nce

ntr

atio

n (

g/L)

rainfallwater dischargesediment concentrationsediment concentration (without the first rainfall event) D50 = 20 µm

56


Psem_2D application

Erosion and deposition pattern on the plot at the end of thetwo consecutive rainfall events (time = 135 min after the

beginning of the simulation)

-0.007 m

-0.006 m

-0.005 m

-0.004 m

-0.003 m

-0.002 m

-0.001 m

0 m

0.001 m

0.002 m

0.003 m

0.004 m

0.005 m Deposition

D50 = 20 µm

Erosion

57


Psem_2D application

Computed flow depths

time = 124 mintime = 27 min

0 m

0.0004 m

0.0008 m

0.0012 m

0.0016 m

0.002 m

0.0024 m

0.0028 m

0.0032 m

0.0036 m

0.004 m

0 m

0.0004 m

0.0008 m

0.0012 m

0.0016 m

0.002 m

0.0024 m

0.0028 m

0.0032 m

0.0036 m

0.004 m

58


Psem_2D application

Hydrograph and related sedimentographs for different particle size diameter

020406080

100120140160180200

0 20 40time (min)

rain

fall

inte

nsity

(mm

/h)

0

20406080100120140160180200

wat

erdi

scha

rge

(mm

/h) a

ndse

dim

entc

once

ntra

tion

(g/L

)

rainfall water dischargesediment concentration D50=12µm sediment concentration D50=20µmsediment concentration D50=100µm sediment concentration D50=200µmsediment concentration D50=500µm sediment concentration D50=1000µm

59


Psem_2D application

Contribution of the different processes to the sediment yield

-40

-20

0

20

40

60

80

12µm 15µm 20µm 100µm 200µm 500µm 1000µm

Median diameter D50

mas

s (k

g)

EntrainmentF Detachment

R Re-detachmentR Detachment

Deposition

60


Psem_2D application

Interrill versus Rill erosion: what does it change in terms of processes ?

Hyetograph

0

20

40

60

80

100

120

140

160

180

2000 20 40

time (min)

rain

fall

inte

nsity

(mm

/h)

61


Psem_2D application

Comparaison interrills rills contributing processes to the total sediment yield

plot size0%

20%

40%

60%

80%

100%

1m*1m plot 15m*5m plot

Tota

l sed

imen

tm

ass

EntrainmentF Detachment

R Re-detachmentR Detachment

29 %

95 %64 %

Deposition represents 11.4 % of the total mass eroded

Deposition represents 0.7 % of the total mass eroded

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Some key issues Runoff production limited to excess rainfall

Sources and sinks of sediment vary with the magnitude of the events

The soil erodibility coefficients have not yet been quantitatively related to a measurable soil property and must therefore be determined empirically or calibrated

Model calibration, a lot of parameter to determine

More complex models increase data requirement and …

Increase data and model uncertainty, which affects model results

Propagation of errors in input data

Model structural errors

Uncertainty associated with evaluation of model parameters

Problem of the model evaluation (spatial field data) “the right answer for the wrong reason”

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As a conclusion Future research

Overland flow hydraulicsTo improve the prediction of the flow resistance from surface roughness To analyse the respective effects of roughness and micro topography

Modelling erosionTo validate the model for complex microrelief and natural rainfall events : new experimentsTo improve the representation of the flow detachment at the subgrid level To implement a multiclass sediment representationTo test alternative parametrisation of the transport capacity

Unit stream powerGovers equation (1990)

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M. Esteves, G. Nord · Model description: Soil erosion ¾Soil detachment by rainfall Detachment...

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