# IJEGE-11_BS-Gregoretti-et-alii

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*DOI: 10.4408/IJEGE.2011-03.B-048*

**GIS-BASED CELL MODEL FOR SIMULATING DEBRIS FLOW**

**ROUTING AND DEPOSITION PHASES ON A FAN**

runoff descending from the upstream cliffs. The routing

path of debris flows is usually obliged and coincides

with the channel in the upper part of the fan but in the

medium part it can deviates (t

*et alii*, 2007)

can spread (i

*et alii*, 1998; R

tial, by debris flows. Therefore hazard maps are built

both using data from surveys of areas flooded by de-

bris flows and through the simulation of potential sce-

narios. The models (methods) used for the simulation

of a potential scenario are empirical (a

*et alii*, 2000; a

*et alii*, 2009), SPH (P

*et alii*, 2008) and automata cellular (d

was proposed by z

*et alii*, (1970) for simulate

was successively adapted to simulate flood and runoff

routing in urban areas (R

*et alii*, 2009; C

*et alii*,

flow characteristics (channel flow, weir flow, floodplain

flow). Moreover, a cell model was also used to simulate

**ABSTRACT**

on a fan. Flow pattern is discretized by square cells,

2m size, which coincide with the DEM cells and the

mixture is assumed a monophasic continuum. Flow ex-

change between adjacent cells is ruled by uniform flow

or broad-crested weir laws and by continuity equation.

Flow occurs from cells with higher surface to those

with lower surface and is simulated by uniform flow

law if the elevation of the formers is higher than the

latter and by broad-crested weir law otherwise. Erosion

and deposition are simulated using the empirical law of

Egashira, adjusted for monophasic continuum. The cell

model is used to simulate debris flow occurred on Rio

Lazer (Dolomites, Eastern Italian Alps) the 4

ing Flo-2D model for a comparison with a widely used

model for debris flow simulation. Results of the two

simulations were compared with extension of deposi-

tion area and the map of measured depths of deposited

sediments. Both the model simulate quite well the ex-

tent of deposition area, whereas the deposited debris

depths are better simulated by the cell model.

**K**

**ey**

**words***: GIS, cell model,fan spreading, hazard map*

**INTRODUCTION**

*C. GREGORETTI, M. FURLAN & M. DEGETTO*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

more depressed cell is inadequate to simulate debris

flow spreading in the deposition areas. Objective of this

paper is a robust model based on strong simplification

of hydraulics (i.e. flow cell) that allows a reliable simu-

lation of routed areas and sediment deposits of debris

flow. In this work the flow cell is used to simulate the

debris flow routing and deposition phases on a fan and

the flow cell scheme of z

*et alii*(1970) is there-

hydraulic links to simulate flow exchange of a cell with

the neighbours. Flow occurs from cell with higher flow

surface elevation towards cell with lower surface eleva-

tion and is simulated by the uniform flow equation in

the case of flow from higher elevation cell to lower el-

evation cell and by the weir equation otherwise.

by uniform flow and weir equation laws, requiring the

respect of continuity equation. The governing equa-

tions, then, are those of continuity for each cell and

discharge relationships between linked cells. Eight

possible flow directions are assumed (Figure 1) as in

the FLO-2D model and a possible lattice geometry in

automata cellular models (s

1. The solid-liquid mixture is assumed continuum

4. There are eight possible flow directions;

5. Flow section between cells are considered rectan-

7. Exchange flow of a cell with the neighbouring

where flow paths to the outlet are, a priori, determined

using topographic gradient and the routing is ruled by

De Saint Venant equation without inertia terms (J

*et*

*alii*, 2005). The proposed model does not distinguish

the cells but the hydraulic links that depend on both the

bottom and flow surface elevation between neighbour-

ing cells and is not limited by the topographic gradient.

The mathematical structure of the model is analogous

to that used by cellular automata models and that of

FLO-2D model even if the latter does not simulate the

sediment entrainment and deposition.

