Download Modelling a Debris Flow with PCRaster

Modelling a Debris Flow with PCRaster
Tirano – 22/07/2008
Targets of Presentation
1. PCRaster model I’ve done
2. How I think to improve it and ask you some
suggestions about it
Geographical Setting
Comunità Montana
Valtellina di Tirano
Tresenda
Debris Flows in Tresenda
2
1
3
Debris Flows in Tresenda
►
►
►
►
►
Three debris flows in May 1983 caused by the
collapse of a series of dry retaining walls
13 victims in Tresenda and 4 in Valgella
Damages to buildings and infrastructures
In 2003 a very similar event occurred
damaging the houses, obstructing a water
channel and determining the accumulation of
material along the national road and its
closing.
Number 1 is that I use to calibrate the model
PCRaster Model
Initial parameters
► Maps
► Principal points in the dynamic section
►
1.
2.
Moving of material respect to the slope
Material which deposits – material which
flows
3. Erosion in presence of a dry-stone retaining
wall
Rheological Parameters
► CohMob=
debris flow cohesion [kN.m-2]
► PhiMob= angle of internal friction of
moving mud
► GammaDf= unit weight of debris flow
[kN.m-3]
Maps
Fail depth
Dem
Terraced area
Script
Dynamic Section – Principal Points
1
2
4
7
In the dynamic section I want to calculate how the
debris flow moves from a geometrical point of view.
In every cell the material can come from every
upstream cell (Hn > H ) and can move to every
downstream cell (Hn < H ). The quantity of material
that arrives in each cell is proportional to the slope
between the central cell and the cell N.
3
6
8
9
►
H_mov_N = H_mov * FractToN
ƒ
ƒ
ƒ
FractToN = SlopeN / SlopeTOT
SlopeN = (Z - Zn)/DL
SlopeTOT= Slope1 + Slope2 + … + Slope9
Script
Dynamic Section – Principal Points
1
2
4
7
3
6
8
9
H_mov is the part of material existent in the cell
that can move and it is calculated from:
= H - H_crit
► H_mov
where H is all the thickness of debris and H_crit is
the height of material that deposits in the cell,
calculated from Factor of Safety formula with FS =
1 and solved respect to H:
►
H_crit = if (SinBeta eq 0 or CosBeta eq 0 or
TanBeta eq 0, then H, else CohMob /
(GammaDf*SinBeta*CosBeta * (1(TanPhi/TanBeta))))
where Beta is the slope [deg] calculated cell by cell
with the PCRaster’s operator slope from the DEM
map.
Script
Dynamic Section – Principal Points
H_loop= H -abs(H_mov_1) -abs(H_mov_2) abs(H_mov_3) -abs(H_mov_4) -abs(H_mov_6) abs(H_mov_7)
-abs(H_mov_8)
-abs(H_mov_9)
+shift0(H_mov_9, 1, -1) +shift0(H_mov_8, 1, 0)
+shift0(H_mov_7, 1, 1) +shift0(H_mov_6, 0, -1)
+shift0(H_mov_4, 0, 1) +shift0(H_mov_3, -1, -1)
+shift0(H_mov_2, -1, 0) +shift0(H_mov_1, -1, 1)
+Erosion_loop
At the end in each cell the new value of H is equal
to the value of H of the precedent loop – FlowOut +
FlowIn (both calculate with the FractTo method) +
Erosion. Erosion:
► Erosion= if (Slope gt 0.05, if (H_mov gt 0.1, if
(TerracedArea, 1, 0.05), 0), 0))
►
1
2
4
7
3
6
8
9
Results
c = 0.6 kN/m2; φ= 10°; γ= 16 kN/m3
c = 0.26 kN/m2; φ= 17°; γ= 16 kN/m3
Results
► The
first result is the best fit one but it
presents a too small friction angle
► The second result has a more correct
value of φ but the geometry isn’t perfect
► I think to solve this problem by the
introduction of velocity because I
suppose it is created by the values of Hcrit
based only on slope.
How to Improve the Model
1.
2.
3.
4.
5.
6.
7.
In order to obtain the hazard an consequently the
risk connected with the event I need the velocity of
the flow. For this reason I will try to implement in the
model the equations of Takahashi (1991) following this
scheme:
Velocity of the debris flow
Critical slope
Critical velocity
Compare velocity of df with the critical one
Deposition rate
Height of the deposit
Height of the flowing mud
How to Improve the Model
1
1
⎤ 3
2⎡
3
c
⎤
ρ
2 ⎡ gsinθ ⎛
⎞
⎛
⎞
*
1)U =
⎜ c + ⎛⎜⎝1 − c ⎞⎟⎠ ⎟⎥ ⎢⎜ ⎟ − 1⎥h 2
⎢
5d ⎣ asinα ⎝
σ ⎠⎦ ⎢⎝ c ⎠
⎥⎦
⎣
(σ − ρm )cL
2)tanθc =
(σ − ρm )cL + ρm
1
ρm ⎞⎤ ⎡⎢⎛ c*DL
2 ⎡ gsinθc ⎛
3)Uc =
⎜ cL + (1 − cL ) ⎟⎥ ⎜⎜
⎢
5dL ⎣ asinα ⎝
σ ⎠⎦ ⎢⎝ cL
⎣
2
4)U − − − − − Uc
⎛
U
5)i = δd ⎜⎜ 1 −
⎝ rrUc
6)hstop = i ⋅ t
⎞ cL∞ − cL qt
⎟⎟
⎠ c*DL dL
7)hmov = h − hstop
(1), (2), (3), (5) from Takahashi (1991)
⎞
⎟⎟
⎠
1
3
⎤ 3
− 1⎥h 2
⎥
⎦
How to Improve the Model
a: constant to describe the constituve relations
c: volume concentration of the solid in the flow
c*: packing concentration of the solid
cL: volume concentration of the coarse fraction in the total volume
c*DL: volume concentration of the coarse fraction in the static bed after
deposition of debris flow
d: particle diameter
dL: mean diameter of the coarse particle in debris flow which is supported
by the dispersive pressure
h: total height of the flow in each cell
i: rate of deposition
qT: unit width discharge of debris flow
rr: constant. Its value is more or less 1/3
α: angle of particle encounter analogous to kinetic friction angle
δd:costant
ρ: density of fluid
ρm: apparent density of debris flow
σ: density of the particle
θ: slope angle
Discussion
► I’m
not sure about the FractTo method that
I use, based on slopes between cells. Have
you any idea?
► Do
you think there are other things I should
modify?