Empirical and graphical models in
zoning
Harald Norem
Statens vegvesen, Vegdirektoratet
The problem
To estimate the maximum
run-out distances for
avalanches with a certain
return period
The run-out distance is
defined as the farthest runout of debris materials.
The run-out of the snow
cloud, suspension flows and
suspended particles in
water are not included
Definition of run-out distances
Run-out distances are
defined by either:
1. The distance from the
starting point to the stop of
the debris of the landslide, L.
2. The angle of the line
connecting the start position
and the end position.
Tan a=H/L
a
H
L
Run-out distance as a function of return period
5 år
10 år 30 år
50 år
100 år
300 år
1000 år
Magnitude of avalanches versus return periods
(Fitzharris 1981)
Type of models
Topographical models
– Based on only topographical parameters
Dynamical models
– Estimate the velocity and the run-out distance based on the
profile of the slide path and selected values for size and
properties of slides
Nearest neighbour method
– Comparing recorded run-outs for slopes having similar
topography
Graphical methods
– Graphical interpretation of dynamical models
Requirements for run-outs models
The models should be based on
objective criterias, as:
– Topographical parameters
– Climatological parameters
– Height, width and length of the released
avalanche are selected based on topography or
climate
– Parameters describing the physical properties
of the dimensioning slide are fixed numbers or
may be a function of topography or climate
Simple run-out models
Scheidegger 1973
Run-out angles versus
volumes
Rickenmann 1999
H/L=1,9M-1/6H0,17
Lower bound for snow avalanches?
Run-out distance for submarine slides
Edgers and Karlsrud (1982)
100
10
Run-out distance versus volume of the
avalanche, Ryggfonn
Topographical models
NGI a-b model
•The model is based on an archive of approx. 200 avalanches with known
long run outs (long return periods, 100-300 years))
•The standard deviation is 2,3, which means 225 m with a=25o and H=1000 m.
Topographical models
The topographical model are based on
the following assumptions:
Within long return periods all kinds of major
avalanches will occur, and the climate thus plays
a less important role than topography for the
run-outs. The model can thus be used for a wide
variety of climatic conditions
The topography may be described by some
objective parameters
Topographical parameters
= gradient in the starting area
H
Best fit parabel, y’’H
10o
b= Gradient to the 100 point
a
L
Equations:
a= 0,96b- 1,4o
a=0,92b-0,00079H+0,024y’’Hθ+0,04
b=300
θ=350 H=1000m y’’=0,0003 m-1
a=28,8-1,4=27,4
a=27,6-0,79+0,25+0,04=27,1
s=2,30, r=0,92
s=2,28 r=0,92
Factors favouring long run-outs
(Small a-values)
Gentle slopes in the starting zone (<350)
Gentle and parabolic avalanche tracks
Dry snow in the run-out area (Cold climates in
the run-out area, in Norway; in the high
mountains and in North-Norway)
High frequency of intense snow deposits in the
starting zone.
Topographic features having minor
effects on the run-out:
Scars and bowls in the starting zone
Channelized flows
Dynamical models
Dynamical models may
either be:
1400
50
Velocity
45
1200
40
1000
Height
1 D (Center of mass or
blankets) Only velocities
arecalculated)
35
30
800
25
600
20
Avalanche path
15
400
2 D (Velocity and flow height)
Road 1560
10
200
5
3 D (Velocity, flow height and
width)
0
0
0
200
400
600
800
1000
1200
1400
1600
1800
Distance
Main weaknesses:
Parametric values are usually not allowed to change during the flow
and changes due to humidity (water content), entrainment, pore
pressures and granular distribution are not included.
Dynamical models
Requirements for dynamical models used for
estimating run-out distances:
– The topography needs to be well defined
– The physical parameters defining the
properties of the avalanche should be selected
due to objective criterias.
– The physical parameters may be selected due
to:
• topographic parameters
• Climate
• Selected fixed values
Dynamical models
So far, there has been made only a few
investigations to test the accuracy of these
models to compare them with the database of
recorded avalanches.
The investigations indicate that the accuracy is
equal or partly lower than the topographic model
Forces acting on the avalanche
h
g
F=F +
F=Fri
C F =H
ksjon
d
astigh
etsua
vheng
ig
Aksel
er
kraft: erende
P=g
hsina
friksjo
n + ha
a
stighe
tsavh
engig
friksj
on
Velocity dependent stresses are necessary to have the
avalanche to stop in sloping terrain
Velocity dependent stresses are necessary to avoid unrealistic
high velocities in the steep part of the path
Existing models
Velocity independent stresses:
Coulomb friction: tc=mghcos a (common for all models)
Velocity dependent stresses:
Voellmy:
PCM:
NIS:
td=gv2/ξ
td=Dv2
td=mm(v/h)2
Terminal velocity
Voellmy:
PCM:
NIS:
v=(ξh(sina-mcosa))½
v=(M/D(sina-mcosa))½
v=2/3·(gh3(sina-mcosa)/(m-mn2))½
Effects that are hardly not
incorporated in the models
Entrainment/ deposition
Boundary conditions
Cohesion
Change of the physical properties during the flow
Active/passive stresses during the flow
Pressures at the front of the avalanche
Effects of varying properties normal the flow
Slab thickness as function of the
gradient in the starting zone
Skjærspenning: t=gz·sina
z
zkr
Kritisk
dybde
a
Dybde til kritisk snølag (m)
Skjærfasthet: tf=c+mgz·cosa
6
5
Kohesjon: 5 kN/m2
Friksjonskoeff: 0,3
4
3
Kohesjon: 2 kN/m2
Friksjonskoeff: 0,2
2
1
0
25
30
35
40
45
Helling (grader)
h =
sin 40  0,31cos 40
sin   0,31cos 
50
55
60
Slab length as function of the height
difference
100 m or H/6
H
Comparing topographical and
dynamical models
atop-aRec
2
-2
Acceptable
-2
2
aNIS-aRec
Comparing topographical and
dynamical models
atop-aRec
aNIS-aRec
The basic idea of the nearest
neighbour model:
Is it possible to find avalanche paths in the
”avalanche library” that are almost identical to
the avalanche path investigated?