**THE CELL MODEL**

*OVERVIEw*

*et alii*, (1970), proposes to represent a basin through

homogeneous compartments, channels, floodplain gal-

leries, weirs which are in turn represented by cells. Each

cells interacts with the neighbouring cell by hydraulic

links (Saint Venant equations, with or without inertia

terms, broad-crested weirs, orifices, gates laws) that are

chosen on the base of the cell type: two “channel” cells

interact using the De Saint Venant equation, a “chan-

nel” cell interacts with a “floodplain” cell by the weir

equation, and so on. The cell model is able to reproduce

multiple flow patterns as those of urban areas and over-

comes the difficulty of implementing the usual numeri-

cal 2D models based on the shallow water equations in

a complex environment of streets, buildings, elevated

terrains and so on. On the other hand J

*et alii*(2005)

outlet considering only a flow path departing from each

cell along the steepest slope, that is, towards the sur-

rounding cell of lowest altitude. The model proposed by

J

*et alii*(2005) is therefore an hybrid between one

inflow could come from more than one of the neighbour-

ing cells but the outflow is only to the lowest altitude

cell. Moreover, this technique needs the pre-processing

of DEM at the purpose to eliminate holes that can inter-

rupt the flow path between a cell and the outlet.

which integrate the shallow water equations, meet

some difficulties that are due to the irregular and slop-

ing flow pattern and the presence of civil structures. On

the other hand the distributed cell model considering

*Fig. 1 - Scheme of the possible*

*flow directions*

**GIS-BASED CELL MODEL FOR SIMULATING DEBRIS FLOW ROUTING AND DEPOSITION PHASES ON A FAN**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

which sin q

between flow surface elevations of the two cells in-

stead of the difference h

and should avoid the use of a diminution discharge co-

efficient in the case of drowned “weir”. The weighting

functions depend on the topographic and flow surface

drops respectively and are used to narrow the flow

width because of the flow is not one-dimensional.

*EROSION AND DEPOSITION*

*U*

*i,k*

*Q*

*i,k*

*θ*

*i,k*

uniform flow and the angle between the line joining the

centres of flow surfaces of cell i and k diminished of the

adverse topographic slope between the two cells in the

case of weir flow;

*θ*

*LIM*

deposition respectively:

*θ*

*LIM-E*

*θ*

*LIM-D*

*θ*

*i,k*

*θ*

*LIM-E*

*U*

*i,k*

*U*

*LIM-E*

*U*

*LIM-E*

occurs for

*θ*

*i,k*

*<*

*U*

*LIM-D*

*U*

*i,k*

*U*

*LIM-D*

*U*

*LIM-D*

*ALGORITHM STRUCTURE*

*CONTINUITY EQUATION*

step n∆t (t = (n-1) ∆t) depends on the flow depths of

cells. In differential form, the continuity equations is:

cell k, negative otherwise.

*DISCHARGE EXCHANGE RELATIONSHIP*

BETwEEN CELLS

BETwEEN CELLS

elevation: uniform flow equation if cell elevation is

higher than that of the surrounding one (Figure 2a);

modified broad-crested equations if cell elevation is

lower than that of the surrounding one (Figure 2b). The

discharge equations in the two case are respectively:

*q*

i and k (atan(z

*Fig. 2 - Scheme of the possible flow between two adjacent cells*

*C. GREGORETTI, M. FURLAN & M. DEGETTO*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

volumes of all the cells and the corresponding bed

erosion/deposition velocities, flow surface elevations

of all the cells are simultaneously updated summing,

for each cell, the outflow, inflow, deposited and eroded

volumes just computed. The output boundary cells ex-

change flow discharge both with the surrounding cells

and with the extern. The flow exchange with the extern

is simulated without weighting functions by equations

(2) and (3) along flow directions by which the bound-

ary cell receives flow discharges according to uniform

flow and broad-crested weir equations respectively. In

other words, the flow directions from the boundary cell

toward the extern coincide with those from inner cells

to the boundary cell. Moreover there are other two pa-

rameters hROUT and hER. The first, hROUT, is the

minimum flow depth for routing to avoid the routing

of very small flow depth (inferior to 0.01 m) that is a

physical non sense. This parameter is somehow com-

parable to the roughness height. The second, hER, is

analogous to the previous one and is a lower bound for

the flow erosion capacity. At the end of the simulation

flow depths inferior to hROUT are assumed deposited.

technique used in the cellular automata models. On

the base of that written above there is a strong simi-

larity between this cell model and cellular automata

models. In fact this model corresponds to a cellular

automata model without substates with local rules

given by equations (1), (2) and (3) and mobilisation

condition given by flow level larger than that in the

surrounding cells and a flow depth larger than h

**RIO LAZER BASIN AND DEBRIS FLOW**

**OCCURRED THE 4**

**TH**

**OF NOVEMBER 1966**

equal to 1608 m a.s.l.. Its extension is 1.57 km

m a.s.l. and joins Cismon torrent at 742 m a.s.l. between

the built-up areas of Siror and Tonadico (figure 3). The

4th of November 1966 a debris flow initiated at 850 m

a.s.l. after high intensity rainfall and routed along the

main channel. Just downstream the wooded area (figure

3), it spilled out the channel and flooded the entire fan

depositing sediments of about 80800 m

cells) and routing cells. At the first time step, only the

input boundary cells are activated by filling it with

the volume of the input hydrograph corresponding

to the first time step. At the second time step, flow

routing from input boundary cells towards those sur-

roundings occurs. At the third time step, flow routing

occurs from input boundary cells and those surround-

ings and from the cells routed at the previous time

step towards those surrounding the last ones. The

coordinates of cells routed for the first time during a

same time step are stored in a vector. So at each time

step corresponds a vector containing the coordinate of

activated cells, that is those routed for the first time.

Flow routing is computed sequentially from the input

boundary cells followed from the first order routed

cells (cells routed at the second time step) and so on.

Input boundary cells cannot be routed by other cells

but receive flow only by the input hydrograph and are

not subjected to erosion and deposition. The time step

is computed according to the CFL condition with the

Courant number equal to 0.95. This last constraint

does not origin from numerical instabilities problems

but is used at the purpose of respecting the physics of

routing. The numerical scheme is explicit, that is the

quantity at the time t + ∆t is computed by the values

of the quantities at time t. Therefore, equation (1) after

the integration in time, for the generic cell i, becomes:

roundings cells are computed according to equations

(2) and (3); once all the flow discharges are computed

the cell outflow volume is checked and in the case it

results lower than the cell flow volume at the begin-

ning of the time step, all the flow discharges are pro-

portionally diminished to obtain the equality between

outflow volume and flow volume of the cell at the be-

ginning of time step. For each cell, then, the erosion

bed velocities corresponding to the flow discharges are

computed and summed according to equation (6). Pos-

itive value of i

flow cell volume at the beginning of time step and the

outflow volume computed for the present time step, it

is assumed a deposited volume equal to the difference

**GIS-BASED CELL MODEL FOR SIMULATING DEBRIS FLOW ROUTING AND DEPOSITION PHASES ON A FAN**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

0.75 and 1.5 m respectively). Most of deposited sediment

(70686 m

that was built (pixel size 2 m) and the deposit map.

The digital elevation model was built on the base of

3000 topographic measurements covering an area of

0.2 km2. The input hydrograph was built combining

the measured volume of deposited sediments with the

runoff computed by an hydrological model. The total

volume is 93900 m

hydrograph is entered are ten and located in the upper

part of the watershed (the input hydrograph is distrib-

uted over a 20 m length). Therefore the total volume

is equally divided for 10 and assigned to each of the

hydrograph of the ten cells and is showed in figure 4.

**FLO-2D SIMULATIONS**

simulate erosion or deposition for the grain-inertial be-

haviour of mixture. Simulations were carried out using

the values of parameters of the rheological quadratic

law given by O’ Brien and Julien (1985) correspond-

ing to Aspen Natural Soil (Flo-2D user manual) which

allowed the best reconstruction of the occurred event.

In this case the sediment concentration was assumed

equal to 0.45 instead of 0.86 that causes the deposition

of most of sediments in the upper part of the watershed.

ology corresponds to the Natural Aspen Soli with a

0.45 solid concentration value. Using roughness coef-

ficient values K

mass after, 4 hour, has velocities lower than 0.001 m/s

(Figure 5) and is assumed deposited. Figure 6 shows

the comparison between the measured deposition

depths and the simulated flow depths (as FLO-2D does

not simulate the deposition, flow depth is considered

deposition depth if velocity is less than 0.001 m/s).

*Fig. 3 - Arial photo of Rio Lazer flooded area with su-*

*perimposed the sediment deposits maps and the*

*DEM contour*

*Fig. 4 - Reconstructed hydrograph for each*

*of the ten inlet cells*

*Fig. 5 - Flow depth (0<h<4 m) simulated by FLO-2D af-*

*ter 0.25, 1, 2 and 4 hours*

*Fig. 6 - Comparison between the measured (left) and sim-*

*ulated (righjt) deposition depths*

*C. GREGORETTI, M. FURLAN & M. DEGETTO*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

debris deposits are absent. This fact could be due to the

initial assumption of monophasic flow. Considering a

bi-phase flow after sediments deposition, flow is largely

constituted of fluid that spills out from the border of the

considered basin. The monophasic assumption causes

the continuous deposition until the basin border.