What caracterizise simular avalanche paths?
Nearest neighbour model
= gradient in the starting area
H
Dy
Best fit parabel, y’’H
Parameters used:
10o
b= Gradient to the 100 point
a
L
- bangle (within 20)
•Gradient in the starting zone
•Height difference
•Height in the run-out zone
•Deviation between the recorded path and the
parabolic curve
Z
Comparing avalanche paths, Austria
Avalanche 06 Sill
Nearest neighbour model
The model picks up the 5
slide paths that are most
similar to those investigated
Nearest neighbour model
Experiences from the nearest
neighbour model
Norway
•
The avarage standard deviation between the 5 avalanche
paths and the recorded run-out for the one investigated
was 1,86, which is an improvement for the ab and the
NIS, which is 2,2o and 2,3o respectively
Austria
•
•
The ab model gave lower standard deviations than the
PCM and the NIS models
The nearest neighbour model was promising, and the
average run-outs for the 5 most simular paths were close
to the one investigated.
Scope of graphical models
Simplified methods to evaluate:
Run-out distances
Velocities along the avalanche path
The dimensioning velocities for the design of
protecting measures
The Profile
concept
of the energy line
of Nakkefonna with calculated velocity and energy line
Bernoulli equation
1400
50
Calculated velocity
45
1200
40
1000
35
Height
Energy line
30
800
25
Avalanche profile
600
Kinetic energy, v2/2g
20
0,40
15
400
Potential energy, z
Road X=1560
10
200
5
0
0
0
200
400
600
800
1000
Distance
1200
1400
1600
1800
Ideas for an alternative model
Is it possible to define a certain point on the energy line
and the gradient of this line for most kinds of slides?
Bernoulli Equation
Z
+
p/g
Height
+
Pressure height
v2/2g
=konstant (no loss of energy)
Velocity height
Loss of energy
from 1 to 2
v2/2g
1
p/g
v2/2g
2 p/g
Z
Z
Energy line with only Coulomb friction
m·Dx
m
1
2
Dx
Loss of energy from 1 to 2: F·Ds=mghcosa·(Dx/cosa)
Loss of energy per unit weight: mDx
The energy line is a straight line when there are only Coulomb
friction and the gradient of the line is equal m
Effects of the friction terms
Snow avalanches
1200
1000
m=0,3
M/D=2000
v-max=53 m/s
Height
800
75 %
Coulomb
600
m=0,1
M/D=120
v-max=16 m/s
400
25 %
Velocity
200
0
0
200
400
600
800
1000
Distance
1200
1400
1600
1800
2000
Effects of the friction terms
Debris flows
1200
1000
m=0,3
M/D=2000
v-max=53 m/s
Height
800
25 %
Coulomb
75 %
Velocity
600
m=0,1
M/D=120
v-max=16 m/s
400
200
0
0
200
400
600
800
1000
Distance
1200
1400
1600
1800
2000
Radarmålinger,
Gubler(1986)
(1987)
Radarmålinger, Gubler
Numerisk modellering, Norem (1992)
Numerisk modellering (Norem 1992)
Energilinje
1
0,4
Model for snow avalanches
Avalanche paths:
The energy model
Idealized parabolic profiles
Gradient of the energy line:
Most major avalanches has a
gradient close to 0,4, maybe
as low as 0,35
Velocity at the 20o point:
V=Ch(singmcosg)½
Flow height:
h =
sin 40  0,31cos 40
sin   0,31cos 
f
Energy line
g
1,0
20o
hk
0,4
Establishing an energy line
Estimating velocity at the road
f
Velocity at the road:
v = 2 gh
g
Energy line
1,0
hk
h
0,4
Road
Estimated run-out
Energy model versus a-b model
Represents different b
and  values
Sensitivity analyses
Snow avalanches
60 m/s
60 m/s
50 m/s
0,4
0,4
0,35
5o
106 m
193 m
Debris flows
15 m/s
10 m/s
15 m/s
0,3
0,26
0,2
33 m
47 m
Adjustments to different return periods
+1/2s ?
-s (One standard deviation)
a-b-model
5 år
10 år 30 år
50 år
100 år
300 år
1000 år
Conclusions
The ”avalanche librarys” are mainly based on major snow
avalanches having return periods varying between 100 to
300 years
There are no references to variations of the relative run-out
for different return periods
The ab-model is very useful for a first estimate of the runout for major slides
Best results are probably obtained by comparing the reults
of several models
Adjustments should be made due to climate and return
periods