**DISCUSSION OF THE SIMULATIONS**

**RESULTS**

is shown in Table 1.

simulated by the two models. The first comparison is

direct and the second one is carried out verifying that

the simulated value is included in a fixed interval. The

measured depths have been divided in four intervals: d

< 0.1 m, 0.1 < d < 0.5 m, 0.5 < d < 1 m and d > 1.0 m.

(91% of inundated area in the case of Flo-2D and

95.9% in the case of the cell model) even if they

ed sediment depth distribution is somehow reversed

respect to that measured. The measured sediment

depths are larger upstream and decrease downstream

while the simulated sediment depths are lower up-

stream and increase downstream. This fact is due to

the missing of a direct deposition mechanism in the

FLO-2D model for which flow depths, when veloci-

ties are lower than 0.001 m/s, become larger when

slope decreases, in the present case downstream.

**CELL MODEL SIMULATIONS**

simulations. The value of the conductance coefficient

C was assumed constant and equal to 3 according to

G

resistance is equivalent in the two cases. The values of

parameters of the best simulation, that is, with the best

agreement with measured deposits, are K = 0.1,

*θ*

*LIM*

*θ*

*LIM*

*U*

*LIM-E*

*U*

*LIM-D*

ue

*θ*

*LIM*

The value of parameter K is slightly inferior to the minus

value of those used for 1D simulation by Brufau

*et alii*

(2000): 0.2 ≤ K ≤ 1. A change of hROUT in the range

0.01-0.05 m does not imply substantial modification of

results. The simulation time is about 0.39 h (about 24

minutes). Figure 7 is analogous to figure 5 and shows the

inundated areas and flow depth values at different times.

logical law. Flo-2D uses a viscous flow law while cell

model uses a grain-inertial flow law. The routing time of

cell model simulation is more physically plausible than

the routing time of Flo-2D simulation that is too large.

Figure 8 shows the comparison between the measured

deposit depths and those simulated by the cell model.

The extension of the simulated area satisfactory coin-

cides with the deposition area, as in the case of Flo-2D,

and the simulated sediment depth distribution somehow

agrees with that measured. The measured and simulated

sediment depths are both larger upstream and decrease

downstream. The cell model, as Flo-2D, simulates dep-

*Fig. 7 - Flow depth (0<h<4 m) simulated by cell model*

*after 0.065, 0.13, 0.26 and 0.39 hours*

*Fig. 8 - Comparison between the measured (left) and sim-*

*ulated (righjt) deposition depths*

**GIS-BASED CELL MODEL FOR SIMULATING DEBRIS FLOW ROUTING AND DEPOSITION PHASES ON A FAN**

simulated in the case of the cell model and the 14.5

% in the case of Flo-2D model. Considering only the

area with deposit depths larger than 0.5 m, that corre-

sponds to the 87 % of measured sediment volume, the

cell model provides 47 % of deposition area correctly

simulated while Flo-2D provides 27.1 % of it.

the cell model is able to simulate right deposition

depth in all the inundated area while Flo-2D model

predicts the deposits only in a intermediate position.

This is the reason why Flo-2D simulates correctly

only the 27.4% of the area with deposits larger than

0.5 m that are located in the upper part.

depths are shown along with zones with uncorrect-

ed simulated deposit depths, zones with measured

deposit but not simulated and zones with simulated

deposit but no measured. It can be observed that

the blue areas corresponding to correctly simulated

deposit depths are distributed on all the inundated

areas in the case of the cell model while this does

not occurs in the case of Flo-2D.

139% of inundated areas respectively). Neverthe-

less cell model simulates more correctly the deposit

depths than the Flo-2D (50.5 % against 18% of the

measured area: the half of the simulated deposition

depths by the cell model in the measured deposition

area are correct whereas it occurs only for more than

a fourth of the simulated deposition depths by Flo-2D

in the measured area. Moreover as regard the total

*Tab. 1 - Comparison of the extension of measured deposi-*

*tion area and deposition depths with those simu-*

*lated by the two models*

*Fig. 9 - Comparison between the measured deposit depths (middle) and those simulated by Flo-2D (left) and cell model (right)*

*Fig. 10 - Comparison between measured and simulated deposition depths (left Flo-2D simulation, middle measured depths and*

*right cell model simulation): gray deposition depths measured but not simulated; green deposition depths simulated but*

*not measured; red deposition depths with simulated uncorrected values; blue deposition depths correctly simulated*

*C. GREGORETTI, M. FURLAN & M. DEGETTO*

of the two model are due to the different rheologi-

cal laws implemented by the models. Flo-2D models

debris flow as viscous continuous and this implies

larger deposition depths in the downstream part of

deposition area as it occurs in many real cases but

not in the present one. Cell model that includes the

rheological law in the conductance coefficient C is

not subjected to any chain under this point of view.

The fact that Flo-2D cannot directly simulate the

deposition phase forbids the correct simulation of de-

posit depths in the upstream part (most of sediment

of debris flow deposited just after the spilling out of

the channel during the event). On this point of view,

cell model appears more suitable to simulate debris

flow deposition phase on a fan. Moreover the rout-

ing times of cell model are more physically realistic

than those of Flo-2D when simulating the deposition

phase of the debris flow occurred on Rio Lazer.

results of the simulations of the two models. The cor-

rect hydrograph to be used with Flo-2D should be little

lower than that of Figure 4 because the debris flow vol-

ume and the deposition volume must coincide. As the

difference between the hydrograph volume and deposit

volume is little it means that the use of an hydrograph

with a little smaller volume does not change signifi-

catively the results of the Flo-2D simulation. The

percentage of deposition area correctly simulated and

the percentage of measured area with deposit depths

correctly simulated should change just a bit while the

simulated deposition area should decreases.

el. Table 2 is analogous to table 1 and compares the

performance of flow cell model for three different

cell size ∆x: 1, 2 and 4 m.

centage of measured area with deposit depth larger

than 0.5 m correctly simulated and percentage of

deposition simulated but not measured. In the other

case the differences in the performances are negligi-

ble. This leads to affirm a very slightly influence of

cell size in the simulation results.

**CONCLUSIONS**

sign hazards maps. The simplifications at the base of

the model do not strictly respect the physics of routing

when considering the routing times but allow a simula-

tion of deposition depth quite realistic. Debris flow oc-

curred the 4th of November 1966 on Rio Lazer torrent

has been simulated with satisfactory results. The same

event has been simulated by the commercial model

Flo-2D for comparison with a largely used model for

debris flow simulation. The two simulations provide

both nearly equal and different results. Both the mod-

els simulate a deposition area larger than that meas-

ured which cover the 91% of the measured area for

Flo-2D and 95.9 % for the cell model. However, cell

model provides a better simulation of the deposit depth

about more than two times respect to Flo-2D (50.5 %

against 18% of measured area). Moreover, the correct-

ly simulated deposition depths are distributed all over

the deposition area while in the case of Flo-2D simu-

lation, they are grouped in a unique zone of it. This

fact is due to the missing of direct deposit mechanism

in the Flo-2D model that indirectly simulates it when

velocity reduces to the order of 0.001 m/s. This obliges

the use of viscous flow, a prori excluding the granular

inertial flow, with large routing times. On this point of

view, cell model appears more suitable than Flo-2D for

simulating debris flow. It must be added that Flo-2D,

due to the viscous flow rheological law should better

simulate the extension of inundated areas. For a bet-

ter comparison the two models should be both tested

in cases where the deposition depths are larger on the

downstream part of deposits of occurred debris flows.

*Tab. 2 - Comparison of the extension of measured deposi-*

*tion area and deposition depths simulated by cell*

*model using different cell sizes*

**GIS-BASED CELL MODEL FOR SIMULATING DEBRIS FLOW ROUTING AND DEPOSITION PHASES ON A FAN**

*U*

*i,k*

*U*

*LIM-D*

*U*

*LIM-E*

∆x = cell size;

*q*

*i,k*

*θ*

*i,k*

*θ*

*LIM-D*

*θ*

*LIM-E*

**ACKNOWLEDGEMENTS**

of Education and Research grant PRIN 2007 and Euro-

pean grant PARAmount (imProved Accessibility: Reli-

ability and security of Alpine transport infrastructure

related to mountainous hazards in a changing climate),

within the Alpine Space Programme 2007-2013.

**NOTATIONS**

A = cell area;

h

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